55
I Latin American Workshop on Optimization and Control Quito, July 2008
I Latin American Workshop on
Optimization and Control
Quito, July 2008
Contents
Program 5
Organizing Committee 9
Local Organizing Committee 10
Scientific Committee 11
Sponsors 12
List of Speakers 13
Abstracts 17
Information and tips 41
Touristic information 45
Map Reference 51
3
4
Program
Monday, July 7
8:30–9:30. Registration
9:30–10:30. Opening Ceremony
10:30–11:15. Invited talk: A sensitivity analysis of a class of variational inequal-itiesMichel Thera. Universite de Limoges. France
11:15–11:30 Coffee break
11:30–12:00. An augmented Lagrangian method for equilibrium problemsAlfredo Iusem. Instituto de Matematica Pura e Aplicada. Brasil
12:00–12:30. Generalized Nash games and equilibrium problemsWilfredo Sosa. Instituto de Matematica y Ciencias Afines. Peru
12:30–13:00. Variable metric proximal decomposition methodsLisandro Parente. Universidad Nacional de Rosario. Argentina
13:00–15:00. Lunch pause
15:00–15:45. Invited talk: Control optimo de problemas elıpticos con restric-ciones puntuales sobre el gradiente del estadoEduardo Casas. Universidad de Cantabria. Espana
15:45–16:00 Coffee break
16:00–16:30. Error estimates for the FEM approximation of semilinear controlproblems with state constraints and finite dimensional controlsPedro Merino. Escuela Politecnica Nacional. Ecuador
16:30–17:00. Optimal control on nillpotent Lie groupsFelipe Monroy. UAM Azcapotzalco. Mexico
17:00–17:30. A semismooth Newton algorithm for Bingham fluidsSergio Gonzalez. Escuela Politecnica Nacional. Ecuador
17:30–18:15. Invited talk: Semismooth Newton methods for portfolio optimiza-tionRoland Griesse. Technische Universitat Chemnitz. Alemania
18:30–19:30. Cocktail de bienvenida
5
Tuesday, July 8
9:00–9:45. Invited talk: Compact convex sets of constant width (in 2D) or ofconstant thickness (in 3D) : new stuff from old onesJean-Baptiste Hiriart Urruty. Universite Paul Sabatier Toulouse. France.
9:45–10:00 Pause.
10:00–10:30. QVI systems related to some production optimization problemsElina Mancinelli. Universidad Nacional de Rosario. Argentina.
10:30–11:00. A symmetry preserving alternating projection method for matrixmodel updatingJoali Moreno. Universidad Central de Venezuela. Venezuela.
11:00–11:30. A Kantorovich type results for vector optimizationLuis Grana Drummond. Universidade Federal do Rio de Janeiro. Brasil.
11:30–11:45. Coffee break
11:45–12:15. Um metodo de regiao de confianca para minimizacao irrestrita semderivadasLiliana Jimenez Urrea. Universidade Estadual de Campinas, Brasil.
12:15–12:45. Un metodo que no usa derivadas para resolver un problema deprogramacion no lineal con restricciones linealesTulio Lopez. Universidad del Cauca. Colombia.
12:45–13:15. Condiciones de existencia en optimizacion fuzzyAndres Baez. Universidade Estadual de Campinas. Brasil.
13:15–15:00. Lunch pause
15:00–15:45 Invited talk. Numerical and optimization methods for active vi-bration control and model updating in vibrating structures: linking control toindustryBiswa Datta. Northern Illinois University. USA
15:45–16:00. Coffee break
16:00–16:30. Numerical approximation of the LQR problem in a highly dampedwave equationDante Kalise. Universidad Tecnica Federico Santa Marıa. Chile
16:30–17:00. Efficient solution of large scale matrix equations arising in LQR/LQGdesign for parabolic PDEsHermann Mena. Escuela Politecnica Nacional. Ecuador.
17:00–17:30 Neural network control of material sinthetization systemDarıo Fajardo. Universidad de Narino. Colombia.
6
Wednesday, July 9
9:00–9:45. Invited talk. Recent advances for location routing problemsFrederic Semet. Universite de Valenciennes. France.
9:45–10:00 Pause.
10:00–10:30. Set packing and set covering problemsLuis M. Torres. Universitat Magdeburg, Germany, and Escuela Politecnica Nacional, Ecuador.
10:30–11:00. Isomorfismo de grafos e varianzas do PQAPaulo Boaventura-Netto. Universidade Federal do Rio de Janeiro. Brasil.
11:00–11:30. Polyhedral considerations for a flower production and packingproblemDiego Recalde. Escuela Politecnica Nacional. Ecuador.
11:30–11:45. Coffee break
11:45–12:15. Line planning in QuitoRamiro Torres. Escuela Politecnica Nacional. Ecuador.
12:15–12:45. Programacion estocastica y algoritmos en lınea para el PGSCMarıa Soto. Banco Central del Ecuador. Ecuador.
12:45–15:00. Lunch pause
15:00–15:45 Invited talk. Feedback Solutions to Quantum Control ProblemsKazufumi Ito. North Carolina State University. USA
15:45–16:00. Coffee break
16:00–16:30. N. Sukhomlin: Nuevo metodo de resolucion exacta del problemainverso de modelos de tipo de Black-Scholes-Merton con volatilidades deter-minısticasNikolay Sukhomlin. Universidad Autonoma de Santo Domingo. Rep. Dominicana.
16:30–17:00. An integer programming approach for the single source minimumcost unsplittable flow problemMarıa Fernanda Salazar. Escuela Politecnica Nacional. Ecuador.
17:00–17:30 Plan maestro de abastecimiento de alimentos para GuayaquilCarlos Cepeda. ESPOL. Ecuador
19:00–22:00. Conference Dinner
7
8
Organizing Committee
• Juan Carlos De Los Reyes (Chairman)Escuela Politecnica Nacional, [email protected]
• Lisandro ParenteUniversidad Nacional de Rosario, [email protected]
• Polo VacaEscuela Politecnica Nacional, [email protected]
9
Local Organizing Committee
• Juan Carlos De Los ReyesEscuela Politecnica Nacional, [email protected]
• Sergio GonzalezEscuela Politecnica Nacional, [email protected]
• Pedro MerinoEscuela Politecnica Nacional, [email protected]
• Polo VacaEscuela Politecnica Nacional, [email protected]
10
Scientific Committee
• Laura AragoneUniversidad Nacional de Rosario, ARGENTINA
• Eduardo CasasUniversidad de Cantabria, SPAIN
• Roberto CominettiUniversidad de Chile, CHILE
• Martin GrotschelTechnische Universitat Berlin, GERMANY
• Jean-Baptiste Hiriart UrrutyUniversite Paul Sabatier, FRANCE
• Kazufumi ItoNorth Carolina State University, USA
• Pablo LotitoUniversidad Nacional del Centro de Buenos Aires, ARGENTINA
• Boris MordukhovichWayne State University, USA
• Luis Miguel TorresUniversitat Magdeburg, GERMANY, and Escuela Politecnica Na-cional, ECUADOR
11
Sponsors
• Sociedad Ecuatoriana de Matematica (SEdeM).
• Escuela Politecnica Nacional, Ecuador.
• Universidad Nacional de Rosario, Argentina.
• International Mathematical Union (IMU).
• Universidad Andina Simon Bolıvar.
• Corporacion Centro Nacional de Control de Energıa (CENACE).
• CLYAN Services World
• UniBanco.
• Hansel y Gretel.
12
List of Speakers
• Baez Sanchez, Andres David.Universidade Estadual de Campinas, [email protected]
• Boaventura-Netto, Paulo Oswaldo.Universidade Federal do Rio de Janeiro, [email protected]
• Casas, Eduardo.Universidad de Cantabria, [email protected]
• Cepeda de la Torre, Carlos Alberto,Escuela Politecnica del Litoral, [email protected]
• Datta, Biswa.Northern Illinois University, [email protected]
• De Los Reyes, Juan Carlos,Escuela Politecnica Nacional, [email protected]
• Fajardo Fajardo, Darıo Fernando.Universidad de Narino, [email protected]
• Gonzalez, Sergio.Escuela Politecnica Nacional, [email protected]
• Grana Drummond, Luis Mauricio.Universidade Federal do Rio de Janeiro, [email protected]
• Griesse, Roland.Technische Universitaet Chemnitz, [email protected]
13
• Hiriart Urruty, Jean-Baptiste.Universite Paul Sabatier Toulouse, [email protected]
• Ito, Kazufumi.North Carolina State University, [email protected]
• Iusem, Alfredo.Instituto de Matematica Pura e Aplicada, [email protected]
• Jimenez, Miguel.Universidad Nacional de Piura, [email protected]
• Jimenez Urrea, Liliana.Universidade Estadual de Campinas, [email protected]
• Kalise, Dante.Universidad Tecnica Federico Santa Marıa, [email protected]
• Lopez Erazo, Tulio Emiro.Universidad del Cauca, COLOMBIA [email protected]
• Mancinelli, Elina.Universidad Nacional de Rosario, [email protected]
• Mena, Hermann.Escuela Politecnica Nacional, [email protected]
• Merino, Pedro.Escuela Politecnica Nacional, [email protected]
• Monroy Perez, Felipe.UAM Azcapotzalco, [email protected]
• Moreno, Joali.Universidad Central de Venezuela, [email protected]
14
• Parente, Lisandro.Universidad Nacional de Rosario, [email protected]
• Recalde, Diego.Escuela Politecnica Nacional [email protected]
• Salazar, Marıa Fernanda.Escuela Politecnica Nacional, [email protected]
• Semet, Frederic.Universite de Valenciennes, [email protected]
• Sosa, Wilfredo.Instituto de Matematica y Ciencias Afines, [email protected]
• Soto, Marıa.Banco Central del Ecuador, [email protected]
• Sukhomlin, Nikolay.Universidad Autonoma de Santo Domingo, REP. [email protected]
• Thera, Michel.Universite de Limoges, [email protected]
• Torres, Luis Miguel.Universitat Magdeburg, GERMANY.Escuela Politecnica Nacional, [email protected]
• Torres, Ramiro.Escuela Politecnica Nacional, [email protected]
15
16
Abstracts
QVI systems related to some production optimization problems
In this paper we study the Quasi Variational Inequalities (QVI) Systems whichappear in the optimization of some problems related to a multi-item singlemachine, where at any time the machine is either idle or producing any of m
different items.Our aim is to obtain an optimum production schedule for each of the cases
we propose, therefore we define a policy control as a pair (production state,time) which states at each instant of time which item must be produce. Thecost to be optimized takes into account a running cost and the productionswitching costs, as the integral cost functional has an infinite horizon a discountfactor is used to guarantee that the integral converge.
