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Configuration Analysis and Design by Using Optimization Tools in Mathematica
Frank J. Kampas
János D. Pintér

Introduction

Quantitative decisions related to various engineering, economic, and scientific investigations are frequently modeled by applying optimization concepts and tools. In mathematical terminology, the decision maker or modeler wants to find the "absolutely best" decision vector that corresponds to the minimum (or maximum) of a suitable objective function, while it satisfies a given collection of feasibility constraint functions. The objective function expresses an overall--scalar or scalarized--performance measure, such as profit, utility, loss, risk, or error. The model constraints originate from physical, technical, economic, or possibly some other, considerations.

A large number of such models naturally belong to the realm of "traditional" continuous optimization, notably, linear and convex nonlinear programming. These subjects are covered by most operations research/management science texts [1-21].

There exist numerous practically important optimization problems in which the phenomena and processes modeled are highly nonlinear and/or are evaluated computationally. In many cases, the necessary structural (convexity) requirements of traditional optimization are not satisfied or may not be simply verifiable. Nonconvex objective functions often possess a multitude of local optima; nonconvex feasible sets also may lead to a similar situation, even for convex objective functions. The subject of global optimization is the theory and application of seeking the "absolutely best" solution in models that have a number of local optima. For discussions of highly nonlinear models and of global optimization, consult, for example, [22-44].

Global optimization (GO) models and strategies have been analyzed since the 1950s. For example, think of the origins of "naturally inspired" genetic and evolutionary searches, simulated annealing, or random search methods. To our best knowledge, the first systematic collection of English GO papers is found in the volumes edited by Dixon and Szegö [45]. Around the same time, several topical books in Russian also became available. See, for example, the related review and references in [35].

As of 2004, there exist well over one hundred books, as well as many thousands of research articles, that discuss GO theory, models, methods, software, and applications. In addition to the works listed earlier, we refer the reader to the Handbook of Global Optimization volumes edited by Horst and Pardalos [46] and by Pardalos and Romeijn [47]. The Journal of Global Optimization, which has been published since 1991, is devoted entirely to the subject.

The Continuous Global Optimization Model

For the purposes of this article, we assume that discrete optimization variables are absent and that all model functions are continuous. This leads to the corresponding specific case of the mathematical programming (MP) model, referred to as the continuous global optimization (CGO) problem. In order to introduce this model form, we use the following notations.

Applying this notation, the CGO model can be expressed as

The set of feasible solutions to this model is defined by

Note that D could be empty: for example, one may think of a system of nonlinear equations that has no solutions within the range specified by . In many practical models, however, D is nonempty, and this property is explicitly postulated in most theoretical analyses of optimization models.

The globally best solution of the CGO model is an such that for all .

The locally best solution of CGO is an , such that for all , Here is a suitable constant defining the Delta-neighborhood of . (Here typically the Euclidean norm is used, but other norms could be used as well.)

Obviously, , and the relation becomes equality only when is also a globally optimal solution.

According to the classical theorem of Weierstrass, a continuous function attains its minimum on a closed, bounded set D. This fact is often used to guarantee the existence of the global solution in various special cases of the CGO model. Observe at the same time that, in its full generality, this model encompasses a large class of difficult instances.

Figure 1 illustrates the potential difficulty of continuous optimization models. Although the feasible set is just a two-dimensional interval, the objective function is highly multimodal. Hence, purely local scope search methods, as a general rule, even in much simpler cases, will not find the best solution. Note that this model is due to Trefethen, and it has been used also in our software tests: consult, for example, [38].

Figure 1. A box-constrained GO model in two dimensions.



     
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