223
8
Geometric Programming
8.1 Introduction
Geometric programming can be applied to optimization problems in which
the objective function and constraints have a special structure. The conven-
tional format of the objective function and constraints can be converted into the
format required for geometric programming. Once the problem is written in
the required format, it is much easier to solve the optimization problem using
geometric programming than using nonlinear programming (NLP) methods
described in previous chapters. The geometric programming technique pro-
posed by Zener, Dufn, and Peterson can solve large-scale optimization prob-
lems with high reliability and efciency. Geometric programming is applied to
various disciplines such as inventory model (Abuo-El-Ata et al. 2003), structural
optimization (Hajela 1986), communication systems (Chiang 2005), very-large-
scale integration (VLSI) design (Chu and Wong 2001), and so on.
In geometric programming, the objective function is written in posynomial
form:
f cx x x x
a a
a
n
a
n
( )x =
1 2 3
1 2
3
(8.1)
where c is a positive constant, the exponents a
i
are real numbers, and x
i
are
the design variables that can take positive values. It is important to note that
in polynomials, c can take both positive and negative values. For example,
f x x x x( )x = 5 2 3
1
2
2
2
1 2
is a polynomial, while
f x x x x( )x = + +
2 5 4
1
2
2
2
1
2
2
1
is a posynomial.
If the objective function is obtained in polynomial form, then it has to be
transformed into a posynomial before geometric programming techniques
can be used. For example, the maximization function
f x x( )x
=
1
2
2
can be
224 Optimization: Algorithms and Applications
transformed into a posynomial form minimization function
f x x( )x
=
1
2
2
1
.
It is very interesting to note that in geometric programming, the objective
function is evaluated rst and then optimal design variables are obtained.
That is, the optimized value of the objective function can be obtained with-
out knowing the optimal value of design variables. Thus, the solution to
geometric programming problems does not depend on the initial guess. In
this chapter, both unconstrained and constrained optimization problems are
solved using geometric programming. The chapter concludes with a practi-
cal application of geometric programming. The road map for this chapter is
given in Figure 8.1.
8.2 Unconstrained Problem
Consider minimization of the function
f U c x
j
j
N
j i
a
j
N
i
n
ij
( ) ( )x x= =
= =
=
1 1
1
(8.2)
where x
i
, c
j
> 0. The minimum or maximum of the function can be obtained
using the rst-order condition
=
f
x
i
0 (8.3)
Unconstrained problems
Dual problem
Constrained optimization
Application
Geometric programming
FIGURE 8.1
Road map of Chapter 8.
225Geometric Programming
The solution of this equation leads to the orthogonality condition
j
N
j ij
w a
=
=
1
0
*
(8.4)
and the normality condition
j
N
j
w
=
=
1
1
*
(8.5)
where
w
U
f
j
j
*
*
( )
=
x
*
(8.6)
The procedure for obtaining the optimal value of the objective function is
to write the function as
f
U
w
U
w
U
w
w w
n
n
*
*
*
*
*
*
*
* *
=
1
1
2
2
1 2
w
n
*
(8.7)
where the values
w
j
*
are obtained by solving the orthogonality and normal
equations.
The quantity N − (n + 1) is called as the degree of difculty in geometric pro-
gramming, where n is the number of design variables and N is the number
of posynomial terms in the objective function. If the degree of difculty is
zero, then the problem has a unique solution. If the degree of difculty is
positive (number of equations obtained through orthogonality and normal-
ity condition being less than the number of variables), some variables have to
be expressed in terms of other variables to obtain the solution. In geometric
programming, we do not have the negative degree of difculty.
Using f* and
U
j
*
, optimal values of the design variables can be evaluated
using the expression
U w f c
j j
i
n
i
a
ij
* *
(
*
)= =
=
1
1
*
x
(8.8)

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