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Introduction. Optimization: Produce best quality of life with the available resources Engineering design optimization: Find the best system that satisfies given requirements Analysis versus design Analysis: determine performance of given system

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Introduction

Introduction

  • Optimization: Produce best quality of life with the available resources

  • Engineering design optimization: Find the best system that satisfies given requirements

  • Analysis versus design

    • Analysis: determine performance of given system

    • Design: Find system that satisfies given requirements. Design involves iterations in which many design alternatives are analyzed.


Introduction

  • Objective function: measures performance of a design or a decision

  • Constraints: Requirements that a design must satisfy

  • Numerical optimization can be the only practical approach for most real-life problems


General optimization problem statement

General optimization problem statement

  • Find design (decision) variables, X

  • To minimize objective function, F(X)

  • so that

    • g(X) no greater than zero (inequality constraints)

    • h(X)=0


Example tubular column optimization

Example: tubular column optimization

  • Design a column to minimize the mass so that the column does not fail under a given applied axial load

  • Three failure modes--three constraints: yielding, Euler buckling, local buckling

  • May not have unique optimum

  • At the optimum some constraints are active, i.e. applied stress is equal to failure stress


Active constraints

Active constraints

g3=0

x2

Feasible region

Weight increases

g2=0

Optimum

x1


A taxonomy of optimization problems

A taxonomy of optimization problems

Multiple

objectives

Dynamic

Static

One objective

Deterministic

Non deterministic


Taxonomy

Taxonomy

  • Deterministic: know values of all input variables

  • Non deterministic: Only probability distribution of input variables known

  • Static: Solve one optimization problem

  • Dynamic: Solve sequence of optimization sub problems (e.g. chess)

  • Single objective

  • Multiple objectives


Iterative optimization procedure

Iterative optimization procedure

  • Most real life optimization problems solved using iterations

  • Two steps is each iteration

    • Find search direction

    • One dimensional search -- find how far to go in a given direction


Necessary and sufficient conditions unconstrained minimization

Necessary and sufficient conditions, unconstrained minimization

Gradient =0 at X*

Gradient =0 at X*, Hessian pos. def. at X* , X* local min

Gradient =0 at X*,

Hessian pos. def. everywhere ,

X* global min


Necessary condition for local optimum constrained minimization example

Necessary condition for local optimum, constrained minimization. Example

F=constant

F=constant

F

F

Feasible

sector

B

Feasible sector

A

g2=0

g1

g2=0

-F

g1=0

g1=0

-F

g2

g1

g2

B is not a local minimum,

the feasible sector and the

usable sector intersect

A is local minimum, there is no

feasible and usable sector


Necessary condition for local optimum constrained minimization continued

Necessary condition for local optimum, constrained minimization (continued)

  • For the example, there exist two non negative numbers 1 and 2 such that:

    F+ 1g1+ 2g2=0

    General case: There are non negative numbers j0, j=1,…,m

    F+ jgj+ k+mhk+m=0

    where the first sum is for j=1,…,m and the second for k=1,…,l


Introduction

Sufficient conditions global optimum

K-T conditions satisfied

Local optimum

Global optimum

Design space convex,

K-T conditions satisfied


Sensitivity analysis

Sensitivity analysis

  • Allows one to find the sensitivity derivatives of the optimum solution and the optimum value of the objective function with respect the a problem parameter without solving the optimization problem many times.

  • Useful for finding important constraints and important design variables.

  • Very high sensitivity of objective function wrt design parameters; poor design


Equations for sensitivity analysis sensitivity derivatives of design variables

Equations for sensitivity analysisSensitivity derivatives of design variables

  • A: second order derivatives of objective function and active constraints (size nxn)

  • B: columns are gradients of constraints (size nxm)

  • c: second order derivatives of objective function and constraints wrt design variables and design parameter (size nx1)

  • d: derivatives of constraints wrt design parameter (size mx1)

  • X and : derivatives of design variables at optimum and Lagrange multipliers wrt parameter


Equations for sensitivity analysis continued

Equations for sensitivity analysis (continued)

  • Chain rule for sensitivity derivatives of objective function

  • Sensitivity derivatives useful for predicting effect of small changes in problem parameters on solution


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