C++ Neural Networks and Fuzzy Logic: Preface

C++ Neural Networks and Fuzzy Logic

by Valluru B. Rao

MTBooks, IDG Books Worldwide, Inc.

ISBN: 1558515526   Pub Date: 06/01/95

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Chapter 15

Application to Nonlinear Optimization

Introduction

Nonlinear optimization is an area of operations research, and efficient algorithms for some of the problems in

this area are hard to find. In this chapter, we describe the traveling salesperson problem and discuss how this

problem is formulated as a nonlinear optimization problem in order to use neural networks (Hopfield and

Kohonen) to find an optimum solution. We start with an explanation of the concepts of linear, integer linear

and nonlinear optimization.

An optimization problem has an objective function and a set of constraints on the variables. The problem is to

find the values of the variables that lead to an optimum value for the objective function, while satisfying all

the constraints. The objective function may be a linear function in the variables, or it may be a nonlinear

function. For example, it could be a function expressing the total cost of a particular production plan, or a

function giving the net profit from a group of products that share a given set of resources. The objective may

be to find the minimum value for the objective function, if, for example, it represents cost, or to find the

maximum value of a profit function. The resources shared by the products in their manufacturing are usually

in limited supply or have some other restrictions on their availability. This consideration leads to the

specification of the constraints for the problem.

Each constraint is usually in the form of an equation or an inequality. The left side of such an equation or

inequality is an expression in the variables for the problem, and the right−hand side is a constant. The

constraints are said to be linear or nonlinear depending on whether the expression on the left−hand side is a

linear function or nonlinear function of the variables. A linear programming problem is an optimization

problem with a linear objective function as well as a set of linear constraints. An integer linear programming

problem is a linear programming problem where the variables are required to have integer values. A nonlinear

optimization problem has one or more of the constraints nonlinear and/or the objective function is nonlinear.

Here are some examples of statements that specify objective functions and constraints: