Lecture 5. Matheuristic Nested Stochastic Decompositionand Yn is a feasible set of mixed 0-1...

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Lecture 5. Matheuristic Nested StochasticDecomposition

Laureano F. EscuderoUniversidad Rey Juan Carlos (URJC), Mostoles (Madrid),

laureano.escudero@urjc.es

Laureano F. Escudero Universidad Rey Juan Carlos (URJC), Mostoles (Madrid), laureano.escudero@urjc.esNSD

Program. Course on Stochastic-Equilibrium in NetworkExpansion Planning under uncertainty, SE-NEP

Lecture 1. Introduction to Network Expansion Planning.Deterministic version.Lecture 2. Stochastic multistage NEP: Strategic scenariotrees, multi-period Tactical scenario graphs andOperational two-stage trees. Scenario reduction. RiskNeutral.Lecture 3. Stochastic-Equilibrium in Network ExpansionPlanning under Uncertainty, SE-NEP.Lecture 4. Risk averse policies in stochastic optimization.Lecture 5. Matheuristic Nested StochasticDecomposition.

Outline

Aim and ScopeMath Optim under uncertaintyMatheuristic Nested Stochastic DecompositionComputational experienceConclusions

Outline. Computational experience

SARM-RN. Stochastic Airline Revenue Mngmnt.LP model. State vars: continuous linking multistage.SPP-RN-P. Stochastic Production planning.MIP model. State vars: continuous linking two consecutivestages. Versions Serial and cooperative Parallelization.STSCP-TSD. Stochastic Tactical supply chain planningunder uncertainty w/ a time-inconsistent stochasticdominance risk averse measure.MIP model. State vars: continuous linking multistage.STSCP-ECSD. Stochastic Tactical Supply Chain Planningunder uncertainty w/ a time-consistent stochasticdominance risk averse measure.MIP model. State vars: continuous linking multistage.MS-BILEVEL. Stochastic Equilibrium-based TollAssignment Network Expansion Planning. M01QP model.State vars: 0-1 step linking two consecutive stages.

Stochastic Optimization. Some elements

A period of a given time horizon is a time unit where therealization of the uncertain parameters takes place.

A scenario is a realization of the uncertain parameters alongthe periods of a given time horizon.

A scenario group for a given period is the set of scenarios withthe same realization of the uncertain parameters up to theperiod:Non-anticipativity principle should be satisfied.

A scenario group has one-to-one correspondence with a nodein the scenario tree.

t = 1 t = 2 t = 3 t = 4 t = 5 t = 6 t = 7

T = 1, . . . , 7

Ω = Ω1 = 18, 19, . . . , 33

A6 = 6, 4, 3, 2, 1

t6 = 5

σ6 = 4

S6 = 10, 11, 18, 19, 20, 21

S61 = 10, 11

N = 1, . . . , 33

N6 = 10 . . . , 17

Ω7 = 22, . . . , 25

Figura: Multi-period scenario tree

Scenario tree notation

T , set of the time periods along the horizon, T = |T |.Ω, set of scenarios.N , set of nodes in the tree.

Ωn ⊆ Ω, set of scenarios in a group with one-to-onecorrespondence with node n, for n ∈ N .

Nt ⊂ N , set of nodes for stage t , for t ∈ T .An, set including node n and its ancestor nodes, for n ∈ N .Sn, set including its successor nodes to node n, for n ∈ N .

Note that Sn = ∅ for n ∈ NT .Sn

1 ⊆ Sn, immediate successor nodes to node n, for n ∈ N .σn, immediate ancestor node of node n, for n ∈ N . Note:

σn =null, for n ∈ N1.tn, period to which node n belong to, for n ∈ N .

wω, probability assigned to scenario ω ∈ Ω.wn =

∑ω∈Ωn wω, weight or probability of node n ∈ N .

Multiperiod Mixed 0-1 DEMfor Network Expansion Planning (NEP)

z = max∑n∈N

wn(anxn + bnyn)

s.t. xn ∈ 0,1nx(n), xσn ≤ xn ∀n ∈ N

Anxn + Bnyn = hn, yn ∈ Y n ∀n ∈ N ,

where xn is a vector included by the so-named 0-1 step vars,and Y n is a feasible set of mixed 0-1 vectors.

Let xn, yn: Value vectors of the vectors xn, yn, resp., in a (full orpartial) solution, if any, for n ∈ N .

Nested Stochastic Decomposition methodologoy.Some refs.

NSD versions for stage-independent uncertainties:SDDP for SLP, for solving large-sized LP instances,providing lower & upper bounds:Pereira & Pinto WRR’85, MP’91 for Risk Neutral, and manyothers.Philpott & de Matos EJOR’12; Philpott, de Matos & FinardiOR’13 with time-consistent and coherent measuresSDDiP for SMIP, for solving large-sized mixed 0-1instances with 0-1 state vars (finite convergence withprobability one).Zou, Ahmed & Sun MPB’18

Finite convergence with probability one NSD version forstage-dependent uncertainties:

NSD for MSMIP, with 0-1 state vars.Zou, Ahmed & Sun, OptimOnline’16.

