Engineering Optimization: Theory and Practice, Fourth Edition

76 Classical Optimization Techniques

Heremis less than or equal ton; otherwise (ifm>n), the problem becomes overdefined

and, in general, there will be no solution. There are several methods available for the

solution of this problem. The methods of direct substitution, constrained variation, and

Lagrange multipliers are discussed in the following sections.

2.4.1 Solution by Direct Substitution

For a problem withnvariables andmequality constraints, it is theoretically possible

to solve simultaneously themequality constraints and express any set ofmvariables

in terms of the remainingn−mvariables. When these expressions are substituted into

the original objective function, there results a new objective function involving only

n−mvariables. The new objective function is not subjected to any constraint, and

hence its optimum can be found by using the unconstrained optimization techniques

discussed in Section 2.3.

This method of direct substitution, although it appears to be simple in theory, is

not convenient from a practical point of view. The reason for this is that the con-

straint equations will be nonlinear for most of practical problems, and often it becomes

impossible to solve them and express anymvariables in terms of the remainingn−m

variables. However, the method of direct substitution might prove to be very simple

and direct for solving simpler problems, as shown by the following example.

Example 2.6 Find the dimensions of a box of largest volume that can be inscribed

in a sphere of unit radius.

SOLUTION Let the origin of the Cartesian coordinate systemx 1 ,x 2 ,x 3 be at the

center of the sphere and the sides of the box be 2x 1 , 2x 2 , and 2x 3. The volume of the

box is given by

f (x 1 , x 2 , x 3 )= 8 x 1 x 2 x 3 (E 1 )

Sincethe corners of the box lie on the surface of the sphere of unit radius,x 1 ,x 2 , and

x 3 have to satisfy the constraint

x 12 +x 22 +x^23 = 1 (E 2 )

This problem has three design variables and one equality constraint. Hence the

equality constraint can be used to eliminate any one of the design variables from the

objective function. If we choose to eliminatex 3 , Eq. (E 2 ) ivesg

x 3 =( 1 −x 12 −x^22 )^1 /^2 (E 3 )

Thus the objective function becomes

f (x 1 , x 2 )= 8 x 1 x 2 ( 1 −x^21 −x 22 )^1 /^2 (E 4 )

which can be maximized as an unconstrained function in two variables.