Using the dynamical programming methods we arrive to Hamilton-Jacobi-Bellman equations associated to these problems which are given by QVI sys-tems. Our central result is the characterization of the optimal cost functionas the unique solution in the viscosity sense of this QVI systems.
We focus our attention on two particular cases: the ergodic case and thepiecewise deterministic one. In Gonzalez R.L.V., Rofman E., Sur des solutions
non bornees de l’equation de Bellman associee aux problemes de commutation
optimale avec constraints sur l’etat. Comptes Rendus Acad. Sc. Paris, (1993),a deterministic problem akin to these ones was studied.
In the ergodic case the objective is to find an optimal production schedulethat minimizes the average cost for an infinite horizon. By using dynamic pro-gramming techniques and taking into account the switching cost, it is possibleto find an optimal feedback policy, in terms of any solution in the viscositysense of a first order QVI system. This system is obtained considering a se-quence of optimization problems with non zero discount rate. We prove theexistence of solution of this QVI system and the uniqueness of the optimalaverage cost.
Secondly we deal with the case where the demand varies randomly accord-ing to a piecewise deterministic process. The demand changes are describedby a Poisson processes and the demand value takes a finite number of values.This type of problems belong to the class of optimal control problems of jump-ing processes with state space constraints. We describe the optimal schedulingproblem of a multi-item single machine production system where, besides theinner production, external purchases are allowed to cope with the demands.We analyze the HJB equation associated to this problem, considering boththe integral and the differential expression of this equation. We present theexistence, uniqueness and regularity of the optimal cost function.
17
Aragone, LauraUniversidad Nacional de RosarioRosario, [email protected]
Mancinelli, ElinaUniversidad Nacional de RosarioRosario, [email protected]
Condiciones de existencia en optimizacion fuzzy
En un problema de optimizacion usualmente consideramos que todos los datose informaciones correspondientes al problema estan perfectamente determina-dos. Sin embargo en la practica este no es siempre el caso y puede ser necesarioconsiderar incertidumbre en uno o varios aspectos del problema, bien sea porinformacion insuficiente, porque la informacion actual no es objetiva o simple-mente por errores en la medicion de los datos.
El analisis de sensibilidad es un aspecto fundamental en un problema deoptimizacion y en cierta forma ataca un aspecto relacionado con la incertidum-bre en los datos. Otra forma muy frecuente para modelar situaciones de opti-mizacion bajo incertidumbre es ofrecida por la optimizacion estocastica perono siempre las situaciones de incertidumbre que presenta un problema son detipo probabilıstico. Varias herramientas diferentes al analisis estocastico hansido presentadas para adaptarse mejor segun diferentes contextos especıficos,podemos nombrar por ejemplo el analisis intervalar y la teorıa de conjuntosfuzzy.
Desde la aparicion del concepto de conjuntos fuzzy (fuzzy sets) introducidopor Lofti Zadeh en 1965 han aparecido numerosas aplicaciones de este conceptoy de los aspectos teoricos y practicos desarrollados a partir de el, algunoscomponentes de esta teorıa fuzzy son la logica fuzzy, aritmetica fuzzy, teorıade la posibilidad, ecuaciones e inclusiones diferenciales fuzzy entre otros.
La optimizacion fuzzy trata justamente del estudio y analisis de problemasde optimizacion donde uno o varios aspectos del problema estan especifica-dos por algun elemento fuzzy. Con cierta profundidad se han estudiado hastaahora problemas de programacion matematica en los cuales las restriccionesson consideradas como conjuntos fuzzy de R
n o incluso problemas en dondealgunos de los coeficientes del problema son considerados numeros fuzzy. Nu-merosos artıculos y aplicaciones con este enfoque de optimizacion fuzzy hanaparecido en los ultimos 30 anos.
Otro tipo problema interesante, poco estudiado hasta ahora, pero que estacomenzando a recibir mas atencion es el problema de optimizacion sobre es-pacios fuzzy, esto es, problemas de optimizacion donde las variables de de-cision son tambien de tipo fuzzy. Este problema presenta multiples aspectosmatematicos interesantes y consideramos que tiene un amplio potencial deaplicacion.
18
A pesar de que el espacio de decision pueda ser dotado con operaciones desuma y multiplicacion por escalar razonables, estas no forman en general unespacio vectorial. Por otro lado, las posibles metricas de que puede dotarse esteespacio le proporcionan cada una diferentes propiedades topologicas. Parecenatural considerar que dado que los conjuntos fuzzy son funciones de pertenen-cia sobre un conjunto universal, un problema de optimizacion en espacios fuzzypueda verse como un problema de optimizacion sobre un espacio funcional, deforma analoga a lo que sucede en un problema de control optimo. Esta ideadebe considerarse con cuidado por que la estructura algebraica y topologicadel espacio fuzzy no constituye en general un espacio vectorial topologico o unespacio vectorial normado.
En este trabajo pretendemos mostrar una introduccion a la optimizacionfuzzy, con un enfasis especial sobre la optimizacion en espacios fuzzy. Quere-mos mostrar resultados recientes sobre las condiciones de existencia del optimopara este tipo problema y discutir las posibilidades de aplicacion a futuro.
Baez Sanchez, Andres DavidUniversidad Estadual de CampinasCampinas, [email protected]
Isomorfismo de grafos e variancias do PQA
Uma instancia simetrica de ordem n do Problema Quadratico de Alocacao(PQA) possui duas outras instancias a ela associadas: uma instancia relaxada,de ordem n(n − 1)/2, cuja solucao e polinomial, caracterizada pela matrizQ = FDT , onde F e D sao vetores que contem os triangulos superiores dasmatrizes de dados da instancia; e uma instancia-padrao, de ordem n, na qual osvalores dos vetores F e D sao ordenados em ordens opostas. As variancias dosvalores das solucoes dessas instancias podem ser obtidas em tempo polinomial,[GW70], [ABQG02].
O problema do isomorfismo de grafos pode ser modelado como um PQA,cujas matrizes de dados sejam descritivas dos dois grafos em questao. Doisgrafos serao isomorfos se e somente se o valor maximo da funcao objetivo doPQA formado por suas matrizes de adjacencia for igual ao numero comum mde arestas dos dois grafos. Esta verificacao exige, portanto, a resolucao exatade uma instancia do PQA, que e NP-hard.
Em [ABQG02] e definido o isomorfismo de instancias do PQA e se mostraque duas instancias isomorfas possuem a mesma variancia. Para utilizar esteresultado na caracterizacao do isomorfismo de um par de grafos, comparasea instancia a ele correspondente, com instancias construidas com duas copiasdo primeiro, e com duas copias do segundo grafo. Se os dois grafos foremisomorfos, as variancias associadas a essas tres instancias serao iguais.
Esta igualdade de variancias, porem, nao e suficiente para caracterizar o iso-morfismo: dois grafos podem ser nao isomorfos e apresentar variancias iguaisneste teste. Foram utilizadas, entao, valoracoes invariantes das arestas, ouseja, valoracoes dependentes das propriedades do grafo e nao de sua rotulacao.
19
Testes realizados com a valoracao dada pela contagem de percursos de com-primento 2 entre vertices ligados por arestas que correspondem ao numero deciclos de comprimento 3 de que cada aresta participa mostraram ser a mesmacapaz de distinguir entre grafos nao isomorfos de ate 500 vertices, com apenasuma troca de aresta dentre um maximo de 30.000 arestas. O custo computa-cional desta valoracao e O(n2), assim como o custo da variancia relaxada,enquanto as outras duas sao O(n4).
A um par de grafos valorado como acima descrito se pode, ainda, aplicaruma sobrevaloracao correspondente a soma dos valores das arestas adjacentes,em O(m). Outras possibilidades podem ser consideradas, tais como a soma degraus, em grafos nao regulares, ou os valores dos afastamentos dos vertices quedefinem as arestas; enfim, a contagem de percursos pode ser levada a compri-mentos maiores, embora a custo mais elevado.
[GW70] Graves, G.W. e Whinston, A.B. (1970). An algorithm for the qua-dratic assignment problem. Management Science 17, 453-471.
[ABQG02] Abreu, N.M.M., Boaventura-Netto, P.O., Querido, T.M. e Gouvea,E.F.. (2002). Classes of quadratic assignment problem instances: isomorphismand difficulty measures using a statistical approach. Discrete Applied Mathe-matics 124, 113-116.
Boaventura Netto, Paulo OswaldoUniversidade Federal do Rio de JaneiroRio de Janeiro, [email protected]
Control optimo de problemas elıpticos con restricciones puntualessobre el gradiente del estado
Algunos autores han abordado recientemente el analisis numerico de problemasde control optimo gobernados por ecuaciones en derivadas parciales elıpticascon restricciones puntuales sobre las derivadas del estado. Estos trabajos sehan apoyado en los resultados teoricos probados por Casas y Fernandez (Ap-plied Mathematics and Optimimization 27, 35-56, 1993). Aquel estudio secentro en la demostracion de la existencia de un multiplicador de Lagrangeasociado a las restricciones sobre el estado y las correspondientes condicionesde optimalidad de primer orden. En esta conferencia volvemos sobre el mismoproblema y aportamos nuevos resultados que pueden conducir a mejoras sig-nificativas en las estimaciones del error de las aproximaciones numericas. Elproblema se formula en los siguientes terminos.