Some refs. on NSD methodology (c)

Matheuristic NSD versions for mixed 0-1 state vars withstage-dependent uncertainties, for MSMIP:

Case w/ influential vars from immediate ancestor node:NSD-RN. Aldasoro, LFE, Merino, Monge & Perez TOP’14,among others. Continuous state vars.NSD-BILEVEL-NEP, for Stochastic Equilibrium in NetworkExpansion Planning. 0-1 state vars.LFE, Monge & Rodrigez-Chıa. Submitted, 2019.

Case w/ influential var from ancestor nodes:NSD-TSD, with Time in-consistent Stochastic Dominancefunctional.LFE, Monge & RomeroMorales COR’15NSD-ECSD, with Time consistent Expected ConditionalStochastic Dominance functional.LFE, Monge & RomeroMorales COR’18

NSD matheuristic, Nested Stochastic Decomposition

NSD starts by partitioning the set of time periods intomodeler-driven so-named stages each one is a subset ofconsecutive periods), creating thus a collection ofsubtrees/subproblems.The subproblems are not independent but each one islinked by the 0-1 step vars to the immediate ancestorsubproblem.NSD solves the subproblems at each iteration, byexecuting first a front-to-back scheme and after aback-to-from scheme.

Front-to-Back (FtB)

Back-to-Front (BtF)

t = 1 t = 2 t = 3 t = 4 t = 5 t = 6 t = 7

e = 1 e = 2 e = 3

First stage Intermediate stages Last stage

T = 1, . . . , 7,E = 1, 2, 3

Ω = Ω1 = 18, 19, . . . , 33

A6 = 6, 4, 3, 2, 1

t6 = 5

σ6 = 4

S6 = 10, 11, 18, 19, 20, 21

S61 = 10, 11

G2 = 4, 5, 6, 7, 8, 9

R2 = 4, 5

C4 = 4, 6, 7

L15 = 28, 29

Figura: Partitioning periods in stages

Scenario subtrees supporting the NSD subproblems.Sets

E , stages in the time horizon.Te, set of periods in stage e, such that

T =∑

e∈E Te, Te ∩ Te′ = ∅ : e 6= e′.Ge ⊆ N , set of nodes in stage e, for e ∈ E .Re ⊆ Ge, root nodes of the subtrees in stage e, for e ∈ E .Cr ⊆ Ge, nodes that belong to the subtree rooted with node r , for

r ∈ Re,e ∈ E .Lr ⊆ Cr , leaf nodes in subtree related to set Cr , for r ∈ Re, e ∈ E .

Note: The 0-1 step x`-vars are the state ones, i.e., they arethe only vars in a subproblem supported by the subtreerooted with node r that have nonzero elements inconstraints associated with the root nodes in theimmediate successor subproblems to leaf node `

Those subtrees are defined by the nodeset⋃

r ′∈S`′1Cr ′ , for

` ∈ Lr , r ∈ Re, e ∈ E : e < E .Laureano F. Escudero Universidad Rey Juan Carlos (URJC), Mostoles (Madrid), laureano.escudero@urjc.esNSD

T =∑

r ′∈S`′1Cr ′ , for

T =∑

r ′∈S`′1Cr ′ , for

T =∑

r ′∈S`′1Cr ′ , for

T =∑

r ′∈S`′1Cr ′ , for

T =∑

r ′∈S`′1Cr ′ , for

NSD matheuristic. EFV curves

Let the so-named Expected Future Value (EFV) curve ofthe subproblem supported by the subtree given by nodesetCr ′ , whose root node r ′ is an immediate successor of leafnode `, for r ′ ∈ S`1, ` ∈ Lr , r ∈ Re, e ∈ E : e < E , whereremember Lr ⊆ Cr .

EFV curves estimate the impact of the state vars(decisions) of the leaf nodes in the objective function ofsuccessor subproblems.

(.), current solution of the variables’ vector (.).

For easing the exposition, let a set of variables in the originalmodel (1) be notated as vector Z n, for n ∈ N . It can beexpressed

Z n ≡(yg ∀g ∈ An; xg ∀g ∈ An \ n

EFV curve λ′r ′(x `), forr ′ ∈ S`1, ` ∈ Lr , r ∈ Re, e ∈ E : e < E

λ′r′(x`), (assumed convex) curve of the expected future

objective function value (EFV) in the set of scenarios Ωr ′ ,due to the leaf node-related variable x`. It is difficult tocompute.NSD approach approximates λ′r

′(x`) with the piecewise

linear convex EFV function λr ′(x`).

