Search

Sensitivity analysis, the quantitative analysis of the perturbation map, is of paramount interest in optimization theory and has applications in several branches of pure and applied mathematics. During the last five decades, substantial progress has been made in sensitivity analysis for optimization problems with scalar objectives. On the other hand, the differentiability issues of the perturbation map for vector optimization problems and set optimization problems are rather involved and they require modern tools from variational analysis. The main difficulty here stems from the fact that the perturbation map for such problems is, in general, set-valued.

In this chapter we present a generalization of the Hahn–Banach–Kantorovich extension theorem to K-convex set-valued maps, as well as Yang’s extension theorem. We also present classical separation theorems for convex sets, the core convex topology on a linear space, and a criterion for the convexity of the cone generated by a set.

Tangent cones of first-order and tangent cones and tangent sets of higher-order play a very important role in set-valued optimization. For instance, derivatives and epiderivatives of set-valued maps are commonly defined by taking tangent cones and tangent sets of graphs and epigraphs of set-valued maps. Moreover, properties of tangent cones and tangent sets are quite decisive in giving calculus rules for derivatives and epiderivatives of set-valued maps. Furthermore, optimality conditions in set-valued optimization are also most conveniently expressed by using tangent cones and tangent sets. Sensitivity analysis, constraints qualifications, and many other issues in set-valued optimization heavily rely on tangent cones and tangent sets.

To each type of efficiency for optimization problems it is possible to associate notions of conjugate and subdifferential for vector valued functions or set-valued maps. In this chapter we study the conjugate and the subdifferential corresponding to the strong efficiency as well as the subdifferentials corresponding to the weak and Henig type efficiencies. For the strong conjugate and subdifferential we establish similar results to those in the convex scalar case, while for the other types of subdifferential we establish formulas for the subdifferentials of the sum and the composition of functions and set-valued maps.

In this chapter, first, we give an introduction to order relations and cone properties. Then we present a detailed overview of solution concepts in vector-valued as well as set-valued optimization. We introduce and discuss the following solution concepts for set-valued optimization problems

In this chapter we introduce nonlinear scalarization methods that are very important from the theoretical as well as computational point of view. We introduce different scalarizing functionals and discuss their properties, especially monotonicity, continuity, Lipschitz continuity, sublinearity, convexity. Using these nonlinear functionals we show nonconvex separation theorems. These nonlinear functionals are used for deriving necessary optimality conditions for solutions of set-valued optimization problems in Sect. 12.8 and in different proofs, especially in the proof minimal point theorems in Chap. 10. Moreover, we study characterizations of solutions of set-valued optimization problems by means of nonlinear scalarizing functionals. Finally, we present the extremal principle by Kruger and Mordukhovich and discuss its relationship to separation properties of nonconvex sets. This extremal principle will be applied in Sect. 12.9 for deriving a subdifferential variational principle for set-valued mappings and in Sect. 12.11 in order to prove a first order necessary condition for fully localized minimizers of set-valued optimization problems.

In this chapter we present existence results for minimal points of subsets of the Cartesian product of a complete metric space and a topological vector space with respect to order relations determined by generalized set-valued metrics; such results are useful for deriving EVP type results for vector and scalar functions. Then we derive EVP results of Ha’s type as well as an EVP result for bi-set-valued maps. We end the chapter with an application to error bounds for set-valued optimization problems.

In this chapter we present solution procedures for solving set-valued optimization problems. In Sect. 14.1 a Newton method for solving general set-valued optimization problems is shown. For a special class of set-valued optimization problems where the objective map is polyhedral, convex and set-valued, we present an algorithm in Sect. 14.2.

We consider the optimization problem $$f(x): = (c,x) + \sum\limits_{i = 1}^n {\alpha _i \parallel A^i x - a^i \parallel ^{\beta _i } \to \mathop {\min }\limits_{x \in D} ,} $$ which is an extension of a problem studied by Idrissi, Lefebvre and Michelot [7]. This class of problems contains many practically important special cases so as approximation, location and optimal control problems, perturbed linear programming problems and surrogate problems for linear programming. Necessary and sufficient optimality conditions are derived using the sub-differential calculus. A proximal point algorithm is modified by the method of partial inverse [13] in order to solve the optimality conditions.

It is a great pleasure for us to praise Rosalind Elster’s long-standing professional and courageous commitment to improve the journal Optimization to a high-level international journal. Our journal...

We introduce concepts for approximate minimal and nondominate solutions of vector optimization problems with variable ordering structure. We discuss basic concepts of approximate solutions, where the variable ordering structure is defined by sets with certain properties. Furthermore, we introduce a scalarization method by means of nonlinear functionals and present a characterization of approximate minimal and nondominate solutions by using this scalarization method.

We study a strictly robust optimization problem in the context of a nonlinear scalarization method. We introduce a strictly robust multicriteria optimization problem and discuss its relation to a well-known scalar strictly robust optimization problem by using the nonlinear scalarization concept. Furthermore, we propose an unrestricted multicriteria optimization problem and note that its set of weakly Pareto optimal solutions contains all solutions of the scalar strictly robust optimization problem.