(P )
Min J(u) =
∫
Ω
L(x, yu(x))dx +N
2
∫
Ω
u2(x)dx
α ≤ u(x) ≤ β a.e en Ω, |∇yu(x)| ≤ δ ∀x ∈ K
20
donde yu es la solucion del problema de Dirichlet
Ay + a0(x, y) = u en Ω
y = 0 sobre Γ,
Ω siendo un abierto acotado de Rn, n = 2 o 3, Γ es la frontera de Ω, A un
operador elıptico de segundo orden en Ω, K ⊂ Ω es un conunto compacto,−∞ < α < β < +∞ y δ > 0.Para este problema deducimos las condiciones suficientes de optimalidad de se-gundo orden y un resultado de regularidad inesperado para el control optimo.
Casas, EduardoUniversidad de CantabriaSantander, [email protected]
Plan maestro de abastecimiento de alimentos para Guayaquil
En el presente trabajo se plantea la utilizacion de un modelo conceptual parael Abastecimiento de Alimentos en la Ciudad de Guayaquil. Este modelo estabasado en la creacion de entidades logısticas (puntos de transferencia, nodoslogısticos y plazas logısticas) cuyo proposito fundamental es mejorar la organi-zacion en terminos de distribucion y comercializacion de los diversos productosque demanda la poblacion de la ciudad. En base a estas entidades, se realizanlos analisis respectivos para determinar la configuracion optima del sistema.Para ello, se emplea el concepto de la programacion matematica, disenando unmodelo que permite minimizar los diferentes costos que intervienen en todo elSistema de Abastecimiento planteado: costos de construccion, de operacion,de transporte, y de produccion. El modelo fue disenado sobre las distintascadenas agroindustriales de alimentos elegidas para el estudio, teniendo sub-modelos para cada una de las cadenas homologadas con cinco eslabones co-munes: Produccion, Transporte, Transformacion, Distribucion al consumidory Flujos internos en la ciudad. Se lo ha construido de tal forma que genera losflujos y volumenes de las cadenas de acuerdo con una demanda previamentesectorizada dentro de la ciudad, la misma que es estimada en base a unapoblacion proyectada y a una canasta basica de consumo de alimentos al cabode un horizonte de tiempo preestablecido. Finalmente, segun los escenariosconsiderados, se plantea la utilizacion de esta herramienta para la localizacionoptima de las plazas de mercado existentes actualmente en la ciudad y laspropuestas en el estudio.
Cepeda de la Torre, Carlos AlbertoEscuela Politecnica del LitoralGuayaquil, [email protected]
21
Numerical and optimization methods for active vibration controland model updating in vibrating structures: linking control to in-dustry
The use of active feedback control strategy is an effective way to stabilize andcontrol dangerous vibrations, such as resonance and other forms of instability,in vibrating structures, including bridges, highways, buildings, and space andair crafts. These structures are distributed-parameter systems. However, be-cause of practical considerations such systems are very often discretized intosystems of matrix second-order differential equations using finite-element tech-niques; control is then designed and implemented on this discretized systemand, finally applied to a real-life structure. Unfortunately, the existing vibra-tions control techniques, even for these simplified models, have some severenumerical difficulties and engineering limitations. The vibration industrieshave approached vibration control problems in an ad hoc way and many ofthe current engineering practices are not supported by strong mathematicalfoundations. In the last few years, the speaker and his collaborators havedeveloped a practical computational approach for feedback control in finiteelement structures. The distinctive features of this approach are (i) controlcan be designed directly on the finite model without requiring transformationto a standard first-order state-space form, (ii) the algorithms require knowl-edge of only a small number of eigenvalues and eigenvectors of the associatedquadratic matrix pencil, and (iii) above all, no a priori reduction of the orderof the model or controller is required, irrespective of the size of the systems.Exploiting the parametric nature of the approach, optimization-based numeri-cally robust algorithms have also been developed to design feedback controllerswhich remain as insensitive as possible with respect to small perturbations inthe data. This approach has also been successfully applied to another relatedindustrial problem, namely, the finite-element model updating problem whichconcerns updating an analytical finite element model with experimental dataso that the updated model can be used with confidence for future design andconstructions. The minimal computational and engineering requirements ofthis new approach make it readily applicable to feedback control design andmodel updating in even very large practical-life structures. In this talk, theserecent advances will be reviewed and a brief discussion will be presented onfuture directions of research in this area.
Datta, BiswaNorthern Illinois UniversityIllinois, [email protected]
Optimal control of partial differential equations with affine controlconstraints
The numerical solution of PDE optimal control problems involving affine point-wise control constraints is investigated. Optimality conditions are derived and
22
a semi-smooth Newton method is presented. Global and local superlinear con-vergence of the method are obtained for linear problems. Differently from boxconstraints, in the case of general affine constraints a proper weighting of thecontrol costs is essential for superlinear convergence of semi-smooth Newtonmethods. This is also demonstrated numerically by controlling the two dimen-sional Stokes equations with different kinds of affine constraints.
De Los Reyes, Juan CarlosEscuela Politecnica NacionalQuito, [email protected]
Neural network control of materials sinterization system
This report describes the development of control strategies based on neuralnetwork for a complex thermal system on range of 20C to 1400C. We use anadaptive neural network identifier with parallel structure and neurocontrolleron line with adaptive learning. To control the temperature of the oven, a5-3-1 neural network structure, was formed by process control signal. Thesystem dynamic was identified with a neural 6-1 structure neural network andWe used an estimate system’s output for including in controller. Both tan-gent sigmoidal activation function. All system is proved with a typical curvefor material’s sinterization in physical analysis. We applied adaptive feedbackback-propagation algorithm, the input to the controller was selected before testestimate and measure signal output and some past system’s input. Inverse Di-rect strategy control with system identification was used on development. Thiswork was made in Universidad de Narino, San Juan de Pasto Colombia by In-telligent Systems and Instrumentation Research Group.
Fajardo Fajardo, Darıo FernandoUniversidad de NarinoSan Juan de Pasto, [email protected]
Ruiz Rosero, Juan PabloUniversidad de NarinoSan Juan de Pasto, [email protected]
Martınez Benavides, John JairoUniversidad de NarinoSan Juan de Pasto, COLOMBIAjohn [email protected]
23
A semismooth Newton algorithm for Bingham fluids in channels
In this work, we are concerned with the boundary value-problem describingthe flow of a Bingham fluid in a bounded channel Ω ⊂ R
d, d ∈ 1, 2, consid-ering multiple boundary conditions. We analyze the variational setting of thisproblem given by a variational inequality of the second kind. We prove thatthis inequality is a necessary optimality condition of a minimization problemin a convex set of H1(Ω). Then, by using Fenchel’s duality theory, we deeplyanalyze the problem and obtain a primal-dual optimality system relating thesolutions for the primal and dual problems. Unfortunately, this system is ill-conditioned since the dual problem has not unique solution. Thus, we proposea Tikhonov regularization, which is a penalty-type smoothing, to overcomethis issue. Penalty-type smoothings imply a local regularization of the originalproblem, and, differently from global smoothings, stay as close as possible tothe original problem formulation and reduce in this manner the ill-conditioning.
Since semi-smooth Newton methods have been successfully applied to infi-nite dimensional complementarity problems like the Signorini or contact prob-lem and to simulation and control of fluids, we propose a second order algo-rithm, based on these methodology, to numerically solve the problem. Wedeeply analyzed the constructed algorithm. Particulary, we prove that it islocally superlinear convergent. Finally, we approximate the problem by usinga conforming finite element discretization, and we numerically test the behav-ior of the algorithm when used to study two problems: the flow of a Binghamfluid in a reservoir with a forcing term and the flow of a Bingham fluid in agiven geometry of R
2, considering homogenous Dirichlet boundary conditions,non-homogeneous Dirichlet boundary conditions and stress free boundary con-ditions.
Gonzalez, SergioEscuela Politecnica NacionalQuito, [email protected]
De los Reyes, Juan CarlosEscuela Politecnica NacionalQuito, [email protected]
A Kantorovich-type result for vector optimization
For an unconstrained multiobjective problem, we propose a Newton-like method(together with a globalization strategy). The method neither scalarizes theoriginal problem nor uses ordering information. Under the assumptions ofstrict convexity and twice continuous differentiability of all criteria, as in thereal valued case, the method locally superlinearly converges to an optimum(a Pareto point). Also as in the scalar case, whenever the Hessians of all ob-jectives are Lipschitz continuous, the order of convergence is quadratic. As
24
a by product, local existence of efficient point is obtained under semi-localassumptions.
We also consider an extension of the method for the vector optimizationcase, i.e., when the partial order is induced by a closed convex cone. Assum-ing that the objective function is twice continuously differentiable and locallystrongly convex with respect to the ordering cone, we show, by means of aKantorovich-like technique, that the method locally converges to an efficientpoint with quadratic order.
Grana Drummond, Luis MauricioUniversidade Federal do Rio de JaneiroRio de Janeiro, [email protected]
Semismooth Newton methods for portfolio optimization
We are considering a continuous-time market model of one bank account andone stock. Trading the stock incurs transaction costs proportional to the trad-ing volume. Given the interest rate r, the stock trend parameter µ, the trans-action costs γ and the volatility σ, the values X0 and X1 of the bank accountand the stock, respectively, evolve according to
dX0(t) = rX0(t) dt − (1 + γ) dL(t) + (1 − γ) dM(t)(1a)
dX1(t) = X1(t) µ dt + X1(t) σ dW (t) + dL(t) − dM(t).(1b)
L(t) and M(t) denote the cumulative purchases and sales of the stock and thusdetermine the trading strategy. The objective is to maximize the expected util-ity of the total wealth at a given terminal time T . Admissible trading strategiesensure positive total wealth at all times. The associated value function satisfiesa Hamilton-Jacobi-Bellman equation of the type
(2) maxVt + LV, LBV, LSV = 0 on (−1
γ, 1
γ) × (0, T ),
subject to terminal and boundary conditions. Here, LB, LS and L are first andsecond order linear differential operators. The boundaries of the sets wherethe maximum is attained by either one of the terms determine the optimaltrading policy. In the presentation, we address semismooth Newton methodswhich allow for the efficient numerical solution of (2) and give some examples.