xℓ1b

xℓ3b

λr′3(x) = µr′3 + πr′3x

λr′1 (x) = µr′1 + πr′1x

λr′2(x) = µr′2 + πr′2x

λr′

Objective function value F ′r of subproblem defined bynode set Cr , r ∈ Re, e ∈ E

F ′r = max∑`∈Lr

w `[∑

n∈A`

(anxn + bnyn) +∑

r ′∈S`1

w r ′

w `λ′r′(

s.t. Anxn + Bnyn = hn, yn ∈ Y n ∀n ∈ Cr

xn ∈ 0,1nx , xσn ≤ xn ∀n ∈ Cr

Zσr= Zσr

xσr − xσ

Elements definition of EFV function λr ′, forr ′ ∈ S`1, ` ∈ Lr , r ∈ Re, e ∈ E : e < E

Let Q` denote the set of indices q, such that the referencelevels indexed with q define the EFV function λr ′ .Each qth reference level is composed by vector(

x`q; µr ′q , πr ′q ∀r ′ ∈ S`1

where vector x`q

is the value of the vars in vector x`q, for

q ∈ Q`, retrieved from the subproblem related to scenario noder , andconstant µr ′q and vector πr ′q define a linear function of vector x`

obtained from the solution of the subproblem related to theimmediate successor node r ′ of scenario leaf node `, i.e.,q ∈ Q`, r ′ ∈ S`1, ` ∈ Lr .

λr ′ = minq∈Q`

µr ′q + πr ′q x`

EFV function λr ′, forr ′ ∈ S`1, ` ∈ Lr , r ∈ Re, e ∈ E : e < E

λr ′ = minq∈Q`

µr ′q + πr ′q x`

It is required that it is a valid cut for EFV curve λ′r′(·) as well as

a tight one, such that λ′r′(x`

q) = µr ′q + πr ′q x`

Objective function value F r of subproblem defined bynode set Cr , r ∈ Re, e ∈ E : e < E , where EFV curveλ′r(x `) is replaced with EFV (piecewise linear) functionλr ′(x `), for r ′ ∈ S`1, ` ∈ Lr

F r = max∑`∈Lr

w `[∑

n∈A`

(anxn + bnyn) +∑

r ′∈S`1

w r ′

w `λr ′] (4a)

s.t. Anxn + Bnyn = hn, yn ∈ Y n ∀n ∈ Cr (4b)xn ∈ 0,1nx , xσ

n ≤ xn ∀n ∈ Cr (4c)

Zσr= Zσr q

x` − xσr q

= 0 (4e)λr ′ ≤ µr ′q + πr ′q x` ∀q ∈ Q`, r ′ ∈ S`1, ` ∈ Lr . (4f)

xℓ1b

xℓ3b

λr′3(x) = µr′3 + πr′3x

λr′1 (x) = µr′1 + πr′1x

λr′2(x) = µr′2 + πr′2x

λr′

NSD approach. Front-to-Back scheme from stagee = 1 to last stage e = E

Subproblems (4) from stage 1 to last one E are solved bypassing the values of linking x-vars onto the subproblemsin the next stage.

λ-values are zero in iteration q = 1.

In case of infeasibility of subproblem (4) for r ∈ Re, e ∈ E :merging scheme of stages e-1 and e.

Building a feas solution for original model (1)(xn, yn ∀n ∈ N ).

Obj fun value F =∑

n∈N wn(anxn + bnyn)

In that case, a testing is performed to analyze whether itimproves the incumbent solution.

Front-to-Back (FtB)

Back-to-Front (BtF)

t = 1 t = 2 t = 3 t = 4 t = 5 t = 6 t = 7

e = 1 e = 2 e = 3

First stage Intermediate stages Last stage

T = 1, . . . , 7,E = 1, 2, 3

Ω = Ω1 = 18, 19, . . . , 33

A6 = 6, 4, 3, 2, 1

t6 = 5

σ6 = 4

S6 = 10, 11, 18, 19, 20, 21

S61 = 10, 11

G2 = 4, 5, 6, 7, 8, 9

R2 = 4, 5

C4 = 4, 6, 7

L15 = 28, 29

Figura: Partitioning periods in stages

NSD approach. Front-to-Back scheme from stagee = 1 to last stage e = E (c)

Upper bound of the solution value of the original model (1):solution value F r of model (4) for r = 0 (i.e., e = 1).

OG: Optimality gap of a given feasible solution value F ofthe original model (1) with respect to F 0 at iteration, say q.

NSD approach. Stopping criteria

1 OG is not greater than a given tolerance, say ε12 The relative change in the upper bounds bound F 0 and

the relative change in the obj fun value F ,both between the last two consecutive iterations q − 1 andq are not greater than a given tolerance, say ε2.

3 An upper bound on the number of iterations is reached,say q.

4 Computing time limit is reached.

NSD approach. Back-to-Front scheme from last stagee = E to stage e = 2

It aims to refining the EFV functions λr ′ in the subproblemssupported by the subtrees rooted with the nodeset r ′ inRe, e ∈ E \ 1 around the feasible soln (Z n, xn ∀n ∈ NT )built in the front-to-back scheme at the current iteration q.

