Engineering Design by Geometric Programming

Chia-Hui Huang

doi:10.1155/2013/568098

Abstract

A geometric program (GP) is a type of mathematical optimization problem characterized by objective and constraint functions, where all functions are of signomial form. The importance of GP comes from two relatively recent developments: (i) new methods can solve even large-scale GP extremely efficiently and reliably; (ii) a number of practical problems have recently been found to be equivalent to or approximated by GP. This study proposes an optimization approach for solving GP. Our approach is first to convert all signomial terms in GP into convex and concave terms. Then the concave terms are further treated with the proposed piecewise linearization method where only binary variables are used. It has the following features: (i) it offers more convenient and efficient means of expressing a piecewise linear function; (ii) fewer 0-1 variables are used; (iii) the computational results show that the proposed method is much more efficient and faster than the conventional one, especially when the number of break points becomes large. In addition, the engineering design problems are illustrated to evaluate the usefulness of the proposed methods.

Highlights

Compared with the conventional piecewise linearization methods [30,31,32,33,34,35,36], the number of newly added binary variables in the proposed method for a piecewise linear function with m break points is significantly reduced from m − 1 to ⌈log2(m − 1)⌉
This paper proposes an optimization approach for solving geometric programming problems
Our approach is first to convert all signomial terms in geometric program (GP) into convex and concave terms by the proposed methods

Summary

Introduction

Obtaining the optimal solutions for GP is not straightforward because the signomial terms in the objective function and constraints cannot be solved directly. Coello and Cortes [24] proposed a genetic algorithm with an artificial immune system to solve a GP in engineering optimization This method can only obtain the local optima. Lin and Tsai [14] introduced a generalized method to find multiple optimal solutions of signomial discrete programming problems with free variables. By means of variable substitution and convexification strategies, a signomial discrete programming problem with free variables is Mathematical Problems in Engineering first converted into another convex mixed-integer nonlinear programming problem solvable to obtain an exactly global optimum.

Proposed Methods

Numerical Examples

Findings

Conclusions