4GC10 Third year design project L.F.P. Etman - Materials …piet/edu/ogo1415/pdf/optimslides.pdf ·...

Problem formulation Visualization & Terminology Formulation classes Optimization algorithms Practicalities

Introduction to optimization4GC10 Third year design project

L.F.P. Etman

Department of Mechanical EngineeringEindhoven University of Technology

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Outline

1 Problem formulation

2 Visualization & Terminology

3 Problem formulation: classes and properties

4 Optimization algorithms

5 Practicalities

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Design optimization

Design optimization is the selection of the best designwithin the available means

[Papalambros & Wilde 2000: Principles of optimal design]

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Design optimization

1 Parametrize your design in terms of design variables, that is,define which design parameters you wish to optimize

2 Define objective function in terms of design variables(to minimize or maximize)

3 Define constraint functions in terms of design variables,which must be satisfied

4 Determine design variable values which minimize (maximize)the objective while satisfying all constraints


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Mathematical problem formulationMinimize

xf (x) x = (column) vector of design variables

subject to hj (x) = 0 j = 1; : : : ;mh

gk (x) ≤ ck k = 1; : : : ;mg

x ∈ X ⊆ Rn

In words:

• Find the values for the design variables x such that the objectivecriterion f (x) is minimized while satisfying equality constraintshj (x) = 0; j = 1; : : : ;mh and inequality constraintsgk (x) ≤ ck ; k = 1; : : : ;mg .

Negative null form:

• Reformulate gk (x) ≤ ck into gk (x) ≤ 0

Maximization:

• Minimizex

− f (x)


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Two-bar truss example

F

SS

h

d

Minimizex

f (x)

subject to g(x) ≤ 0x ∈ R+n

Constant parameter:

• width S

Design variables:

• height h

• diameter bars d

Objective criterion:

• mass

Inequality constraints:

• maximum height

• limit on stresses in bars

• limit on buckling forces in bars

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Two-bar truss example

minx

f (x) =��

2�2 x22

qS 2 + x2

1

s.t. g1(x) =1

Hmaxx1 − 1 ≤ 0

g2(x) =2F�2

��y

pS 2 + x2

1

x22 x1

− 1 ≤ 0

g3(x) =32F�4

�3E

�S 2 + x2

1

�3=2

x42 x1

− 1 ≤ 0

� : x1; x2 > 0;

• x1 = h is height, x2 = �d is scaled diameter (such that both designvariables operate in same order of magnitude).

• Height constraint g1, stress constraint g2 and bucking constraintg3, are analytically expressed in terms of design variables.

• Constraints are scaled (divided by the bound value).

• � is material density, E is modulus of elasticity, �y is yield strength.

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Visualization of a one-variable optimization problem

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x

f

f(x)

x

g

g3(x)

0

g6(x)

• Suppose, we have an optimizationproblem in one variable x:

• objective function f (x) is(quadratically) increasing in x,

• one constraint function g6(x) is(linearly) increasing in x,

• another constraint function g3(x)is (linearly) decreasing in x.

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Visualization of a one-variable optimization problem

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x

f

f(x)

x

g

g3(x)

0

g6(x)

g3 = 0

g6 = 0

• There is a certain value of x abovewhich g6(x) > 0 (bottom picture)

• Similarly, there is a value of xbelow which g3(x) > 0.

• Feasible values of x satisfy bothg3(x) ≤ 0 and g6(x) ≤ 0 (greenarea).

• Constraint g3 imposes a lowerbound on x,

• Constraint g6 imposes an upperbound on x.

• The optimum is attained for the xvalue that yields the lowestobjective function value whilesatisfying all constraints (redcircle)

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Constrained versus unconstrained optimumi

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x

f f(x)

g = 0

• constrained optimum

• bounded optimum

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x

ff(x)

g2 = 0g1 = 0

• unconstrained optimum

• interior optimum

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Multimodality (multiple local minima)

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x

f

f(x)

global

local

x

ff(x)

globallocal

• An optimization problem (unconstrained or constrained) may havemultiple minima.

• The minimizer which has the lowest objective function value (whilebeing feasible) is referred as the global minimizer.

• The other minimizers are referred to as local.

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Visualization of a two-variable optimization problem

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x1

x2

objective function

6 78

9 10

f

x1

x2

constraint g1

g1= 0

g1< 0

feasible

• Consider an optimization problem in two design variables x1 and x2.

• The objective function is a 3-D surface, which may be visualizedusing contour lines of equal objective function value.