The subproblems in a given stage are solved, passing therefinement of the EFV functions onto the subproblems inthe previous stage.

NSD approach. On EFV function λr ′, forr ′ ∈ S`1, ` ∈ Lr , r ∈ Re, e ∈ E : e < E

Generating and appending a new set S`1 of linear functionsλr ′q

λr ′ ≤ µr ′q + πr ′q x` (5)

to the collection (4f):

λr ′ ≤ µr ′q + πr ′q x` ∀q ∈ Q` \ q, r ′ ∈ S`1, ` ∈ Lr

in model (4) related to the solution x`q, retrieved from the

front-to-back scheme, where q is the last iteration.

On obtaining vector (µr ′q , πr ′q)

It is worth to pointing out that some state-of-the-art optimizers(as CPLEX and others) do not provide the duals of theconstraints for fixed vars in MIP models.So, some alternatives:

Benders cut, B (Benders NM’62).

Lagrangean cut, L(Zou, Ahmed & Sun OptimOnline’16, MPB’18).

Strengthened Benders cut, SB(Zou, Ahmed & Sun OptimOnline’16, MPB’18).

Heuristic Benders cut, HB.

A mixture of SB cuts for later sages and L cuts for earlierstages, in particular, stage e = 1. Work-in-Progress.

Approximation schemes for EFV curve λ′r ′(x `): L-,SB-, B- and HB-based cuts for obtaining EFV functionλr ′(x `)

L: Valid and tight

SB: Valid and no-guarantee it is tight.

B: Valid and ’almost’ sure it is not tight.

HB: No guarantee it is valid, and sure it is tighter than B.

That is, there it not any guarantee that the EFV functionλr ′(x`) does not cut any feasible solution of the originalmodel (1).

So, no guarantee that the NSD incumbent solution is theoptimal one for, nor an upper bound is obtained.

HB-based cut for approximating EFV curve λ′r ′(x `) viaEFV function λr ′(x `)

No guarantee that the EFV function λr ′(x`) does not cutany feasible solution of the original model (1).

So, no guarantee that the NSD incumbent solution is theoptimal one for, nor an upper bound is obtained.

xℓ1b

xℓ3b

λr′3(x) = µr′3 + πr′3x

λr′1 (x) = µr′1 + πr′1x

λr′2(x) = µr′2 + πr′2x

λr′

On obtaining (µr ′q , πr ′q). Benders cut

Vector πr ′q for r ′ ∈ S`1: Duals, say, πr ′qLP related to constraint

(4e) xσ` − x`

q= 0 in the soln of the LP relaxation of

subproblem (4) indexed with r ′ in the back-to-front scheme,

where the current soln x`q

is retrieved from thefront-to-back scheme.

Constant µr ′q is computed as F r ′qLP − π

r ′qLP x`

q, by considering

the first order of Taylor expansion around the objectivefunction value F r ′q

Notice that the scheme gives a valid cut but it tightnesscannot be guaranteed.

On obtaining (µr ′q , πr ′q). HB cut

Benders cut could be weak enough for problems where theLP feasible set is not too close to the MIP one.

In that case, some MIP-based scheme is required.

HB cut uses the LP dual vector πr ′qLP as in the Benders

scheme,

but constant µr ′q is computed as F r ′qH − π

r ′qLP x`

where F r ′qH is the objective function value of subproblem (6)

for r ′ ∈ Re, e ∈ E \ 1.

Since F r ′qLP ≥ F r ′q

H , then µr ′q in B is higher than in HB.

On obtaining (µr ′q , πr ′q). HB cut (c)

Let subproblem (4) supported by the subtree with nodeset Cr ,being now rooted with node r ′, for r ′ ∈ Re, e ∈ E \ 1, whosesolution value is now denoted as F r ′q

F r ′qH = max

∑`′∈Lr ′

w `′[ ∑

n∈A`′

(anxn + bnyn) +∑

r ′′∈S`′1

w r ′′

w `′λr ′′]

s.t. Anxσn

+ Bnxn + Cnyn = hn, yn ∈ Y n ∀n ∈ Cr ′

xn ∈ 0,1nx , xσn ≤ xn ∀n ∈ Cr ′

Z ` = Z `q

x` − x`q

= 0λr ′′ ≤ µr ′′q + πr ′′q x`

′ ∀q ∈ Q`′ , r ′′ ∈ S`′1 , `′ ∈ Lr ′ .(6)

On obtaining (µr ′q , πr ′q). HB cut warning

There it not any guarantee that the estimation λr ′ (5) doesnot cut any feasible solution of model (3) and, then, anyfeas soln in the original one (1).