Griesse, Roland.Technische Universitaet ChemnitzChemitz, [email protected]
25
Compact convex sets of constant width (in 2D) or of constant thick-ness (in 3D) : new stuff from old ones.
“When Minkowski’s theory of convexity appeared, some mathematicians
said that he discovered a nice mathematical joy which, unfortunately, is quite
useless. About a century passed, and now the theory of convex sets is a very
important applied branch of mathematics.”
V. Boltyanski, in Geometric methods and optimization problems (1999).
The “geometrical” convexity (the one treating of the structure of convex bod-ies, their smoothness, volume,...) and the “functional” convexity (the onetreating of properties of convex functions, their use in optimization,...) aretwo areas of mathematics, well delineated but different, with, for each of them,their research and working communities, mathematical questions, journals, etc.There however is a contact point which allows to link the two fields, that ofvariational problems concerning convex bodies. These questions lie at the con-fluence of three domains of mathematics: Variational analysis and calculus,Optimal control, Shape optimization. In this communication, essentially ped-agogical and of synthesis, we present some properties, mainly of a “variational”type, of convex bodies of constant width (in 2D) or of constant thickness (in3D), laying particular emphasis on the fundamental differences between 2D and3D. We thus arrive at the front of some open problems (some are long-standingones), for which results and techniques from Variational calculus, Optimal con-trol, or Shape optimization, have failed to provide answers, up to now. Weshall present in detail the conjecture on convex bodies of constant thickness in
3D of minimal volume. The most conclusive results on that subject, in the 3Dcontext, have been obtained recently by T. Bayen, Th. Lachand-Robert, E.Oudet (two papers published in 2007).
“ La convexite dans le plan et dans l’espace presente un sujet passionnant,
la convexite, a la fois par sa simplicite, sa naturalite et sa puissance, pour au
moins trois raisons :
- au niveau des questions que l’on peut se poser naturellement, geometriques,
visibles ;
- du fait que la convexite est une notion qui se rencontre dans de nom-
breuses branches des mathematiques ;
- du fait de son utilite, de sa force, dans de nombreuses applications.”
M. Berger in Convexite dans le plan et au-dela, 2 volumes, collectionOpuscules, editions Ellipses (2006).
Hiriart-Urruty, Jean-BaptisteInstitut de Mathematiques, Universite Paul SabatierToulouse, [email protected]
26
Feedback Solutions to Quantum Control Problems
Control of quantum systems described by Schroedinger equation is considered.Feedback control laws are developed for the orbit tracking via a controlledHamiltonian. Asymptotic tracking properties of the feedback laws are ana-lyzed. Numerical integrations via time-splitting are also analyzed and used todemonstrate the feasibility of the proposed feedback laws. Also, the recedinghorizon feedback law is analyzed and it improves the performance of the pro-posed feedback laws.
Ito, KazufumiNorth Carolina State [email protected]
An Augmented Lagrangian method for equilibrium problems
We propose an Augmented Lagrangian method for equilibrium problems, whosefeasible sets are given by a finite set of convex inequalities. In each step, theprimal variables are updated by solving an unconstrained equilibrium prob-lem, while the dual variables are updated through a closed formula. In theparticular case of optimization problems, we recover the proximal AugmentedLagrangian method proposed by Rockafellar in 1976. Under monotonicity as-sumptions on the bifunction which defines the equilibrium problem, we estab-lish global convergence of the generated sequence to a solution of the problem.
Iusem, AlfredoInstituto de Matematica Pura e AplicadaRio de Janeiro, [email protected]
Um metodo de regiao de confianca para minimizacao irrestrita semderivadas
Em otimizacao, um problema se diz irrestrito se seus parametros podem as-sumir quaisquer valores. Para o caso de nosso interesse neste trabalho, min-imizamos uma funcao objetivo de varias variaveis, sobre todo o espaco R
n.Nossa motivacao para estudar possıveis algoritmos para resolver estes proble-mas, e a grande demanda de profissionais de varias areas por tais ferramen-tas. Nas aplicacoes apresentadas em Powell [2003], calcular o valor da funcaoF (x), dado o vetor x, e muito caro, e os valores das derivadas de F em xnao sao facies de encontrar, ou porque F (x) resulta de algum fenomeno fısicoou quımico, ou, mais comumente, porque este e o resultado de uma complexasimulacao de um computador. O objetivo deste trabalho e estudar a estruturado modelo quadratico Q, construıdo por interpolacao por um numero definido
27
de pontos que sao determinados de uma forma particular, considerando val-ores da funcao F . Metodos de regiao de confianca sao utilizados, no entanto,quando uma aproximacao para F (x); x ∈ R
n, e construıda a partir dos val-ores da funcao objetivo. Entao o seguinte vetor de variaveis e gerado usual-mente procurando o mınimo da aproximacao Q, em uma conveniente regiaode R
n. A aproximacao e chamada ”funcao modelo” contida no espaco linearM das funcoes de R
n a R com dimensao m = 1
2(n + 1)(n + 2). Cada modelo e
definido por condicoes de interpolacao da forma Q(xi) = F (xi), i = 1, 2, . . . , m,onde xi sao os pontos de interpolacao que estao em posicoes que garantem anao singularidade do sistema formado por as condicoes, e m esta no intervalo[n + 1, m]. O metodo utilizado trabalha com posicoes inicias dos pontos deinterpolacao, o ajuste do raio da regiao de confianca, o calculo do vetor ext, ea selecao de um ponto para ser substituıdo para avaliacao das singularidades eerros das equacoes de interpolacao do modelo, utilizando funcoes de Lagrange.Em Powell [2003], e provado que a cota de erro pode controlar o ajuste doraio de uma regiao de confianca, de forma a se obter excelentes resultadosde convergencia num algoritmo para problemas de minimizacao irrestrita. Oalgoritmo proposto em Powell [2002, 2003], sera testado em alguns problemasda colecao Hock-Schittkowski [1981], onde observa-se que a escolha de M eimportante para identificar o trabalho em cada iteracao. A comparacao dosresultados numericos do metodo e feita com um metodo de busca padrao deLewis-Torczon [2000], o qual define em cada iteracao um conjunto padrao dedirecoes de busca no ponto atual na procura de melhores valores para F .
Jimenez Urrea, LilianaUniversidade de CampinasCampinas, [email protected]
da Rocha Lopes, Vera LuciaUniversidade de CampinasCampinas, [email protected]
Numerical approximation of the LQR problem in a highly dampedwave equation
The aim of this work is to obtain optimal order error estimates for the LQR(Linear-quadratic regulator) problem in a highly damped 1-D wave equation.We consider a finite element discretization of the system dynamics and a con-trol law constant in the spatial dimension. To solve the LQR problem, weseek a feedback control which depends on the solution of an algebraic Riccatiequation. Optimal error estimates are proved in the framework of the approx-imation theory for control of infinite-dimensional systems. Finally, numericalresults are reported to illustrate that the optimal rates of convergence areachieved.
28
Kalise, DanteUniversidad Tecnica Federico Santa MarıaValparaıso, [email protected]
Hernandez, ErwinUniversidad Tecnica Federico Santa MarıaValparaıso, [email protected]
Otarola, EnriqueUniversidad Tecnica Federico Santa MarıaValparaıso, [email protected]
Un metodo que no usa derivadas para resolver un problema de pro-gramacion no lineal con restricciones lineales
En el ano 2002, Marco Sciandrone, Stefano Lucidi y Paul Tseng publican elartıculo titulado: Objetive-Derivative-Free methods for constrained optimiza-tion [2], en el que presentan dos metodos de descenso suficiente para resolverun problema de optimizacion no lineal:
minimizar f(x)(3)
sujeta a: gi(x) ≤ 0, ∀i ∈ I,(4)
donde f : Rn 7→ R (funcion objetivo) y gi : R
n 7→ R (funcion restriccionpara cada i), son continuamente diferenciables en R
n para todo i ∈ I, conI = 1, . . . , m ⊂ N. Los metodos propuestos por estos autores para resolver(1)-(2), no hacen uso del gradiente de la funcion f , ni de aproximaciones almismo. Estos metodos se enfocan alrededor de tres aspectos fundamentales: labusqueda de una matriz de rango completo tal que sus columnas correspondena los gradientes de las restricciones casi activas en el punto actual, la busquedade un conjunto de direcciones que satisfacen alguna condicion, las cuales gen-eran positivamente el cono polihedral convexo en el punto actual y el calculode un paso a ser dado a lo largo de una direccion de descenso.
Los autores, entre otros comentarios, mencionan lo simple que puede re-sultar intentar resolver con estos metodos, problemas de la forma (1)-(2) queinvolucren restricciones de igualdad. El trabajo que desarrollamos consiste enadaptar el primero de los metodos propuestos por Lucidi, Sciandrone y Tsengpara resolver un problema de programacion no lineal con restricciones linealesde desigualdad y de igualdad. Formalmente consideramos el problema:
minimizar f(x)(5)
sujeta a: gi(x) ≤ 0, ∀i ∈ I(6)
hi(x) ≤ 0, ∀l ∈ E,(7)
dondeI = 1, . . . , m ⊂ N, E = m+1, . . . , m+ p ⊂ N y f : Rn 7→ R, (n ≥ 1)
es una funcion continuamente diferenciable en Rn. Ademas, suponemos que
29
las funciones gi : Rn 7→ R y hi : R
n 7→ R son lineales para todo i ∈ I y todol ∈ E.
El algoritmo final lo implementamos en MATLAB 6.5 y lo usamos pararesolver algunos problemas de la coleccion Hock-Schittkowski [1, 4]. Ademas,comparamos los resultados obtenidos en la solucion de dichos problemas conesta adaptacion, con el metodo de Busqueda patron, propuesto por RobertMichael Lewis y Virginia Torczon [3].
[1] Hock, W. e Schittkowski, K., 1981. ”Test examples for nonlinear program-ming codes”, Springer, Berlin.
[2] Lucidi, S., Siandrone, M., e Tseng, P., 2002. .Objetive-derivative- freemethods for constrained optimization”, Mathematical Programming, pp. 37-59.
