Engineering Optimization: Theory and Practice, Fourth Edition

6 Introduction to Optimization

1.4 Statement of an Optimization Problem

An optimization or a mathematical programming problem can be stated as follows.

FindX=











x 1

x 2

xn











whichminimizesf (X)

subject to the constraints

gj(X)≤ 0 , j= 1 , 2 ,... , m

lj(X)= 0 , j= 1 , 2 ,... , p

(1.1)

whereXis ann-dimensional vector called thedesign vector,f (X)is termed theobjec-

tive function, andgj( X)andlj( X)are known asinequalityandequalityconstraints,

respectively. The number of variablesnand the number of constraintsmand/orp

need not be related in any way. The problem stated in Eq. (1.1) is called aconstrained

optimization problem.†Some optimization problems do not involve any constraints and

can be stated as

FindX=











x 1

x 2

xn











whichminimizesf (X) (1.2)

Such problems are calledunconstrained optimization problems.

1.4.1 Design Vector

Any engineering system or component is defined by a set of quantities some of which

are viewed as variables during the design process. In general, certain quantities are

usually fixed at the outset and these are calledpreassigned parameters. All the other

quantities are treated as variables in the design process and are calleddesignordecision

variablesxi, i = 1 , 2 ,... , n. The design variables are collectively represented as a

design vectorX= {x 1 , x 2 ,... , xn}T. As an example, consider the design of the gear

pair shown in Fig. 1.3, characterized by its face widthb, number of teethT 1 and

T 2 , center distanced,pressure angleψ, tooth profile, and material. If center distance

d, pressure angleψ, tooth profile, and material of the gears are fixed in advance,

these quantities can be calledpreassigned parameters. The remaining quantities can be

collectively represented by a design vectorX= {x 1 , x 2 , x 3 }T= { b, T 1 , T 2 }T. If there are

no restrictions on the choice ofb, T 1 , andT 2 , any set of three numbers will constitute a

design for the gear pair. If ann-dimensional Cartesian space with each coordinate axis

representing a design variablexi(i = 1 , 2 ,... , n)is considered, the space is called

†In the mathematical programming literature, the equality constraintslj( X)= 0 ,j= 1 , 2 ,... , pare often

neglected, for simplicity, in the statement of a constrained optimization problem, although several methods

are available for handling problems with equality constraints.