So, no guarantee that the NSD incumbent solution is theoptimal one for model (1), nor the solution value F 0 is anupper bound.

xℓ1b

xℓ3b

λr′3(x) = µr′3 + πr′3x

λr′1 (x) = µr′1 + πr′1x

λr′2(x) = µr′2 + πr′2x

λr′

On obtaining (µr ′q , πr ′q). L cut

The relaxation of subproblem (6) that is indexed with r ′ in theback-to-front scheme at iteration q consists of the Lagrangeandual problem to be expressed

Lr ′q = minπr ′q F r ′q

where F r ′qL is the solution value of the Lagrangean problem to

be expressed in next slide.

On obtaining (µr ′q , πr ′q). L cut. Lagrangean problem

F r ′qL = max

∑`′∈Lr ′

w `′[ ∑

n∈A`′

(anxn + bnyn) +∑

r ′′∈S`′1

w r ′′

w `′λr ′′

−πr ′q (x` − x`q)]

s.t. Anxσn

+ Bnxn + Cnyn = hn, yn ∈ Y n ∀n ∈ Cr ′

xn ∈ 0,1nx , xσn ≤ xn ∀n ∈ Cr ′

Z ` = Z `q

x` ∈ [0,1]

λr ′′ ≤ µr ′′q + πr ′′q x`′ ∀q ∈ Q`′ , r ′′ ∈ S`′1 , `′ ∈ Lr ′ .

(8)Note: It is well known that the Lagrangean dual soln is the optlone of model (6).

On obtaining (µr ′q , πr ′q). L cut (c)

So, the λr ′q -cut (5) is given by µr ′q := F r ′qL and πr ′q .

It is proved in Zou, Ahmed & Sun OptimOnline’16 thatunder mild conditions the λ-cut is valid and tight in thesense stated in the work.

In this case, µr ′q + πr ′q x`q

is the point in the EFV curveλ′r′(x`) for reference level q.

It is also proved that the collection of those cuts guaranteewith probability one that an optimal soln is obtained in afinite number of iterations to a multistage mixed integerstochastic problem with only 0-1 state vars.

On obtaining (µr ′q , πr ′q). L cut (c)

Unfortunately, the time required to solving the dual problem(7) in the back-to-front scheme for the subproblems in eachstage at any NSD iteration could render the schemeunaffordable for solving large-sized problems.

In any case, it paves the way for the schemes SB andso-named truncated Lagrangean.

On obtaining (µr ′q , πr ′q). SB cut

Notice that the Lagrangean problem (8) yields a valid cutλr ′q (5) for any vector πr ′q , where µr ′q := F r ′q

Vector πr ′q could coming from the Benders cut (i.e., it is thedual vector πr ′q

LP related to constraint x` − x`q

= 0 in thesolution of the LP relaxation of subproblem (6), and

Constant µr ′q := F r ′qL for πr ′q := πr ′q

SB cut is valid and finite, but no guarantee that it is a tightone.In any case, it is an affordable one.

Difference between the B, HB, SB cuts: Constant µr ′q .

PROCEDURE NSDStep 0: (Input)

Set of stages E , set of periods Te ∀e ∈ E , maximumnumber of NSD iterations q, multiple of NSD iterationsaddfq, tolerances ε1, ε2, cut version V = ′L′,′ SB′,′HB′,and maximum number of Lagrangean iterations q andtolerance ε3, for cut version V =′ L′.

Step 1: (Initialization)Set F ∗ := −∞, q := 0.

Set Q` := ∅ ∀` ∈ Lr , r ∈ Re,e ∈ E .

Set e := 1.

Step 2: (Front-to-Back scheme: Solving subproblem (4)supported by the subtree rooted with node r ∀r ∈ Re)

If q > 0 and 0 = q mod addfq then: Remove anynon-active EFV function λr ′q defining constraints (4f) forq ∈ Q` in subproblem (4) and, consequently, update setQ` := Q` \ q, for ` ∈ Lr .Update q := q + 1 and Q` := Q` ∪ q,Obtaining solution value F r and the related vector(Z `q , x`

q ∀` ∈ Lr ).Step 3: (Forward to next stage e + 1)

If e < E then reset e := e + 1 and go to Step 2.

Step 4: (End Front-to-Back scheme. Updating incumbentsolution value of original model (1) and testing stoppingcriteria)

F q :=∑

n∈N wn(anxnq+ bnynq

If q = 1 or F q > F ∗ then F ∗ := F q and(x∗, y∗) := (xnq

, ynq ∀n ∈ N )

If q > 1 then:

If q = q or |Fq−F∗||F∗| ≤ ε1 or

( |Fq−F

q−1||F q−1|

≤ ε2 and |Fq−F q−1||F q−1| ≤ ε2 ) then STOP,

where Fq

is the upper bound of the original model (1)given by the solution value of subproblem (4) in scenarionode r = 0, say F 0, at the current iteration q.

Step 5: (Back-to-Front scheme: Computing the elements ofEFV function λr ′q for node r ′, namely, vector πr ′q andconstant µr ′q ∀r ′ ∈ Re). Note: e > 1.

