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# Introduction - PowerPoint PPT Presentation

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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## PowerPoint Slideshow about ' Introduction' - larissa-rowland

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Presentation Transcript

• 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.

• Find design (decision) variables, X

• To minimize objective function, F(X)

• so that

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

• h(X)=0

• 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 decision

g3=0

x2

Feasible region

Weight increases

g2=0

Optimum

x1

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

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

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+ 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

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