• Similarly, each constraint function is also represented by a surface.

• The contour line for which the contraint bound is reached(g(x) = 0) represents the boundary between the feasible(g(x) ≤ 0) and infeasible domain (g(x) > 0).

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x1

x2

g1

• The contour lines of the objective function and the contour linesrepresenting the contraint boundaries are usually plotted in asingle picture.

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x1

x2

g1

g4

g5

g2 g4 g6

• If there is more than one constraint, for each constraint theconstraint boundary line is included in the picture.

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x1

x2

g1

g4

g5

g2 g4 g6

feasible

domain

F

• The feasible domain represents the design area where allconstraints are satisfied.

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x1

x2

g1

g4

g5

g2 g4 g6

optimum

F

• The optimum is located at the (x1; x2) location in the feasibledomain where the objective function attains the lowest value.

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Constraint activity

A constraint gj (x) ≤ 0 is active at the optimum implies that

1 if gj (x) ≤ 0 is left out, the location of the optimum changes

2 the constraint is satisfied with strict equality: gj (x) = 0

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Formulation classes: linear programming

Linear programming (LP) problem:

• Objective function linear in design variables

• Constraint functions linear in design variables

• Continuous design variables

Example:

minx∈R2

f (x) = −2x1 − x2

s.t. g1(x) = x1 + 2x2 − 8 ≤ 0g2(x) = 2x1 − 2x2 − 3 ≤ 0g3(x) = −2x1 + 1 ≤ 0g4(x) = −2x2 + 1 ≤ 0

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x1

x2

F

g3

g4

g1

g2

f

If (some) design variables can only take discrete values:

• (Mixed-)integer linear programming problem: (M)ILP

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Formulation classes: quadratic programming

Quadratic programming (QP) problem:

• Quadratic objective function

• Linear constraint functions


Example:

minx∈R2

f (x) = 3x21 − 2x1 + 5x2

2 + 30x2

s.t. g1(x) = −2x1 − 3x2 + 8 ≤ 0g2(x) = 3x1 + 2x2 − 15 ≤ 0g3(x) = x2 − 5 ≤ 0

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x1

x2

g3

g1

f

F g2

(Some) design variables discrete:

• (Mixed-)integer quadratic programming problem: (M)IQP

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Formulation classes: nonlinear programming

Nonlinear programming (NLP) problem:

• Non-linear objective function

• Non-linear constraint functions


Example:

minx∈R2

f (x) = 3x1 +√

3x2

s.t. g1(x) = 18x1

+ 6√

3x2− 3 ≤ 0

g2(x) = 5:73− x1 ≤ 0g3(x) = 7:17− x2 ≤ 0

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x1

x2 F

g3

g1

g2

f

(Some) design variables discrete:

• (Mixed-)integer nonlinear programming problem: (M)INLP

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Formulation properties

Design variables:

• Domain: {0;1}; N ; N+; Z; R; R+

Objective and constraint functions:

• Linearity: linear or nonlinear

• Continuity: none/once/twice-differentiable

Optimization problem:

• Objective: single objective or multi-objective

• Constraints: unconstrained or constrained

• Modality: uni-modal or multi-modal

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Optimization in mechanical engineering

Design variables:

• sizing (dimensions)

• shape (geometry of boundary)

• topological (material distribution)

Objective criterion:

• profit (cost, efficiency, weight, ...)

Constraints:

• geometrical (width, length, height, ...)

• structural (stresses, displacements, ...)

• dynamical (accelerations, eigenfrequency, ...)

• physical (temperatures, pressures, ...)

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Analytical versus numerical solution

Some optimization problems can be analytically solved:

Minx

f (x) = 2x1 + x−21 + 2x2 + x−2

2

Many optimization problems can only be numerically solved:

Minx

f (x) = x1 + x2 + x1e−x2 + x22 e−x1

An optimization algorithm is an iterative procedure to solve anunconstrained or constrained minimization (maximization)problem.