Solving the LP relaxation of subproblem (6), by consideringvector x`

qin constraint x` − x`

q= 0, where ` = σr ′ .

Output: Dual vector πr ′qLP of that constraint.

Cut version V =′ L′ (Lagrangean):Set q’:=0.

1 Solving the MIP Lagrangean problem (8).Output: solution value F r ′q

L .Note: πr ′q := πr ′q

LP for the first Lagrangean iteration.2 If dual problem (7) is solved or q′ = q then set µr ′q := F r ′q

L ;else update dual vector πr ′q and q′ := q′ + 1, and go to 1.

Step 5: (Back-to-Front scheme: Computing the elements ofEFV function λr ′q for node r ′, namely, vector πr ′q andconstant µr ′q ∀r ′ ∈ Re) (c). Note: e > 1.

Cut version V =′ SB′ (Strengthened Benders):

Set πr ′q := πr ′qLP .

Solving the MIP Lagrangan problem (8).Output: Solution value F r ′q

Note: The solution of the LP relaxation of subproblem (6)can be considered as a warm start for the MIP one.Set constant µr ′q := F r ′q

Step 5: (Back-to-Front scheme: Computing the elements ofEFV function λr ′q for node r ′, namely, vector πr ′q andconstant µr ′q ∀r ′ ∈ Re) (c). Note: e > 1.

Cut version V =′ HB′ (Heuristic Benders):

Set πr ′q := πr ′qLP .

Solving the MIP subproblem (6).Output: Solution value F r ′q

Note: Its LP solution above could be considered as a warmstart for the MIP one.Set µr ′q := F r ′q

H − πr ′q x`q.

Step 6: (Back-to-Front scheme: Appending the EFVfunction λr ′q ∀r ′ ∈ S`1, ` ∈ Lr defining constraints (5) tosubproblem (4) supported in the back-to-front scheme bythe subtree rooted with node r , ∀r ∈ Re−1)

Appending λr ′q -constraints (5) to system (4f) in thesubproblem.Reset e := e − 1; If e > 1 then go to Step 5;else go to Step 2.

Computational experience

SARM-RN. Stochastic Airline Revenue Mngmnt.LP model. State vars: continuous linking multistage.SPP-RN-P. Stochastic Production planning.MIP model. State vars: continuous linking two consecutivestages. Versions Serial and cooperative Parallelization.STSCP-TSD. Stochastic Tactical supply chain planningunder uncertainty w/ a time-inconsistent stochasticdominance risk averse measure.MIP model. State vars: continuous linking multistage.STSCP-ECSD. Stochastic Tactical Supply Chain Planningunder uncertainty w/ a time-consistent stochasticdominance risk averse measure.MIP model. State vars: continuous linking multistage.MS-BILEVEL. Stochastic Equilibrium-based TollAssignment Network Expansion Planning. M01QP model.State vars: 0-1 step linking two consecutive stages.

Multistage Mixed 0-1 DEM: Risk neutral. Compact representation

Max objective function expected value over scenarios

zRN = max∑n∈N

wn(anxn + bnyn)

s.t.∑

n′∈An

(An′n xn + Bn′

n ) = hn ∀n ∈ N

xn ∈ 0,1nx(n), yn ∈ R+ny(n) ∀n ∈ N .

Computational results.SARM-RN. Stochastic Airline Revenue Management

Risk neutral.PC, 2.33hz, Intel Xeon dual core processor, 8.5Gb RAM,Linux Debian 4.0.LP model. State vars: continuous multistage.Plain CPLEX v9 failed to find a feas sol in several hours.Metaheuristic NSD.|Ω|=6,561 scenarios, m=2,296,300 cons, nc=2,624,400continuous vars.Elapsed time = 71 secs.Optimality gap OG = 1,22 %.Ref. LFE, Monge, Romero-Morales, Wang TS’13.

Computational results.SPP-RN-P. Stochastic Production planning

Serial, Inner, Outer, Outer-Inner asynchronous parallelmetaheuristic NSD.Risk neutral. MIP model. State vars: continuous twoconsecutive stages.T =16 periods, Randomly generated Ω=7,766 scenarios.P86 (P85): m=5.56 (57.8) M cons, n01=1.41 (15.04) M 0-1vars, nc=3.49 (38.5) M continuous vars.P86 (P85): nprob=28,997 (5,177) MIP subproblems,elapsed time=978 (26,180) secs.NSD: incumbent sol value with GG %=0.16 (.).Effciency=61.64 (89.15) %.Plain CPLEX: running out of memory (35Gb) after 8,274secs, sol value with optimality gap OG %=0.78 (.).Ref. Aldasoro, LFE, Merino, Monge, Perez TOP’14.

