4m020 design tools - optimization problem formulation and...

Problem formulation Truss exercise 1 Visualization & Problem properties Truss exercise 2

4M020 Design toolsOptimization problem formulation and visualization

L.F.P. Etman

Department of Mechanical EngineeringEindhoven University of Technology

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Outline

1 Problem formulation

2 Two-bar truss exercise: one variable minimization

3 Visualization & Problem properties

4 Two-bar truss exercise: two variable minimization

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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 Select design variables

2 Select objective criterion in terms of design variables(to minimize or maximize)

3 Determine constraints in terms of design variables, whichmust be satisfied

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


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Mathematical problem formulation

Minimizex

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

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

gk (x) ≤ 0 k = 1; : : : ;mg

x ∈ X ⊆ Rn


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Negative null form

Minimizex

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

subject to h(x) = 0 h = [h1;h2; : : : ;hmh ]T

g(x) ≤ 0 g = [g1; g2; : : : ; gmg ]T

x ∈ X ⊆ Rn

Other formulations:

• positive null form (g(x) ≥ 0)

• negative unity form (g(x) ≤ 1)

• positive unity form (g(x) ≥ 1)

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Two-bar truss example

F

SS

h

d

Minimizex

f (x)

subject to h(x) = 0g(x) ≤ 0x ∈ X ⊆ Rn

• Analysis equations

• Constant parameters

• Design variables

• Objective function

• Constraint functions

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Two-bar truss exercise

F

SS

h

d

Minimizex

f (x)


Exercise: single variable minimization

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Visualization of a one-variable optimization problem

ii

“plot1Db˙temp” — 2006/6/26 — 16:21 — page 1 — #1 ii

ii

ii

x

f

f(x)

x

g

g3(x)

0

g6(x)

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Visualization of a one-variable optimization problem

ii

“plot1De˙temp” — 2006/6/26 — 16:21 — page 1 — #1 ii

ii

ii

x

f

f(x)

x

g

g3(x)

0

g6(x)

g3 = 0

g6 = 0

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Visualization of a one-variable optimization problemii

“plot1Df˙temp” — 2006/6/26 — 16:21 — page 1 — #1 ii

ii

ii

x

f

f(x)

g3 = 0

g6 = 0

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Constrained versus unconstrained optimumi

i“opt˙bnd˙temp” — 2006/6/15 — 21:36 — page 1 — #1 i

i

ii

ii

x

f f(x)

g = 0

• constrained optimum

• bounded optimum

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“opt˙unc˙temp” — 2006/6/15 — 21:35 — page 1 — #1 ii

ii

ii

x

ff(x)

g2 = 0g1 = 0

• unconstrained optimum

• interior optimum

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Multimodality (multiple local minima)ii

“multi˙temp” — 2006/6/15 — 17:12 — page 1 — #1 ii

ii

ii

x

f

f(x)

global

local

x

ff(x)

globallocal

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Visualization of a two-variable optimization problemi

i“plot2Da˙temp” — 2006/6/15 — 21:15 — page 1 — #1 i

i

ii

ii

x1

x2

objective function

6 78

9 10

f

x1

x2

constraint g1

g1= 0

g1< 0

feasible

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Visualization of a two-variable optimization problem

ii

“plot2Db˙temp” — 2006/6/15 — 21:29 — page 1 — #1 ii

ii

ii

x1

x2

g1

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ii

“plot2Dc˙temp” — 2006/6/15 — 21:16 — page 1 — #1 ii

ii

ii

x1

x2

g1

g4

g5

g2 g4 g6

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ii

“plot2Dd˙temp” — 2006/6/15 — 21:29 — page 1 — #1 ii

ii

ii

x1

x2

g1

g4

g5

g2 g4 g6

feasible

domain

F

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ii

“plot2De˙temp” — 2006/6/15 — 21:26 — page 1 — #1 ii

ii

ii

x1

x2

g1

g4

g5

g2 g4 g6

optimum

F

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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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Two-bar truss exercise

F

SS

h

d

Minimizex

f (x)


Exercise: optimization problem formulation and visualization withtwo design variables

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4m020 design tools - optimization problem formulation and...

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