Engineering Optimization: Theory and Practice, Fourth Edition

104 Classical Optimization Techniques

4.x 1 +x 2 − 001 = 0 , x 1 +x 2 +x 3 − 501 =0: The solution of these equations

yields

x 1 = 05 , x 2 = 05 , x 3 = 05

This solution can be seen to satisfy all the constraint Eqs.(E 7 ) ot (E 9 ) The.

values ofλ 1 , λ 2 , andλ 3 corresponding to this solution can be obtained from

Eqs.(E 15 ) sa

λ 1 = − 20 , λ 2 = − 20 , λ 3 = − 100

Since these values ofλi satisfy the requirements [Eqs.(E 10 ) ot (E 12 ) , this]

solution can be identified as the optimum solution. Thus

x∗ 1 = 05 , x 2 ∗= 05 , x 3 ∗= 05

2.6 Convex Programming Problem

The optimization problem stated in Eq. (2.58) is called aconvex programming problem

if the objective functionf (X)and the constraint functionsgj( X)are convex. The

definition and properties of a convex function are given in Appendix A. Suppose that

f (X)andgj(X),j= 1 , 2 ,... , m, are convex functions. The Lagrange function of

Eq. (2.61) can be written as

L(X,Y,λ)=f (X)+

∑m

j= 1

λj[gj(X)+yj^2 ] (2.78)

Ifλj≥ , then 0 λjgj( X)is convex, and sinceλjyj= from Eq. (2.64), 0 L(X,Y,λ)

will be a convex function. As shown earlier, a necessary condition forf(X) to be a

relative minimum atX∗is thatL(X,Y,λ)have a stationary point atX∗. However, if

L(X,Y,λ)is a convex function, its derivative vanishes only at one point, which must

be an absolute minimum of the functionf(X). Thus the Kuhn–Tucker conditions are

both necessary and sufficient for an absolute minimum off(X) atX∗.

Notes:

1.If the given optimization problem is known to be a convex programming prob-

lem, there will be no relative minima or saddle points, and hence the extreme

point found by applying the Kuhn–Tucker conditions is guaranteed to be an

absolute minimum of f(X). However, it is often very difficult to ascertain

whether the objective and constraint functions involved in a practical engineer-

ing problem are convex.

2.The derivation of the Kuhn–Tucker conditions was based on the development

given for equality constraints in Section 2.4. One of the requirements for these

conditions was that at least one of the Jacobians composed of themconstraints

andmof then+mvariables (x 1 , x 2 ,... , xn;y 1 , y 2 ,... , ym) e nonzero. Thisb

requirement is implied in the derivation of the Kuhn–Tucker conditions.