SPP-RN-P. Stochastic Production Planning.Computing HW/SW

MPI: Message Passing Interface.Big computing cluster, SGI/IZO-SGIker at UPV/EHU, used16 xeon cores (8 or 12 treads each), 48 Gb each.CPLEX v12.5 for solving independent scenario cluster MIPsubproblems

Computational results.STSCP-TSD. Stochastic Tactical Supply ChainPlanning w/ time-inconsistent SD risk averse

PC with a 2.5 GHz dual-core Intel Core i5 processor, 8 Gbof RAM and the operating system was OS X 10.9.Metaheuristic S-NSD. MIP model. State vars: continuousmultistage.P3 (P12): T =7 (10) periods, E=3 (3) stages, Ω=64 (512)scenarios.P3 (P12): m=7,827 (212,544) cons, n01=1,408 (36,864)0-1 vars, nc=4,653 (124,596) continuous vars.P3 (P12): nprob=544 (1,258) MIP subproblems, elapsedtime=610 (6,540) secs, GG %=1.82 (.) optimality gapversus plain use CPLEX v.12.5,CPLEX v12.5: P3 (P12): Elapsed time 17,480 secs (> 8 h,no sol found).Ref. LFE, Monge, Romero-Morales COR’15.

Computational results.STSCP-ECSD. Stochastic Tactical Supply ChainPlanning w/ time-consistent SD risk averse

MIP model. State vars: continuous multistage.

10 end-products, 20 subassemblies, 30 markets, 40 rawmaterials, 25 resources, 7 linear piecewise segments forraw material supplying cost,T=9 periods, 511 scenario nodes, 255 scenarios, breakstage t∗ = 4, Nt∗+1 = 16 scen groups, 2 ECSD profiles pergroup, 3 ECSD bounds per profileDimensions: m=198,584, nc=144,488, n01=36,307,S2=280Time unaffordable for CPLEX use.NSD elapsed time: Up to 7 hours.

LFE, Monge, Romero-Morales COR’18

STSCP-ECSD. Computing HW/SW and main results

Intel i5 processor, 2.5 GHz dual core, RAM 8Gb, OS X 10.9Matheuristic algo NSD-ECSD

C++ experimental code.Stopping tolerance ε = 0,01 % for the relative differencebetween the current feasible solution and the incumbentoneMaximum number of iterations miter = 200

CPLEX v12.5.1, default values but quasi-optimalitytolerance: 0.001

MS-BILEVEL. Stochastic Equilibrium-based TollAssignment Network Expansion Planning, TAP

On pricing-equilibrium for network expansion planning. Amulti-period bilevel approach under uncertainty.

M01QP model.State vars: 0-1 linking two consecutive stages.

LFE, Monge, Rodrguez-Chıa Sub’19.

Multiperiod Mixed 0-1 DEMfor Network Expansion Planning (NEP)

z = max∑n∈N

wn(anxn + bnyn)

s.t. xn ∈ 0,1nx(n), xσn ≤ xn ∀n ∈ N

Anxσn

+ Bnxn + Cnyn = hn, yn ∈ Y n ∀n ∈ N ,

where xn is a vector included by the so-named 0-1 step vars,and Y n is a feasible set of mixed 0-1 vectors.

Let xn, yn: Value vectors of the vectors xn, yn, resp., in a (full orpartial) solution, if any, for n ∈ N .

Computing HW/SW

A processor Intel(R) Xeon(R) CPU E5-2650 v3 @2.30GHz, 20 cores, RAM 62 GiB.

A C++ experimental code.

CPLEX v12.8, using between 15 and 20 cores.

NSD parameters: ε1 = 0,0001, ε2 = 0,0001,iters limit q = 100,Lagrangean iters limit q = 4.

Given the difficulty of the instances, E = T and, so,|Te| = 1, Te = t r, |Cr | = 1, for r ∈ Re, e ∈ E ,

t r , period which scenario node r belongs to.

64 instances

|N|, |A1| and |A2|, number of nodes, network links andlinks by other means, resp., for the transport network.

|N|, number of scenario nodes; |Ω|, number of scenarios(i.e., |NT |).Small-middle instances i1-i32: Up to |N| = 15, |A1| = 8and |A2| = 18, |T | = 5, |N | = 31, |Ω| = 16.m = 30,442; nc = 6,975; n01 = 12,524.

Large instances i33-i64: Up to |N| = 20, |A1| = 12 and|A2| = 25, |T | = 7, |N | = 156, |Ω| = 125.m = 381,108; nc = 84,396; n01 = 156,624.

Results. CPLEX plain use and NSD in in L4, SB, HB

of i and ti , incumbent soln value for original model (1) andcomputing time (secs) for i = 1,3, wherei = 1 (CPLEX plain use w/ computing time as NSD;i = 2 (CPLEX plain use w/ 2 hours computing);i = 3 (NSD).

of 1, CPLEX upper bound of opt sol, in L4, SB.

OG1 %, CPLEX optimality gap, 100.of 1−of 1of 1

, in L4, SB.

of 3, NSD estimation of upper bound of opt sol in L4, SB.

