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Introduction

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

- 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 decision

- 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 decision

- 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

A taxonomy of optimization problems decision

Multiple

objectives

Dynamic

Static

One objective

Deterministic

Non deterministic

Taxonomy decision

- 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 decision

- 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

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

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)

- For the example, there exist two non negative numbers 1 and 2 such that:
F+ 1g1+ 2g2=0

General case: There are non negative numbers j0, j=1,…,m

F+ jgj+ k+mhk+m=0

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

Sufficient conditions global optimum minimization (continued)

K-T conditions satisfied

Local optimum

Global optimum

Design space convex,

K-T conditions satisfied

Sensitivity analysis minimization (continued)

- 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 minimization (continued) 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 minimization (continued) 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