Set-based Robust Optimization of Uncertain Multiobjective Problems via Epigraphical Reformulations

In this paper, we study a method for finding robust solutions to multiobjective optimization problems under uncertainty. We follow the set-based minmax approach for handling the uncertainties which leads to a certain set optimization problem with the strict upper type set relation. We introduce, under some assumptions, a reformulation using instead the strict lower type set relation without sacrificing the compactness property of the image sets. This allows to apply vectorization results to characterize the optimal solutions of these set optimization problems as optimal solutions of a multiobjective optimization problem. We end up with multiobjective semi-infinite problems which can then be studied with classical techniques from the literature

An approximation algorithm for multi-objective optimization problems using a box-coverage

For a continuous multi-objective optimization problem, it is usually not a practical approach to compute all its nondominated points because there are infinitely many of them. For this reason, a typical approach is to compute an approximation of the nondominated set. A common technique for this approach is to generate a polyhedron which contains the nondominated set. However, often these approximations are used for further evaluations. For those applications a polyhedron is a structure that is not easy to handle. In this paper, we introduce an approximation with a simpler structure respecting the natural ordering. In particular, we compute a box-coverage of the nondominated set. To do so, we use an approach that, in general, allows us to update not only one but several boxes whenever a new nondominated point is found. The algorithm is guaranteed to stop with a finite number of boxes, each being sufficiently thin

Proximity measures based on KKT points for constrained multi-objective optimization

An important aspect of optimization algorithms, for instance evolutionary algorithms, are termination criteria that measure the proximity of the found solution to the optimal solution set. A frequently used approach is the numerical verification of necessary optimality conditions such as the Karush-Kuhn-Tucker (KKT) conditions. In this paper, we present a proximity measure which characterizes the violation of the KKT conditions. It can be computed easily and is continuous in every efficient solution. Hence, it can be used as an indicator for the proximity of a certain point to the set of efficient (Edgeworth-Pareto-minimal) solutions and is well suited for algorithmic use due to its continuity properties. This is especially useful within evolutionary algorithms for candidate selection and termination, which we also illustrate numerically for some test problems

Parametergesteuerte Lösung nichtlinearer multikriterieller Optimierungsprobleme

In multi-objective optimization we investigate optimization problems with more than one objective function. As a consequence there is, in general, not only one best solution minimizing all objective functions at the same time, and the solution set called efficient set is very large. Often it is important for the decision maker to have information about the whole efficient set because this provides a useful insight into the problem structure. Thus our aim is to determine an approximation of this set which satisfies certain quality criteria as good as possible. This is achieved by almost equidistant approximation points. Therefore we consider a parameter dependent scalarization approach according to Pascoletti and Serafini. Using new sensitivity results we present an algorithm for controlling the choice of the parameters and with that for generating almost equidistant points of the efficient set. In doing so we allow any partial ordering defined by a convex pointed closed cone in the objective space of the nonlinear multiobjective optimization problem. The effectiveness of this new method is demonstrated at some test problems and what is more we apply it to a recent problem in intensity modulated radiotherapy about prostate cancer treatment. As a further application we develop a new procedure for solving multi-objective bilevel optimization problems and we apply this to a bicriteria bilevel problem in medical technology.Bei multikriteriellen Optimierungsproblemen mit mehreren sich widersprechenden Zielsetzungen gibt es i.Allg. nicht nur eine Minimallösung, die alle Zielfunktionen gleichzeitig optimal erfüllt, sondern die Lösungsmenge, die sog. effiziente Menge, ist sehr groß. Dabei ist es für einen Entscheidungsträger oft wichtig, die gesamte Effizienzmenge zu kennen, da diese wichtige Informationen über das Problem beinhaltet. Das Ziel dieser Arbeit ist es daher eine Approximation der Lösungsmenge zu bestimmen, die bzgl. bestimmter Qualitätskriterien möglichst gut ist, was durch eine Näherung mit nahezu äquidistanten Punkten erreicht wird. Es wird dazu ein parameterabhängiges skalares Ersatzproblem nach Pascoletti und Serafini betrachtet. Aufbauend auf neuen Sensitivitätsergebnissen bestimmen wir einen Algorithmus zur Parametersteuerung und damit zur Generierung nahezu äquidistanter Approximationspunkte. Dabei seien im Zielraum des nichtlinearen multikriteriellen Optimierungsproblems beliebige Halbordnungen induziert durch spitze konvexe abgeschlossene Kegel zugelassen. Die Vorteile dieses neuen Verfahrens demonstrieren wir zunächst an einigen Testproblemen, bevor wir es zur Lösung eines aktuellen Problems aus der Medizin, der optimalen Bestrahlungsplanung zur Behandlung eines Prostatakarzinoms, nutzen. Als weitere Anwendung entwickeln wir eine Lösungsmethodik für nichtlineare multikriterielle Bilevel-Optimierungsprobleme und lösen damit ein bikriterielles Bilevel-Problem aus der Medizintechnik