OG3 %, NSD estimation of the optimality gap, as

100. of 3−of 3of 3

in HB.

GR1, goodness ratio of NSD over CPLEX plain use, as of 3of 1

Cuadro: Average results for small-middle instances i1-i32

Cut option t3 q GR1

Lagrangean 1572 10.4 1.00Strengthened Benders 425 7.9 1.01Heuristic Benders 328 8.2 1.04

Heuristic Benders cut Lagrangean cut Strengthened Benders cut

Figura: Instances i1-i32. NSD optimality gap OG3 % in the cut optionsL4 and SB. Only estimation for HB

Cuadro: Average results for large instances i33-i64)

Cut option t3 q GR1 GR2

Lagrangean 54106 22.4 1.094 1.172Strengthened Benders 11313 9.5 1.174 1.164Heuristic Benders 9256 15.4 1.240 1.159

Heuristic Benders cut Lagrangean cut Strengthened Benders cut0

Figura: Instances i33-i64. NSD optimality gap OG3 % in the cutoptions L4 and SB. Only estimation for HB. Instance i63 was removed

Except for the outliers, the optimality gap OG3 is smaller inL4 than SB, what is not a surprise.

OG3 for HB: A good estimation of OG3

Conclusions. NEP, multi-period Network ExpansionPlanning

Strong step 0-1 vars type(i.e., by occurring in scenario node n):xn ∈ 0,1 where xσ

n ≤ xn for n ∈ N versusimpulse 0-1 ones (i.e., at occurring in scenario node n):xn ∈ 0,1 where

∑n∈Aω

xn ≤ 1 for ω ∈ Ω.

Decomposition methodology, a must for problem solving inlarge-sized instances,due to dimensions of problem network and scenario tree.

Conclusions. NSD, Nested Stochastic Decomposition

NSD: A good tool for helping to solve large-sized verydifficult problems,versus CPLEX, a M01LP/M01QP state-of-the-art solver.

Better for state variables only in consecutive two periods .

NSD performance improving: only step 0-1 vars.

Matheuristic HB type of cuts withw/ a good guarantee estimation of soln optimality gap anda usually high goodness ratio versus CPLEX plain use withmuch less elapsed time.

Conclusions. Future research work

A joint matheuristic algo: NSD plusan `2-norm Regularized scenario Clustering SimplicialDecomposition Progressive algorithm (RCSDPA), see LFE,Garın, Monge & Unzueta Sub’19, for solving theF ′r -related subproblem (4).

REFERENCESJ. Benders. Partitioning procedures for solving mixed variables programmingproblems. Numerische Mathematik 4:238-252, 1962.U. Aldasoro, L.F. Escudero, M. Merino, J.F. Monge and G. Perez. Onparallelization of a Stochastic Dynamic Programming algorithm for solvinglarge-scale mixed 0-1 problems under uncertainty. TOP, 23:703-742, 2015.M.P. Cristobal, L.F. Escudero and J.F. Monge. On Stochastic DynamicProgramming for solving large-scale tactical production planning problems.Computers & Operations Research 36:2418-2428, 2009.L.F. Escudero, M.A. Garın, J.F. Monge and A. Unzueta. On multistage stochasticmixed 0-1 bilinear optimization based on endogenous uncertainty and timeconsistent stochastic dominance risk management. Submitted 2019.L.F. Escudero and J.F. Monge. On capacity expansion planning under strategicand operational uncertainties based on stochastic dominance risk aversemanagement. Computational Management Science, 15:479-500, 2018.L.F. Escudero, J.F. Monge, A.M. Rodrıguez-Chıa. On pricing-based equilibriumfor network expansion planning. A multi-period bilevel approach underuncertainty. Submitted 2019.L.F. Escudero, J.F. Monge, D. Romero Morales and J. Wang. Expected FutureValue Decomposition Based Bid Price Generation for Large-Scale NetworkRevenue Management. Transportation Science, 47:181-197, 2013.L.F. Escudero, J.F. Monge and D. Romero-Morales, An SDP approach formultiperiod mixed 0–1 linear programming models with stochastic dominanceconstraints for risk management. Computers & Operations Research, 58:32-40,2015.

L.F. Escudero, J.F. Monge and D. Romero-Morales. On the time-consistentstochastic dominance risk averse measure for tactical supply chain planningunder uncertainty. Computers & Operations Research, 100:270-286, 2018.

J. Zou, S. Ahmed and X.A. Sun. Stochastic Dual Dynamic intger Programming.Optimization Online, 2016.

J. Zou, S. Ahmed and X.A. Sun. Multistage stochastic unit commitment usingStochastic Dual Dynamic integer Programming. Mathematical Programming,doi.org/10.1007/s10107-018-1249-5, 2018.

Lecture 5. Matheuristic Nested Stochastic Decompositionand Yn is a feasible set of mixed 0-1...

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