[3] Lewis, R. M. e Torczon, V., 2000. ”Pattern search methods for linearlyconstrained minimization”, SIAM Journal of optimization, vol. 10, No. 3, pp.917-941.
[4] Schittkowski, K. 1987. ”More test examples for nonlinear program- mingcodes”, Springer, Berlin.
Lopez Erazo, Tulio EmiroUniversidad del CaucaPopayan, [email protected]
Efficient solution of large scale matrix equations arising in LQR/LQGdesign for parabolic PDEs
Feedback control of systems govern by partial differential equations has beenan engineers desire for many decades now. Linear Quadratic Regulator (LQR)design and Linear Quadratic Gaussian (LQG) design have been investigatedin the literature since the pathbreaking work of Lions, showing that thesemethods are valuable tools in achieving this task.
The solution of large scale Lyapunov and Riccati equations is a majortask in applying the above techniques to semi-discretized partial differentialequations constraint control problems. In this context, the same task appearswhen Balanced truncation model order reduction are applied. The softwareLyaPack has shown to be a valuable tool in the task of solving these equationssince its introduction in 2000.
Here we want to discuss recent improvements and extensions of the un-derlying algorithms and their implementation in the successor library M.E.S.S(matrix equation sparse solver).
Mena, HermannEscuela Politecnica Nacional
30
Quito, [email protected]
Benner, PeterTechnische Universitat ChemnitzChemnitz, [email protected]
Saak, JensTechnische Universitat ChemnitzChemnitz, [email protected]
Error estimates for the FEM approximation of a semilinear ellipticcontrol problem with state constraints and finite dimensional controlspace
We derive error estimates for a finite element based approximation of a semi-linear elliptic optimal control problem with a finite-dimensional control spaceset along with equality and inequality constraints on the state, which are re-quired only in finitely many points of the spatial domain. If pointwise stateconstraints are given in this case, then they will often be active only in finitelymany points. Therefore, finitely many point constraints seem to be inter-esting for numerical computations, if the location of active points has beenfound approximately. Our problem is equivalent to a nonlinear programmingproblem in a finite-dimensional space. We refer Merino et.al. [2007] whereoptimality conditions for semilinear control problems with finite-dimensionalcontrol space and pointwise state constraints in the whole spatial domain arediscussed. Computations indicate that the numerical approximation of theoptimal controls and states is of the order h2, where h > 0 is the mesh size ofthe finite element scheme. In fact, we can show that order the error is equal tothe error made by the finite-element error for the state. Here, no interpolationerror of the optimal control occurs that in the case of control functions whichlimits the order of the order of the approximation. If pointwise state con-straints on the whole domain are given, then there are only a few results. Werefer to Casas [2002] who studied the semilinear problem for functional con-trols and obtain an orther of h2−n/2, with n be the dimension of the domain.In Deckelnick and Hinze [2000] linear-quadratic problems with pointwise stateconstraints are considered and obtain the order h2−n/2. Since the Lagrangemultipliers associated with the (finitely many) pointwise state constraints areregular Borel measures, which appear in the adjoint equation of the controlsystem, it is quite surprising the error near to h2. It is the finite-dimensionalnature of our problem that explains the high order of the error. Numericalexperiments are presented.
[1] Casas, E. (2002). Error estimates for the numerical approximation of semi-linear elliptic control problems with finitely many state contraints. ESAIM:
31
Control, Optimization and Calculus of Variations 8, 345-374.
[2] J. C. de los Reyes, P. Merino, J. Rehberg, and F. Troeltzsch. Optimalityconditions for state-constrained PDE control problems with finite-dimensionalcontrol space]. submitted, 2006.
[3] M. Deckelnick and M. Hinze. Convergence of a finite element approximationto a stateconstrained elliptic control problem. SIAM J. Numerical Analysis,to appear, 2006. Cantabria, 2000.
[4] C. Meyer. Error estimates for the finite-element approximation of an el-liptic control problem with pointwise state and control constraints. WIAS,Preprint 1159, 2006.
Merino, PedroEscuela Politecnica NacionalQuito, [email protected]
Troltzsch, FrediTechnische Universitat BerlinBerlin, [email protected]
Vexler, BorisTechnische Universitat MunchenMunchen, [email protected]
Optimal control on nilpotent Lie groups
Let M be a smooth manifold of dimension n, and let δ be a rank k < n distri-bution of smooth vector fields satisfying the bracket generating condition. Inthis lecture we present the optimal control problem determined by δ, togetherwith the energy functional of δ-horizontal curves, for the case when M is astep-2 nilpotent Lie group and δ is a finite collection of left invariant vectorfields.
The iteration of the Lie bracket of vector fields in δ yields the flag ofmodules of vector fields δ1 ⊂ δ2 ⊂ · · · ⊂ δl · · · ⊂ TM, where δ1 = δ andδi+1 = δi + [δ, δi]. The distribution is said to be bracket generating, if for eachp ∈ mboxM , there exist a positive integer m for which δm
p = TpM.An absolutely continuous curve t 7→ p(t), is said to be δ-horizontal, if
p(t) ∈ δ(p(t)), almost everywhere. A sub-Riemannian metric is defined bya smooth varying inner product p 7→ 〈·, ·〉p in δ(p). For horizontal curvest 7→ p(t), the energy functional is defined as usual,
E(p) =1
2
∫
〈p, p〉.
32
We approach the problem as a sub-Riemannian geodesic problem, that is,the one of minimizing functional E , in the class of δ-horizontal curves. We usethe Pontryagin Maximum Principle and the associated Hamiltonian formalismto derive necessary conditions for geodesics. We integrate some cases of theHamiltonian equations and derive some geometric properties of the geodesicsand small radiispheres.
Monroy Perez, FelipeUAM-AzcapotzalcoMexico D.F., [email protected]
A symmetry preserving alternating projection method for matrixmodel updating
The Matrix Model Updating Problem (MMUP), considered in this paper, con-cerns updating a symmetric second-order finite element model so that the up-dated model reproduces a given set of desired eigenvalues and eigenvectorsby replacing the corresponding ones from the original model, and preservesthe symmetry of the original model. In an optimization setting, this is aconstrained nonlinear optimization problem. Taking advantage of the spe-cial structure of the constraint sets, it is first shown that the MMUP can beformulated as an optimization problem over the intersection of some specialsubspaces and linear varieties on the space of matrices. Using this formulation,an alternating projection method is then proposed and analyzed. The projec-tions onto the involved subspaces and linear varieties are characterized. Tothe best of our knowledge, an alternating projection method for MMUP hasnot been proposed in the literature earlier. A distinct practical feature of theproposed method is that it is implementable using only a few measured eigen-values and eigenvectors. No knowledge of the eigenvalues and eigenvectors ofthe associated quadratic matrix pencil is required. The results of our numer-ical experiments on both illustrative and benchmark problems show that thealgorithm works well. The paper concludes with some future research prob-lems.
Moreno, JoaliUniversidad Central de VenezuelaCaracas, [email protected]
Variable metric proximal decomposition methods
In this work, we extend the general decomposition scheme known as HybridProximal Decomposition Method (HPDM). This extended method deals withtwo kinds of inexactness in both stages of the decomposition scheme and allows
33
for the use of a variable metric in the subproblems. We prove global conver-gence and we characterize the local linear convergence rate. Also, we show thatthe Splitting Method for Composite Mappings proposed by Pennanen and theProximal Alternating Directions Method proposed by He et al. can be put inthis scheme as special cases.
Parente, LisandroUniversidad Nacional de RosarioRosario, [email protected]
Lotito, PabloUniversidad Nacional del Centro de Buenos AiresBuenos Aires, [email protected]
Solodov, MikhailInstituto de Matematica Pura e AplicadaRio de Janeiro, [email protected]
Polyhedral considerations for a flower production and packing prob-lem.
An Ecuadorian company cultivates about ninety varieties of flowers which arethen combined according to certain rules (called recipes) into several kinds ofproducts sold to the customers. We have considered an optimization modelfor assigning flowers to products in such a way that the global revenue ismaximized. Our model is based on the idea of considering each product as anarray of ”slots” that have to be filled with flowers. For this purpose, each slothas a list of ”admissible varieties” associated to it. There are many possibilitiesto assemble a product by selecting flowers for a fixed recipe. For example,several varieties of flowers of the same color are feasible in a recipe. Formixed products which contain several colors, the number of feasible flowercombinations for a single product turns out to be exponential. So, we faced alarge scale integer programming problem.
In this work we present a theoretical analysis of the polyhedron associatedto this problem for which we proved it is NP-hard. We demonstrated, undercertain hypotheses, the existence of some families of facets. Computationalexperiments, using the SCIP v1.00 solver, showed that our families of facetsimproved a twenty percent (in average) the solving time of the integer pro-grams for some specific set of instances.
Recalde, DiegoEscuela Politecnica NacionalQuito, [email protected]
34
An integer programming approach for the single source minimumcost unsplittable flow problem
In the single source unsplittable min-cost flow problem commodities must berouted simultaneously from a common source node to certain sink nodes in agiven digraph. The demand of each commodity must be routed along a singlepath respecting arc capacities and the total cost must be minimized. Sev-eral approximation algorithms have been developed in order to solve differentversions of this problem. Dinitz, Garg, and Goemans proved that any givensplittable flow satisfying certain demands can be turned into an unsplittableflow such that the flow value on any arc exceeds the flow value on that arc,in the given flow, by no more than the maximum demand. Goemans conjec-tured that this result even holds when we include arc costs, then it is requiredthat the cost of the unsplittable flow must not exceed the cost of the givensplittable flow. We consider here an integer programming based approach andformulate a model for finding the minimum cost unsplittable flow of conges-tion bounded by a fixed given parameter. We present a solution algorithmbased on the branch-and-bound framework. As a first step, two alternativesfor the computation of lower bounds are given , namely, Linear relaxation andLagrangian relaxation. Then, a branch and bound scheme for the originalproblem is designed by taking advantage of the inner structure of the integerprogram. Preliminary computational results regarding to Lower bounds as wellas to the whole branch-and-bound algorithm for the integer model are shownand compared with solutions given for integer program solvers. We have usedthe same input data considered in previous investigations. Such tests revealthat most of the time our algorithm achieves the optimal solution or finisheswith a good approximation in a reasonable running time. Moreover, when-ever our algorithm obtains the optimal solution, we compare it with the costof the min-cost (splittable) flow and verify that the Goemans conjecture holds.
