Adaptive Scalarization Methods in Multiobjective Optimization

verfasst von: Dr. Gabriele Eichfelder

Verlag: Springer Berlin Heidelberg

Buchreihe : Vector Optimization

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Inhaltsverzeichnis

Frontmatter

Theory

1. Theoretical Basics of Multiobjective Optimization

Multiobjective optimization is an indispensable tool for decision makers if the benefit of a decision does not depend on one objective only, which then can be mapped by one scalar-valued function, but if several competing objectives are aspired all at once. In this chapter we discuss basic concepts of vector optimization such as minimality notions based on partial orderings introduced by convex cones.

2. Scalarization Approaches

The information provided by this approximation depends mainly on the quality of the approximation. Many approximation points cause a high numerical effort, however approximations with only few points neglect large areas of the efficient set. Thus, the aim of this book is to generate an approximation with a high quality.

A wide variety of scalarizations exist based on which one can determine single approximation points. However not all methods are appropriate for non-convexity or arbitrary partial orderings. For instance the weighted sum method has the disadvantage that it is in general only possible for convex problems to determine all efficient points by an appropriate parameter choice (see [39, 138]). The ε-constraint method as given in (2.1) is only suited for the calculation of EP-minimal points. Yet problems arising in applications are often non-convex. Further it is also of interest to consider more general partial orderings than the natural ordering. Thus we concentrate on a scalarization by Pascoletti and Serafini, 1984 ([181]) which we present and discuss in the following sections. An advantage of this scalarization is that many other scalarization approaches as the mentioned weighted sum method or the ε-constraint method are included in this more general formulation. The relationship to other scalarization problems are examined in the last section of this chapter.

3. Sensitivity Results for the Scalarizations

Numerical Methods and Results

4. Adaptive Parameter Control

In this chapter we use the preceding results for developing an algorithm for adaptively controlling the choice of the parameters in several scalarization approaches. The aim is an approximation of the efficient set of the multiobjective optimization problem with a high quality. The quality can be measured with different criteria, which we discuss first. This leads us to the aim of equidistant approximation points.

For reaching this aim we mainly use the scalarization approach of Pascoletti and Serafini. This scalarization is parameter dependent and we develop a procedure how these parameters can be chosen adaptively such that the distances between the found approximation points of the efficient set are controlled. For this adaptive parameter choice we apply the sensitivity results of Chap. 3. Because many other scalarizations can be considered as a special case of the Pascoletti-Serafini problem, as we have seen in Sect. 2.5, we can apply our results for the adaptive parameter control to other scalarizations as the ε-constraint or the normal boundary intersection problem, too.

5. Numerical Results

For testing the numerical methods developed in Chap. 4 on their effi- ciency, in this chapter several test problems with various difficulties are solved. Test problems are important to get to know the properties of the discussed procedures. Large collections of test problems mainly developed for testing evolutionary algorithms are given in [41, 45, 46, 228]. Some of the problems of these collections test difficulties especially designed for evolutionary algorithms, which are not a task using the scalarization approaches of this book. For instance, arbitrarily generated points of the constraint set are mapped by the objective functions mainly in areas far away from the efficient set and only few of them have images near the efficient set.

Other test problems have a non-convex image set or gaps in the ef- ficient set, i. e. non-connected efficient sets. We examine such problems among others in this chapter. In the remaining chapters of this book, we discuss the application of the methods of Chap. 4 on a concrete application problem in medical engineering as well as on a multiobjective bilevel optimization problem.

6. Application to Intensity Modulated Radiotherapy

As we have already pointed out in the introduction to this book, many problems arising in applications are from its structure multiobjective. Nevertheless these problems are often treated as a single objective optimization problem in practice. This was for instance done in intensity modulated radiotherapy (IMRT). Here, an optimal treatment plan for the irradiation of a cancer tumor has to be found for a patient. The aim is to destroy or at least reduce the tumor while protecting the surrounding healthy tissue from unnecessary damage. For a detailed problem description we refer to [3, 36, 63, 143, 146, 170, 223].

Regarding the natural structure this problem is multiobjective, i. e. there are two or more competing objectives which have to be minimized at the same time. On the one hand there exists the target that the tumor has to be irradiated sufficiently high such that it is destroyed. On the other hand the surrounding organs and tissue, which is also affected by this treatment, should be spared. Thereby the physician has to weight the risk of the unavoidable damage of the surrounding to the tumor against each other. He can decide on the reduction of the irradiation dose delivered to one organ by allowing a higher dose level in another organ.

Multiobjective Bilevel Optimization

7. Application to Multiobjective Bilevel Optimization

Despite multiobjective bilevel optimization has not yet received a broad attention in the literature, these problems are very interesting in the view of applications, see for instance [241]. For getting an idea of this we give an illustrative example. Let us consider a city bus transportation system financed by the public authorities. They have as targets the reduction of the money losses in this non-profitable business as well as the reduction of the number of cars on the streets. The public authorities can decide about the bus ticket price, but this influences the customers in their usage of the buses. The public has maybe several competing objectives, too, as to minimize their transportation time and costs. Hence the usage of the public transportation system can be modeled on the lower level with the bus ticket price as parameter. The solution of the lower level influences then the objective values of the public authorities on the upper level. Such a problem can thus be mapped by multiobjective bilevel optimization.

Multiobjective bilevel optimization problems are also closely related to equilibrium problems and the definition of non-dominated equilibrium solutions, see for instance [8, 30, 35, 171, 177, 205, 240].

Backmatter

Titel: Adaptive Scalarization Methods in Multiobjective Optimization
verfasst von: Dr. Gabriele Eichfelder
Verlag: Springer Berlin Heidelberg
Electronic ISBN: 978-3-540-79159-1
Print ISBN: 978-3-540-79157-7
DOI: https://doi.org/10.1007/978-3-540-79159-1