Choose an optimization algorithm in accordance with theoptimization problem class and properties


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Classification of optimization algorithms

Four classifiers of an optimization algorithm:

• unconstrained / constrained search

• local / global search

• determistic / stochastic search

• 0th / 1th / 2nd -order search

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Unconstrained / constrained search

Unconstrained search:

• aims to minimize a nonlinear, possibly multi-modal, objectivefunction in n-dimensional space

Constrained search:

• aims to minimize a (non)linear objective function inn-dimensional space while accounting for (non)linear equalityand/or inequality constraint functions

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Local / global search

Local search:

• seeks improvement based on the Taylor series expansion;

in a sufficiently small region around x∗ = [x1; x2; : : : ; xn ]∗ a

nonlinear function can be approximated by a quadratic Taylorseries:

~f (x) = f (x∗)+nX

i=1

@f (x∗)@xi

(xi−x∗i )+

12

nXi=1

nXj=1

@2f (x∗)@xi@xj

(xi−x∗i )(xj−x∗

j )

Global search:

• seeks improvement by design space exploration (sampling ofthe full design space)

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Deterministic / stochastic search

Deterministic search:

• gives exactly the same search path when running thealgorithm twice for unchanged algorithmic settings

Stochastic search:

• generates random search paths for every run of the algorithm

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0th / 1th / 2nd -order search

A 0th-order algorithm

• uses only function evaluations of the objective function (andconstraints).

A 1th-order algorithm

• uses function evaluations and gradient evaluations of theobjective function (and constraints),

• the gradient refers to the first-order derivatives with respectto the design variables

A 2nd-order algorithm

• uses function, gradient, and Hessian evaluations of theobjective function (and constraints)

• the Hessian refers to the second-order derivatives withrespect to the design variables

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Newton type of algorithms

Characteristics

• unconstrained/constrained

• local

• deterministic

• 2nd-order

Matlab Optimization toolbox

• fminunc

• fmincon

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x1

x2

x0

x1

x2

x3

s0

s1

s2

s3

Line search

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x1

x2

x0

x1 x2

x30

1

2

3

s0 s1 s2

Trust-region

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Bio-inspired algorithms

Characteristics


• global

• stochastic

• 0th-order

Matlab GA toolbox

• ga

Algorithms

• Particle swarms

• Genetic algorithms (GAs)

• Simulated Annealing

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2

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Direct search algorithms

Characteristics


• local

• deterministic

• 0th-order

Matlab Optimization toolbox

• fminsearch

Matlab Direct Search toolbox

• patternsearch

Algorithms

• Nelder-Mead simplex search

• Generalized Pattern Search

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2

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Gradient

∇f =h@f@x1

; @f@x2

; : : : ; @f@xn

i

Calculating gradients:

• Analytically (by hand)

• Symbolically (e.g. Mathematica)

• Finite differencing (numerical)

• Built-in in the CAE analysis code (e.g. MARC)

Matlab:

• fmincon and fminunc calculate gradients by finitedifferencing if the user does not provide them

• Matlab Symbolic Toolbox

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Hessian

H =

2666664

@2f@x2

1

@2f@x1@x2

· · · @2f@x1@xn

@2f@x2@x1

@2f@x2

2· · · @2f

@x2@xn

.... . .

...@2f

@xn@x1

@2f@xn@x2

· · · @2f@x2

n

3777775

Calculating the Hessian:

• usually does not need to be provided by the user, but isestimated by the optimization algorithm itself (from theiteration history)

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Conditions for optimality

Karush-Kuhn-Tucker (KKT) conditions

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9

8

f

x1

x2

P1

P2

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x1

x2

x1

x2

g

g1

g2

g3

f

g22

3 = 0

g11

f

g

F

F

Unconstrained: gradients objective function zero

Constrained: linear combination of gradients objective andgradients active constraints zero (i.e. gradients Lagrange functionL zero)

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Termination criteria

Condition on optimality

‖∇L(xk+1)‖ < " (Matlab: TolFun)

gj (xk+1) < " and |hj (xk+1)| < " (Matlab: TolCon)

or a condition on change in x

‖xk+1 − xk‖ < " (Matlab: TolX)

or a condition on the number of iterations

k ≤ kmax (Matlab: MaxIter)

or a combination of these, with " > 0

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Nested analysis

• In various engineering optimization problems evaluation of(inequality) constraints (e.g. regarding stresses,displacements, buckling forces, etcetera) requires a call to anexternal computational routine (fem software).

• That is, for every evaluation of the objective function andconstraints, a call is made to the external analysis routine.

• The analysis routine receives the design variable values forwhich the computational analysis has to be carried out, andreturns the desired responses.

• This is referred to as nested analysis• The nested analysis has to be implemented in the objective

function and/or constraint function files.• In your design project, you will be using Matlab for the

structural analysis as well as for the optimization. This meansthe fem analysis becomes a function call within the objectivefunction and/or constraint function m-files.

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Scaling

Scale design variables, objective function and constraintfunctions to avoid numerical difficulties and prematuretermination

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