Ordering structures in vector optimization and applications in medical engineering

This manuscript is on the theory and numerical procedures of vector optimization w.r.t. various ordering structures, on recent developments in this area and, most important, on their application to medical engineering. In vector optimization one considers optimization problems with a vector-valued objective map and thus one has to compare elements in a linear space. If the linear space is the finite dimensional space R^m this can be done componentwise. That corresponds to the notion of an Edgeworth-Pareto-optimal solution of a multiobjective optimization problem. Among the multitude of applications which can be modeled by such a multiobjective optimization problem, we present an application in intensity modulated radiation therapy and its solution by a numerical procedure. In case the linear space is arbitrary, maybe infinite dimensional, one may introduce a partial ordering which defines how elements are compared. Such problems arise for instance in magnetic resonance tomography where the number of Hermitian matrices which have to be considered for a control of the maximum local specific absorption rate can be reduced by applying procedures from vector optimization. In addition to a short introduction and the application problem, we present a numerical solution method for solving such vector optimization problems. A partial ordering can be represented by a convex cone which describes the set of directions in which one assumes that the current values are deteriorated. If one assumes that this set may vary dependently on the actually considered element in the linear space, one may replace the partial ordering by a variable ordering structure. This was for instance done in an application in medical image registration. We present a possibility of how to model such variable ordering structures mathematically and how optimality can be defined in such a case. We also give a numerical solution method for the case of a finite set of alternatives

Numerical procedures in multiobjective optimization with variable ordering structures

Multiobjective optimization problems with a variable ordering structure instead of a partial ordering have recently gained interest due to several applications. In the last years a basic theory has been developed for such problems. The difficulty in their study arises from the fact that the binary relations of the variable ordering structure, which are defined by a cone-valued map which associates to each element of the image space a pointed convex cone of dominated or preferred directions, are in general not transitive. In this paper we propose numerical approaches for solving such optimization problems. For continuous problems a method is presented using scalarization functionals which allows the determination of an approximation of the infinite optimal solution set. For discrete problems the Jahn-Graef-Younes method known from multiobjective optimization with a partial ordering is adapted to allow the determination of all optimal elements with a reduced effort compared to a pairwise comparison

A Solver for Multiobjective Mixed-Integer Convex and Nonconvex Optimization

This paper proposes a general framework for solving multiobjective nonconvex optimization problems, i.e., optimization problems in which multiple objective functions have to be optimized simultaneously. Thereby, the nonconvexity might come from the objective or constraint functions, or from integrality conditions for some of the variables. In particular, multiobjective mixed-integer convex and nonconvex optimization problems are covered and form the motivation of our studies. The presented algorithm is based on a branch-and-bound method in the pre-image space, a technique which was already successfully applied for continuous nonconvex multiobjective optimization. However, extending this method to the mixed-integer setting is not straightforward, in particular with regard to convergence results. More precisely, new branching rules and lower bounding procedures are needed to obtain an algorithm that is practically applicable and convergent for multiobjective mixed-integer optimization problems. Corresponding results are a main contribution of this paper. What is more, for improving the performance of this new branch-and-bound method we enhance it with two types of cuts in the image space which are based on ideas from multiobjective mixed-integer convex optimization. Those combine continuous convex relaxations with adaptive cuts for the convex hull of the mixed-integer image set, derived from supporting hyperplanes to the relaxed sets. Based on the above ingredients, the paper provides a new multiobjective mixed-integer solver for convex problems with a stopping criterion purely in the image space. What is more, for the first time a solver for multiobjective mixed-integer nonconvex optimization is presented. We provide the results of numerical tests for the new algorithm. Where possible, we compare it with existing procedures

Set approach for set optimization with variable ordering structures

This paper aims at combining variable ordering structures with set relations in set optimization, which have been dened using the constant ordering cone before. Since the purpose is to connect these two important approaches in set optimization, we do not restrict our considerations to one certain relation. Conversely, we provide the reader with many new variable set relations generalizing the relations from [16, 25] and discuss their usefulness. After analyzing the properties of the introduced relations, we dene new solution notions for set-valued optimization problems equipped with variable ordering structures and compare them with other concepts from the literature. In order to characterize the introduced solutions a nonlinear scalarization approach is used.Mathematics subject classifcations (MSC 2000): 49J53, 90C29, 90C30, 54C60, 06A7