Optimization of Design

Lecturer：Dung-An WangLecture 1

2

Optimization of Design

Dung-An Wang Phone:04-22840531 ext 365daw@dragon.nchu.edu.tw

LECTURES: Wednesdays 6:20-9pmOFFICE HOURS: Wednesday 9-9:20pmCourse website: web.nchu.edu.tw/~daw/Teaching/Optimization/opti.htm

TEXTBOOK: J.S. Arora “Introduction to Optimum-Design, Third Edition”, Elsevier.

GRADING:Homework (30%), Midterms (20%) Project(30%), Final (20%).HOMEWORK: Homework is due at the begging of the class.

Late homework is not accepted.

3

Objective

Understand optimization approaches to engineeringdesign.

Study the numerical optimization techniques forengineering design.

Practice the systematic process on a semester-longproject.

4

Schedule

Formulation of optimization problems Optimization concepts using the graphical method Optimality conditions for unconstrained and constrained

problems Use of Excel and MATLAB illustrating optimum design of

practical problems Linear programming Numerical methods for unconstrained and constrained

problems Theory and numerical methods for unconstrained

optimization Theory and numerical methods for constrained

optimization Midterm Exam

5

Schedule

Linear and quadratic programming Duality theory in nonlinear programming Rate of convergence of iterative algorithms Derivation of numerical methods Derivation of direct search methods/Project proposal Methods for discrete variable problems Nature-inspired search methods/Process report Multi-objective optimization Final exam/Project presentation

6

Lecture outline

Reading:Ch1 of textToday’s lecture

DESIGN PROCESS BASIC TERMINOLOGY AND NOTATION Derivatives of Functions

7

THE DESIGN PROCESS

Optimization concepts and methods are helpful at everystage of the process.

8

9

BASIC TERMINOLOGY AND NOTATION

A point is an ordered list of numbers, (x1, x2, . . ., xn) isa point consisting of n numbers, and such a point isoften called an n-tuple.

x is interpreted as a point in the n-dimensional space,denoted as Rn

10

Sets S is reads as “S equals the set of all points (x1, x2, x3)

with x3=3.”

11

If a point x is an element of the set S, then we write

S is a subset of T

12

is read “i goes from 1 to n.”

13

dot product

Two vectors are said to be orthogonal (normal) if theirdot product is 0.

If the vectors are not orthogonal, the angle betweenthem

||x|| represents the length of vector x. This is alsocalled the norm of the vector.

14

The length of vector x

The double sum can be written in the matrix form

15

Throughout the text it is assumed that all functions arecontinuous and at least twice continuouslydifferentiable.

A function f(x) of n variables is called continuous at apoint x* if, for

16

Twice-continuous differentiability of a function impliesnot only that it is differentiable two times, but also thatits second derivative is continuous.

17

Derivatives of Functions

first partial derivatives

Gradient of the function f(x).

18

Second partial derivatives for the function f(x):

Matrix of second partial derivatives of f(x)

If f(x) is continuously differentiable two times, thenHessian matrix H(x) is symmetric.

19

Differentiation of each component of the vector g(x)results in a gradient vector, such as

Gradient matrix of g(x)