Engineering Optimization: Theory and Practice, Fourth Edition

8.11 Complementary Geometric Programming 523

Degree of Difficulty. The degree of difficulty of a complementary geometric pro-
gramming problem (CGP) is also defined as

degree of difficulty =N−n− 1

whereNindicates the total number of terms appearing in the numerators of Eq. (8.71).
The relation between the degree of difficulty of a CGP and that of the OGPα, the
approximatingordinary geometric program, is important. The degree of difficulty of a
CGP is always equal to that of the approximating OGPα, solved at each iteration. Thus
aCGP with zero degree of difficulty and an arbitrary number of negative terms can
be solved by a series of solutions to square systems of linear equations. If the CGP
has one degree of difficulty, at each iteration we solve an OGP with one degree of
difficulty, and so on. The degree of difficulty is independent of the choice ofX(α)and
is fixed throughout the iterations. The following example is considered to illustrate the
procedure of complementary geometric programming.

Example 8.6
Minimizex 1

subject to

− 4 x 12 + 4 x 2 ≤ 1

x 1 +x 2 ≥ 1

x 1 > 0 , x 2 > 0

SOLUTION This problem can be stated as a complementary geometric programming
problem as

Minimizex 1 (E 1 )

subjectto

4 x 2

1 + 4 x 12

≤ 1 (E 2 )

x 1 −^1

1 +x 1 −^1 x 2

≤ 1 (E 3 )

x 1 > 0 (E 4 )

x 2 > 0 (E 5 )

Since there are two variables (x 1 andx 2 ) and three posynomial terms [one term in the
objective function and one term each in the numerators of the constraint Eqs. (E 2 ) nda
(E 3 ) the degree of difficulty of the CGP is zero. If we denote the denominators of],
Eqs. (E 2 ) nd (Ea 3 ) sa

Q 1 (X)= 1 + 4 x 12

Q 2 (X)= 1 +x 1 −^1 x 2