Salazar, Marıa FernandaEscuela Politecnica NacionalQuito, [email protected]
Recent advances for location-routing problems
Location-routing problem (LRP) is a generic name for a relatively new class ofNP-hard combinatorial optimization problems at the intersection of locationand transportation analysis. Generally speaking the LRP consists of deter-mining locations for facilities from which customers are served on routes withthe objective of minimizing the overall cost. More precisely the LRP involvesthree types of decisions related to: i) facility location; ii) customer allocationto facilities; iii) vehicle routing. The overall cost includes the costs for usingand operating the facilities and the transportation costs. Many variants of the
35
LRP are addressed in the literature. They mainly differ by the side constraintsimposed and by the number of intermediate facilities considered. In this talkwe focus on some of variants for which we describe mathematical programmingmodels and recent solution methods. In particular, we consider: i) the capac-itated location-routing problem in which capacity restrictions are imposed onthe depots and on the vehicles; ii) the two-echelon location-routing problemin which the customer demands are transported from the depots to customersthrough intermediate facilities.
Semet, FredericUniversite de ValenciennesValenciennes, [email protected]
Generalized Nash games and equilibrium problems
We reformulate the generalized Nash equilibrium problem as an equilibriumproblem so that solving the former problem is reduced to solving the latterproblem. We use Ky Fans Lemma to obtain a new existence result for equilib-rium problems, consequently for the generalized Nash equilibrium problems,which does not invoke monotonicity and convexity of the objective function.
Sosa, WilfredoInstituto de Matematica y Ciencias AfinesLima, [email protected]
Programacion estocastica y algoritmos en-lınea para el PGSC
En este artıculo se aborda el problema de la gestion de fondos que la bancaprivada mantiene en el Banco Central del Ecuador. El problema es formuladocomo un programa lineal estocastico multietapa, donde el parametro sujeto aincertidumbre es el flujo de caja diario. La tecnica de programacion con re-curso resulta inaplicable al momento de resolver instancias reales, debido a queel numero de escenarios crece explosivamente conforme aumenta el numero dedıas dentro del horizonte de estudio. Como alternativa, se proponen algoritmosen-lınea para determinar el monto de inversiones y retiros a realizarse diaria-mente sobre una cuenta en el extranjero, con el fin de maximizar la utilidadesperada dentro de un cierto horizonte de tiempo, mientras el riesgo de que sepresente una situacion de falta de liquidez se mantiene acotado. El desempenode estos algoritmos se estudia mediante simulaciones computacionales sobreinstancias extraıdas del registro historico del BCE.
Soto Lima, Marıa ConsueloBanco Central del EcuadorQuito, ECUADOR
36
Nuevo metodo de resolucion exacta del problema inverso de modelosde tipo de Black-Scholes-Merton con volatilidades determinısticas
El resultado principal de este trabajo es un nuevo metodo de resolucion ex-acta del problema inverso para un gran numero de modelos financieros detipo de Black-Scholes-Merton. Construimos una ecuacion algebraica para lavolatilidad con coeficientes expresados en funcion de las variables observablesen el mercado y una variable que puede ser facilmente calculada usando losdatos del mercado. Resolvemos el problema inverso para algunos modelos convolatilidades variables.
Nuestro metodo se puede aplicar a numerosos modelos de tipo Black-Scholes de valuacion del precio de opciones, el Merton-Black-Scholes-Cox mod-elo estructural de “defaultable” bonos, varios modelos del riesgo de credito,Sharpe ratios, derivados del mercado cambiario, modelo de Black Scholes ajus-tado por la asimetrıa y curtosis, Cox-Ingersoll-Ross modelo, modelo de Vasicek,etc.
De hecho, nuestro metodo esta basado en la existencia de una simetrıaespecıfica. Para todos los modelos de este tipo construimos una ley de conser-vacion que llamamos “Ley de conservacion de strike price” que es intrınsecaa “backward” y “forward” problemas. La resolucion del problema inverso dedichos modelos es posible si el sistema dinamico tiene la simetrıa definida porun especial ”backward” operador que se puede interpretar como el “Operadorde strike price”. Este operador nos ayuda a construir la expresion explıcitadel “backward propagador” de Derman. Dicho tipo de leyes de conservaciontienen el misma papel que las integrales de movimiento en mecanica.
Tambien comprobamos que el problema de la solucion de la ecuacion para-bolica de Black Scholes, junto con la “Ley de conservacion de strike price” esequivalente al problema de la solucion de la ecuacion de Black Scholes juntocon la conocida condicion estandar para el momento de maduracion. Estehecho significa que el conjunto de esta ley de la conservacion y de la ecuacionBlack Scholes constituye una forma implıcita de la solucion clasica de BlackScholes.
El concepto de leyes de conservacion fue originalmente desarrollado porlos fısicos, pero los economistas se han visto considerablemente interesados enel. Argumentamos que la fenomenologica “Sticky Strike Rule” de Derman ynuestra exacta ”Ley de conservacion de strike price” tienen el misma origen.Particularmente, establecemos el papel importante de la elasticidad en todosestos modelos.
Ademas, nuestro metodo permitio descubrir la estructura compleja del “Es-pacio de riesgos” cuyo concepto introducimos. Revelamos que varios modelosactualmente usados en las finanzas cuantitativas incluyen mas de una volatil-idad para un subyacente. Descubrimos la base teorica de la bien conocidaasimetrıa entre los casos de “in-the-money” y “out-of-the-money”.
37
Sukhomlin, NicolayUniversidad Autonoma de Santo Domingo, Pontifıcia Universidad CatolicaMadre y MaestraSanto Domingo, REPUBLICA [email protected]
A sensitivity analysis of a class of variational inequalities
Let us consider a general variational inequality: (VI) find u ∈ K ∩ Dom Φsuch that
〈Au− f ,v − u〉 + Φ(v) − Φ(u) ≥ 0, ∀v ∈ K.
Here K is a nonempty closed convex set of X, f is a fixed element in thetopological dual X⋆ of X, Φ : X → R ∪ +∞ lower semicontinuous convexproper function which is bounded from below and such that K∩ Dom Φ 6= ∅,and A : X → X⋆.
The question of giving sufficient conditions for the existence of a solution ofproblem (VI) is a central problem in the study of variational inequalities. Thisquestion has been at the origin of many contributions during the last years.Several theoretical existence results for variational inequalities in general re-flexive Banach spaces and governed by a general operator A (not necessarilylinear) are well known when a coerciveness condition hold for the operatorA. We can cite for instance the contributions of J.L. Lions, Brezis, Browder.etc. However, the variational formulation of many engineering problems leadsgenerally to non-coercive variational inequalities (e.g. problems in mechanicswhich admits nontrivial virtual rigid body displacement). These problems areformulated by semi-coercive variational inequalities and were studied first byFichera and Lions & Stampacchia, Duvaut & Lions (for problems with fric-tional type functionals). Recently many mathematicians and engineers hasfocused their attention on non-coercive unilateral problems, using several dif-ferent approaches such as the critical point theory, the Leray-Schauder degreetheory, the recession analysis or the regularization method by approximatingnon-coercive problems by coercive ones (see e.g. Adly - Goeleven & Thera,Ang - Schmidt & Vy, Baiocchi - Gastaldi & Tomarelli). The main concernof these contributions is to obtain necessary or sufficient conditions for thesolvability of such problems in a general setting by imposing some compactnesconditions and some compatibility conditions on the term f . More recently,Adly - Ernst & Thera has considered situations in which the existence of thesolution is stable with respect to small uniform perturbations of the data ofthe problem. This result should be of great interest for problems in financeand engineering where the data are known only with a certain precision and itis desired that further refinement of the data should not cause the emptinessof the set of solutions.
In this presentation, we will review, old and new results on existence resultfor the solvability of variational inequalities. We will also focus on a converseof the famous Lions & Stampacchia Theorem of 1967.
38
Thera, MichelUniversite de LimogesLimoges, [email protected]
Line planning in Quito
Line planning is an important step in the phase of strategic planning in apublic transportation system. In the present work, we present an optimizationmodel aiming at the reduction of the global operational costs while ensuringa certain level of quality of service, in terms of available transport capacity.The computational complexity of this model is discussed for some transporta-tion network topologies arising in the context of the Trolebus System, whichis by now the largest public transportation system in Quito, carrying around250, 000 passengers daily. Moreover, we have considered how several otherfactors affect the computational complexity of the model. If either fixed costis allowed or the transportation modes are greater than two, then a reductionfrom the Minimization Knapsack Problem and 3-Dimensional Matching Prob-lem can be used to show that our Line Planning Model is NP-hard. Finally,results of computational tests based on real data are reported.
Torres, RamiroEscuela Politecnica NacionalQuito, [email protected]
Set packing and set covering problems appearing in biological net-works
Given a ground set X, a family F of subsets from X, and a positive weightfunction c : F → R
+, the set packing problem asks for a subfamily F1 ⊂ F ofmutually non intersecting sets such that the sum of their weights is maximized.Conversely, the set covering problem asks for a mimimum weight subfamilyF2 ⊂ F with the property that every element x ∈ X is contained at least inone set of F2. Considering the hypergraph H where the nodes are the elementsfrom X and the hyperedges the sets in F , these problems can be formulated asfinding either a maximum weight matching (set packing) or a minimum weighthyperedge cover (set covering) in H.
Set packing and set covering are fundamental problems in combinatorialoptimization which are related to applications in many areas such as vehi-cle routing, line planning in public transportation, frequency assignment intelecommunications, bandwidth packing, scheduling of airline crews, to cite afew.