We develop a new model for a Private Equity Fund based on stochastic differential equations. In order to find efficient strategies for the fund manager we formulate a multicriteria optimization problem for a Private Equity Fund. Using the e-constraint method we solve this multicriteria optimization problem. Furthermore, a genetic algorithm is applied in order to get an approximation of the efficient frontier.

We introduce a concept for approximately efficient solutions in vector optimization and compare it with another recent concept given in [8]. Further, we study relations between the set of approximately efficient solutions of a vector optimization problem and the approximate solutions of a corresponding parametric surrogate optimization problem. Finally, we prove stability properties for the scalar surrogate problem. Wir fuhren ein Konzept fur Naherungslosungen in der Vektoroptimierung ein und vergleichen dieses mit einem neuen Konzept aus [8]. Weiterhin untersuchen wir Beziehungen zwischen der Menge der Naherungslosungen eines Vektoroptimierungsproblems und den Naherungslosungen eines entsprechenden parametrischen Ersatzproblems. Schlieβlich beweisen wir Stabilitatseigenschaften des skalaren Ersatzproblems.

Deriving existence results and necessary conditions for approximate solutions of nonlinear optimization problems under week assumptions is an interesting and modern field in optimization theory. It is of interest to show corresponding results for optimization problems without any convexity and compactness assumptions. Ekeland’s variational principle is a very deep assertion about the existence of an exact solution of a slightly perturbed optimization problem in a neighborhood of an approximate solution of the original problem. The importance of Ekeland’s variational principle in nonlinear analysis is well known. Especially, this assertion is very useful for deriving necessary conditions under certain differentiability assumptions. In optimal control Ekeland’s principle can be used in order to prove an e-maximum principle in the sense of Pontryagin and in approximation theory for deriving e-Kolmogorov conditions.

In order to show necessary conditions for approximate solutions of vector-valued optimization problems in general spaces, we introduce an axiomatic approach for a scalarization scheme. Several examples illustrate this scalarization scheme. Using an Ekeland-type variational principle by Isac [12] and suitable scalarization techniques, we prove the optimality conditions under different assumptions concerning the ordering cone and under certain differentiability assumptions for the objective function.

This article is a continuation of LA, Tammer C (2007: A new ap- proach to duality in vector optimization. Optimization 56(1-2):221-239) (14). We developed in (14) a duality theory for convex vector optimization problems, which is different from other approaches in the literature. The main idea is to embed the image space R q of the objective function into an appropriate complete lattice, which is a subset of the power set of R q . This leads to a duality theory which is very anal- ogous to that of scalar convex problems. We applied these results to linear problems and showed duality assertions. However, in (14) we could not answer the question, whether the supremum of the dual linear program is attained in vertices of the dual feasible set. We show in this paper that this is, in general, not true but, it is true under additional assumptions.

Das Ziel der vorliegenden Arbeit besteht in der Entwicklung mathematischer Methoden der Standortbestimmung und deren Anwendung bei der Planung und Gestaltung von Bergbaufolgelandschaften. Insbesondere wird ein neuartiger Proximal-Point-Algorithmus entwickelt, der zur Behandlung von Standortproblemen mit gemischten Normen eingesetzt werden kann. Aufgrund der schwachen Bedingungen ist dieser Algorithmus auch zur Losung von Multistandortpro-blemen nutzbar. Zur Losung von vektoriellen Standortproblemen wird ein interaktives Losungsverfahren, welches auf den Proximal-Point-Algorithmus zuruckgreift, vorgeschlagen. Weiterhin wird ein auf Anwendungen in der Landschaftsgestaltung zugeschnittenes Programmpaket vorgestellt. Durch eine Verknupfung mit Geographischen Informationssystemen kann mit diesem Programmpaket auf entsprechende Karten zum betrachteten Gebiet zuruckgegriffen werden.

We introduce several solution concepts for multicriteria optimization problems, give a characterization of approximately efficient elements and discuss a general scalarization procedure. Furthermore, we derive necessary and sufficient optimality conditions, a minimal point theorem, a vector-valued variational principle of Ekeland’s type, Lagrangean multiplier rules and duality statements. An overview on vector variational inequalities and vector equilibria is given. Moreover, we discuss the results for special classes of vector optimization problems (vector-valued location and approximation problems, multicriteria fractional programming and optimal control problems).

This paper deals with several kinds of location problems, necessary optimality conditions and algorithms based on the optimality conditions.

In this paper, we discuss the connection between concepts of robustness for multi-objective optimization problems and set order relations. We extend some of the existing concepts to general spaces and cones using set relations. Furthermore, we derive new concepts of robustness for multi-objective optimization problems. We point out that robust multi-objective optimization can be interpreted as an application of set-valued optimization. Furthermore, we develop new algorithms for solving uncertain multi-objective optimization problems. These algorithms can be used in order to solve a special class of set-valued optimization problems.

International audience; This chapter considers potential games, where agents play, each period, Nash worthwhile moves in alternation, such that their unilateral motivation to change rather than to stay, other players being supposed to stay, are high enough with respect to their resistance to change rather than to stay. This defines a generalized proximal alternating linearized algorithm, where resistance to change plays a major role, perturbation terms of alternating proximal algorithms being seen as the disutilities of net costs of moving.