The analysis of metabolite networks has received increasing attention inthe recent past. A typical approach for metabolic pathway analysis consists in
39
considering the solutions to the system:
Sv = 0, vi ≥ 0, ∀i ∈ I,
where S ∈ Rm×n is the stoichiometric matrix of the network, which repre-
sents a mapping of n reaction rate vectors on m metabolites into a space ofconcentration time derivatives, and I ⊆ R := 1, . . . , n is the set of thermo-dynamically irreversible reactions. This system is obtained from the metabolicbalance equation at the so-called quasi-steady state. A solution v ∈ R
n to thissystem is called a flux vector. Moreover, the set of all feasible solutions definesa polyhedral cone C, which is called the flux cone.
In this talk I will explore, from a polyhedral viewpoint, the structure ofcertain set packing and set covering problems that arise in the context of bi-ological networks; for instance, to create non-conflicting activation rules forreactions involved in a regulatory structure, or to determine minimal inter-vention sets. This is ongoin research that I am currently starting as a partof a postdoctoral stay at the Otto-von-Guericke Universitat in Magdeburg,Germany.
Torres, Luis MiguelUniversitat Magdeburg and Escuela Politecnica NacionalMagdeburg, [email protected]
40
Information and tips
Talks. The LAWOC opening ceremony and monday presentations willtake place at the Paraninfo of the Universidad Andina Simon Bolıvar. The restof the talks will take place in Aula 21 at Universidad Andina Simon Bolıvar.
Coffee breaks. Coffe will be served in the Sala social.
Power Supply. Please note that the voltage in Ecuador is 110 V andAmerican Plugs are needed.
Useful Phone Numbers.
Organization 098122984Emergency 911 Police 101Firemen 102 Airport 2 440 080
International Phone Calls and Internet. Computers with internet ac-cess are available at the computer laboratory of the Universidad Andina SimonBolıvar. A limited amount of wireless internet accounts will also be available.Wireless internet access and computers are also available at the 6th floor ofthe Administration building at EPN Quito.
International calls can be made either from ANDINATEL (see item 29 inthe map) or from internet cafes. Some internet cafes are listed in the mapreference.
Transportation in Quito. Quito has two main bus systems: Trole andEcovıa. Both run on sole set lanes across the city north to south, with extrabus lines running to suburbs from the northern and southern stations. TheTrole has 2 routes running mainly along Av. 10 de Agosto, and Ecovıa crossalong Av. 6 de Diciembre. The fare for both systems is US $0.25.
Taxis are a good transportation alternative. You can get to almost anypoint in town for about $3. Legal taxis (yellow cars) have to be certified bylocal government. That certification placard is prominently displayed on bothsides of and in the upper right windscreen of both taxis and buses. It is notrecommended to use taxis lacking this local government seal of approval. Theywill all pick you up from wherever you’re calling. Here you will find a list ofRadio Taxi companies to use:
Taxi Amigo Tel. 2222222 / 2333333 / 2222220City Taxi Tel. 2633333 / 2630220Central Radio Taxis Tel. 2406806 / 2811111 / 2267900RTR Tel. 2533878 / 2535600JJ Taxi Tel. 2639639 / 2639999 / 2628400.
41
Cash dispensers. Most of the restaurants and services accept credit cards.However, it is recommended to have extra cash. You can get it at places nearbyto the workshop building (see the map references).
Money exchange. If you need to exchange money, there are some placesnear the LAWOC.
• Vazcorp (outside the map). Traveller’s checks in other currencies.Address: Amazonas and Roca (Hotel Alameda Real). Tel. 2225442.
• Multicambios (Item 28 in the reference map).Address: Colon 919 and Reina Victoria, Tel. 2561747; Roca 720, Tel.2567344 and at the airport.
Food and drinks. A practical choice is to take a taxi and go to PlazaFoch. There you can find a lot of restaurants and bars. You can find severalpossible options at different prices.
Restaurants. Specific restaurant choices are given in the following list.
• El Esmeraldas. Ecuadorian seafood.Address: Isabel La Catolica y Luis Cordero.
• Naranjilla Mecanica. Fusion food.Address: Tamayo y General Veintimilla
• Al forno. Italian Pizza.Address: Baquerizo Moreono E7-86 y Almagro.
• Pim’s Grilled food.Address: Isabel La Catolica 915 y Luis Cordero.
• Tibidabo. Catalunian food.Address: Francisco Salazar 934 e Isabela Catolica.
• Antojerıa Chilena. Chilenian food.Address: Isabel La Catolica 1129 y Coruna.
• Time Out. Grilled food and sports bar.Address: Isabel La Carolica y Luis Cordero. Swisshohel.
• El Pobre Diablo. International food and Bar.Address: Isabel La Catolica E12-06 y Galavis esq. La Floresta.
• Swisshotel. International food.Address: Isabel La Catolica E12-06 y Galavis esq.
• La Briciola. Italian food.Address: Toledo 1255 y Luis Cordero
• La Vina. International food.Address: Isabel La Catolica y Luis Cordero.
• La Choza. Ecuadorian food.Address: 12 de Octubre N24-551 y Cordero
• Happy Panda. Chinese food.Address: Isabel La Catolica N94-464
• Quito Deli. Cafeteria.Address: Isabel La Catolica 915 y Luis Cordero.
• Mediterraneum. Seafood.Address:Andalucıa 376 y Cordero
• Entenca. Mediterranean food.Address: Coruna e Isabela Catolica.
42
• Tratoria. Italian food.Whimper N21-29
• Barlovento. Seafood and Ecuadorian food.Address: 12 de Octubre 2511 y Orellana.
• Hunters. Grilled food.Address: 12 de Octubre 2517
• Hotel Quito. International food.Address: Gonzales Suarez N27-142
• Ceviches de la Ruminahui. Ecuadorian seafood.Address: 12 de Octubre y Colon.
• Gran Casa China. Chinese food.Address: Cordero E9-242 y Tamayo.
• Corrientes 348 Argentinean food.Address: Edmundo Carvajal OE4-226 y el Condor.
• Mi Cocina Ecuadorian food.Address: Centro Comercial Megamaxi, 6 de Diciembre y Calle Aleman.106-110
• Amarilo Grilled food and Bar.Address: Juan Gonzales N35-113.
• Rincon Italiano Italian food.Address: Av. Los Shyris41-61 e Isla Floreana.
• Rincon la Ronda Ecuadorian and International food.Address: Bello Horizonte E48-45 y Av. Almagro.
43
44
Touristic information
Quito is the capital of Ecuador and a recommended touristic city. Withattractions ranging from parks to historical centers and museums, there aremany possibilities for sightseeing and visits. Outside the city, there are greenmountains and snow-capped volcanoes to enjoy.
Parque Itchimbıa. The Itchimbıa Park is located at the summit andslope of the Itchimbia hill, at the oriental limit of Quito’s Historic Center at2 900 meters. It is considered as a natural lookout point because of the greatview that you get of Quito’s Historic Center in either way. The total areaof the park is 54 hectares; in it you can find around 400 varieties of flowersand 40 species of birds. One of its main attractions is the Itchimbıa CulturalCenter or Crystal House, which was rebuilt from the old metallic structure ofthe Santa Clara Market, built in Hamburg, Germany, in 1889 and brought toEcuador in pieces.
Hours Mon-Sun 10 a.m. to 6 p.m.Location El Itchimbıa, east of Old TownPrices Musseum tiket $1 pp.
Teleferiqo. One of the city’s most popular attractions is El Teleferiqo,where six-person cable cars transport you up to the side of Volcan Pichinchato 4,050m (13,280 ft.). At the top, you will have a magnificent view of the cityand surrounding mountain peaks and of the Pichincha volcano. The air is thinup there. The ambitious and very modern complex includes an oxygen bar toreplenish the weary traveler. You’ll also find souvenir stands and shops, anda couple of restaurants.
Hours 10 a.m. to 6 p.m.Location El Teleferiqo. Calle Arnulfo Araujo y Av. OccidentalPrices Cable car ride to the top $4 pp.
El Panecillo. From a distance, the hill that hosts a huge statue of thewinged virgin does indeed look like a panecillo (small bread roll). Since it’sdirectly south of the city, this hill was an ideal place to construct the 45m-high (148-ft.) statue. The significance of the Panecillo Hill dates back to Incatimes, when it was known as Shungoloma (Hill of the Heart) and used as aplace to worship the sun. Later the virgin’s statue was constructed in theColonial time. These days, most people come up here for the 360 views ofQuito. Try to get here in the morning (around 10am), before the clouds settlein around the nearby mountains. On a clear day, you can see Cotopaxi in thedistance. This is a relatively quick ride from Old Town, and a taxi should onlycost about $3.
45
Hours Mon-Fri 9am-6pm; Sat-Sun 9am-5pm.Location El Panecillo, south of Old Town.Prices Admission to climb to the top of the monument $1 pp.
Getting to the Equator 0o. The ”Mitad del Mundo” is around 15 kilo-meters from Quito. There is a small town, that owes its name from the factthat it’s fount in the latitude 0, reason why a commemorative monument of30 meters of height has been built. In its upper part, there is a 5 tons globe.One of the attractions and mysteries of the Mitad del Mundo is that duringthe equinoxes (21st of March and 21st of September) people and objects donot project not even the minimum shade. Taxis will take you there for about$10.
Quito’s old Town. Most of the major tourist attractions are in Quito’sOld Town. There you will find all the historic churches and other sites ofcolonial architecture. Named a world heritage site by UNESCO. To get to theold town you can ride a taxi to the Plaza de la Independencia (Independencesquare) then continue by walking. Several restaurants and musseums are ac-conditioned to spend a nice time at the heart of the city.
The Cathedral. Located on the Main Square, the Cathedral has an inter-esting collection of sculptures and paintings from the Escuela Quitena. Amongthe most important ones is the Descending of Christ made by Caspicara. Asin most Ecuadorian and Latin American churches, many styles were used inthe construction of the Cathedral, such as late Gothic in the arches Moorishin the ceilings and Baroque in the main altar. See the Neoclassic style in thechoir, the stone Episcopal chair, the central painting by Manuel Samaniegoand the Statues by Caspicara.
San Francisco Church. It was constructed after the Spanish Conquest andis said to have given the capital of Ecuador the name of San Francisco de Quito.The atrium running along one side of the plaza is opened in the middle to giveway to a beautiful staircase. The facade has a style similar to the Escorial inSpain and in the inside the Baroque style. The coffer ceiling in the marthexhas a rich Moorish style ornamentation with paintings by Miguel de Santiago.It is interesting to note among the ornamentations images of the sun god, theInca divinity. The main altar holds the original masterpiece by Legards: ”LaVirgen de Quito”. This sculpure is the only winged image of the Virgin Mary.
Cantuna Chapel. According to the legend of this Chapel, Francisco Cantuna,the indian who paved the atrium with large stone block, constructed the chapelwith the treasures saved from the Kingdom of Quito. The Calvary woodcarv-ings on the main altar are one of Legarda’s most outstanding masterpieces ofthe colonial times. At this Chapel, another masterpiece to look at, carved byPadre Carlos, is Saint Peter of Alcantara.
San Francisco Museum. Located at the San francisco Convent, in SanFrancisco Square. The zaguan (antique entrance hall), the main cloister, therenacentist stair way, the chorus and the exposition lounge are now the site of
46
a number of selected works of the Franciscan collection. You can find worksof Andres Sanchez Galque (of indigenous origin), Miguel de Santiago, Mateomexıa, the European authors Zurbaran and Bernardo de Bitti, or their repec-tive schools, among others. In the third lounge you can find samples of theEuropean engravings that inspired native painters. Among the sculptures, youcan find the glass eyes characteristic of the XVIII century.
The Museum of the City. The Museum of the City of Quito (Museo dela Ciudad) opened in July, 1998, in the former Hospital of Mercy of our LordJesus Christ in the Old town. The historic building was used as a hospital from1565 to 1974. In the 16th century the hospital provided lodging for the sickawaiting death and also for travelers. After the Bethlemite Brothers acquiredthe hospital around 1700, they added a chaplain, church, drugstore, vegetablegarden and two fountains with fresh water. The brothers also built the richlydecorated chapel, which has nine splendidly carved altars representative of theQuito School of Art. The first Ecuadorian medical doctor, Eugenio de SantaCruz y Espejo, was born in the hospital in 1747 and later practiced there. AFaculty of Medicine and a lecture aula were added in 1826. French scientistscame to teach at the hospital in 1872.
The Museo de la Ciudad shows Quito’s history in chronological order from10,000 BC to today. The city is presented as a center of exchange and of multi-cultural encounters, as a political and administrative center, and as the depar-ture point for expeditions that discovered and explored the Amazonas. Manyexhibits, which include dioramas and lifesize replicas, focus on the changes indaily life through the centuries. The first rooms show the area’s prehistoricinhabitants and Indian communities that preceded the arrival of the Incas in1487. The other exhibitions are organized by century. The original hospital isrecreated with a ward containing beds, where the sick awaited death on boardscovered with hay
The 16th-century Room includes a map of Quito made of inlaid wood,a lifesize Indian hut, a drawing of Plaza San Francisco as the city market,helmets and armor of Spanish conquistadors, and the orders of priests thatcame to Quito. The prosperous 17th century is depicted by the manufactoringand trade in textiles, the studio of Miguel de Santiago of the Quito School ofArt, and houses with bakery and candle shops.
Rooms of the 18th century feature French and German scientists who stud-ied the exact location of the Equator in the Andes and the area’s flora andfauna, period dresses, the rigid class system, a typical kitchen and a roomwhere women sat in Arab fashion on rugs and pillows to chat, sew, embroi-der or smoke. The 19th-century Room shows the influence of the French, thestruggle for independence from Spain and scenes of everyday life.
The House of Urrutia. The house-museum of Marıa Augusta Urrutia islocated in the colonial section of Quito, and was built at the beginning of the20th century, and is an excellent example of the arquitecture of this period, atypical style with inside gardens, corridors and many rooms.
Mrs. Urrutia, came from a very wealthy family and was born in Quitoin 1901. She spent her childhood and adolescent years in Europe, where shedeveloped a taste for French decoration and the art of the great masters. The
47
house is furnished with a collection of Ecuadorian art from the Colonial timesto the 20th century: apart of the paintings and other works of art, you cansee, together the local large bronze pots beside fine French porcelain and silverdinnerware.
Mrs. Urrutia’s husband, Alfredo Escudero, died a few years after the mar-riage. Mrs. Urrutia dressed in black for the rest of her life, until her death in1987. She kept the house exactly as it had been at the time of her husband’sdeath, and as she was very religious, puts all her efforts and wealth to help thepoor, specially the children, and to patronize very good Ecuadorian painters,like Mideros, whose mystic and esoteric type paintings, of great format andmuch use of bright blue, adorn the house. The House of Urrutia is an en-counter with the time, the spirit and the daily life of Quito at the beginningof the 20th Century.
Recoleta de San Diego. The Franciscan Order is linked to Quito fromthe time of the foundation of the city, which, by the end of the 16th Centuryalready had a solid urban structure and an important part of the religiousbuildings: churches, cloisters and monasteries with hundreds of altars, altar-pieces and facades, which required a wealth of artists and capable craftsmenwho studied from 1550 in the schools of San Juan Evangelista and the mostknown of San Andres - where the famous School of art of Quito was born(Quitenian Baroque). These schools were run by the Franciscans.
The devoted parishioners who filled these places of devotion, also demandedthe possibility of doing retreat exercises in places separate from the urban noise.Thus, the house of retreat or the ”Recoleta de San Diego” was built, locatedbetween deep ravines on the slopes of the Pichincha mountain, in the outskirtsof the city, and quickly became a renowned center for retreat for monks andlaymen alike.
Construction started in 1600 in care of Father Bartolome Rubio. Theconstruction includes a cloister of two patios, refectory, sacristy, cells orchards.It has a humble facade that extends in front a patio, similar to a small plaza,in the center of which stands a stone cross. The main door has a cutoff corniceset in a high arch with a square window at the top, and a sober atrium ofpillars ending in pyramidal spires. The original main door is now inside, toprotect it from the elements. The walls of the cloisters show windows of allsizes, created according to the light requirements of the retreat cells.
In its interior the church unites the simplicity of its whitewashed walls tothe serene elegance of the craftsmanship of the presbyter in cedar wood in themudejar style (Moorish), it has floors of stone; the pulpit and altarpieces ofThe Candelaria (virgin of) and the much revered Virgin of Chiquinquira byMenacho an Indian sculptor, are marvelous works of art.
Casa de la Cultura Ecuatoriana. (item 33 in the map) (6 de Diciembreand Patria) This large and modern building is located in the north-central partof the city. It gathers together the best of Ecuadorian art. Each room or spacereflects a different theme, such as ethnography, sculpture, music or painting.From time to time there are shows and exhibitions featuring local and foreignartists. This museum has facilities for artistic performances, exhibitions, filmscreenings, dance shows and other types of events. It is permanently open to
48
the public and allows the visitor to enjoy a wide range of shows in one place.The Central Bank Museum is also located there. Visiting hours: Tuesday-Friday from 09h00 to 17h00; weekends and holidays from 10h00 to 15h00. US$. 2.00. Students: US $. 1.00.
City Council Art and History Alberto Mena Caamano Museum.(Espejo 1147 at the main square). The attractive building in which the mu-seum operates belonged to the Jesuits until 1767 when Charles III of Spainbanished the Jesuit Order from the colonized territories. Later it became thearmy headquarters of the Spain Royal troops sent from Lima to oppress theearly independence effort. This is why the building was known as the RoyalBarrack of Lima (El Cuartel Real de Lima). Opened Tuesday to Saturdayfrom 9h00 - 16h30.
National Museum of Colonial Art. (Cuenca and Mejıa). This museumoperates in a XVII century home of the Marquis of Villacis. The patio has anattractive fountain and garden and the corridors and halls display a collectionof paintings and sculptures of the time. Barguenos, antique bureaus used tostore documents and money, are an interesting exhibit here.
49
50
Map Reference
1 LAWOC.2 Escuela Politecnica Nacional. Ladron de Guevara E11-253
Restaurants.
3 El Pobre Diablo. International food and Bar.4 Tibidabo. Catalunian food.5 Antojerıa Chilena. Chilenian food.6 Naranjilla Mecanica. Fusion food.7 Swissotel. International food.
11 Happy Panda. Chinese food.12 La Choza. Ecuadorian food.12 Mediterraneum. Seafood.13 Barlovento. Seafood and Ecuadorian food.14 Hunters. Grilled food.16 Time Out. Grilled food and sports bar.17 Entenca. Mediterranean food.18 La Vina. International food.19 El Esmeraldas. Ecuadorian seafood.21 Quito Deli. Cafeteria.22 Pim’s Grilled food.23 Tratoria. Italian food.24 Hotel Quito. International food.25 Ceviches de la Ruminahui. Ecuadorian seafood.26 Gran Casa China. Chinese food.35 La Briciola. Italian food.36 Al forno. Italian Pizza.
Cash dispensers.
7 Swissotel.8 Radisson Hotel.9 Banco Pichincha.
15 Supermaxi.27 Banco del Pacıfico.
Internet cafes.
30 Papaya Net. Address: Calama 413 and J.L. Mera.31 Pool Net. Address: Calama 233 and Diego de Almagro.32 Pizza Net. Address: Calama 354 and J.L. Mera.
International calls.
29 Andinatel. Address: Calama 413 and J.L. Mera.
51
Hotels.
7 Swissotel.8 Radisson Hotel.
10 Hostal Santa Barbara.
Supermarket.
15 Supermaxi.
Money exchange.
28 Multicambios. Address: Colon 919 and Reina Victoria; Roca 720.
Museums.
33 Casa de la cultura. See the Toristic information section for otherinteresting choices. (outside the map)
52
53
10
78
129 54
1723
13 14
24
25
26
27
28
31
30
32
34
36
Fig
ure
1.
LAW
OC
neigh
borh
ood
54
1
7
8
6
3
11
2
15
1619
1812
20
21
22
29
35Fig
ure
2.
LAW
OC
neigh
borh
ood
55
