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Convex cones play an important role in nonlinear analysis and optimization theory. In particular, specific normal cones and tangent cones are known to be convex cones, and it is a crucial fact that...

In this special issue, we take the opportunity, on the occasion of his 65th birthday, to acknowledge the outstanding contributions of Professor Alexander Kruger (home page: https://asterius.ballara...

We study the inverse problem of parameter identification in general saddle point problems. For saddle point problems, the use of elliptic regularization is an essential component. Saddle point prob...

This paper deals with abstract quasi-variational-like problems, which cover vector quasi-variational problems with set-valued mappings, for instance. The main objective of this paper is to study existence results for generalized solutions of quasi-variational-like problems extending the approach by Jadamba, Khan and Sama, see (Optim. Lett. 6, 1221–1231, 2012). It is well-known that a quasi-variational-like problem can conveniently be formulated as a set-valued fixed-point problem by using the so-called variational selection. Since the values of the corresponding set-valued mapping are non-convex in general and since the existing fixed-point theorems for non-convex mappings require very stringent conditions on the data of the problem, a few existence results have been obtained in the case where the underlying constraining set is bounded or even compact. In this paper, we focus on the relationship between solutions of a quasi-variational-like problem and generalized solutions of an optimization problem minimizing the difference between inputs and outputs of the variational selection. Doing so, we can solve the latter problem by the well-known Weierstras theorem with much less restrictive assumptions imposed on the given data. This approach allows to consider problems where the variational selection is non-convex and/or the underlying constraining set is unbounded.

International audience; The difficulty of solving a multi-objective optimization problem is impacted by the number of objectives to be optimized. The presence of many objectives typically introduces a number of challenges that affect the choice/design of optimization algorithms. This paper investigates the drivers of these challenges from two angles: (i) the influence of the number of objectives on problem characteristics and (ii) the practical behavior of commonly used procedures and algorithms for coping with many objectives. In addition to reviewing various drivers, the paper makes theoretical contributions by quantifying some drivers and/or verifying these drivers empirically by carrying out experiments on multi-objective combinatorial optimization problems (multi-objective NK-landscapes). We then make use of our theoretical and empirical findings to derive practical recommendations to support algorithm design. Finally, we discuss remaining theoretical gaps and opportunities for future research in the area of multi- and many-objective optimization.

In this paper, we introduce new nonlinear scalarization methods to characterize minimal elements of set-valued problems equipped with variable domination structures. For certain set relations with respect to variable domination structures, we give a characterization of these relations using corresponding nonlinear scalarizing functionals and the properties of these functionals. We describe different kinds of minimal elements to a family of sets with respect to variable domination structures. Furthermore, we investigate a descent method to find approximations of minimal solutions of set-valued problems with respect to variable domination structures using nonlinear scalarization methods. Finally, we apply the results to a set-valued problem in medical image registration.

Inverse problems of identifying parameters in partial differential equations constitute an important class of problems with diverse real-world applications. These identification problems are commonly explored in an optimization framework and there are many optimization formulations having their own advantages and disadvantages. Although a non-convex output least-squares (OLS) objective is commonly used, a convex-modified output least-squares (MOLS) has shown encouraging results in recent years. In this work, we focus on various aspects of the MOLS approach. We devise a rigorous (quadratic and non-quadratic) regularization framework for the identification of smooth as well as discontinuous coefficients. This framework subsumes the total variation regularization that has attracted a great deal of attention in identifying sharply varying coefficients and also in image processing. We give new existence results for the regularized optimization problems for OLS and MOLS. Restricting to the Tikhonov (qua...

When applying a translation invariant function for the separation of sets or the scalarization of problems, then it usually has to be minimized over a given set. In the subsequent chapters, we will characterize solutions to vector optimization problems by minimizers of translation invariant functions on a feasible set. Good-deal bounds of cash streams turn out to be infima of translation invariant functions. In this chapter, we study minimization problems having a translation invariant objective function. Such problems are connected with a generalization of an optimization problem introduced by Pascoletti and Serafini. The existence of optimal solutions and properties of the solution set are investigated. Relationships to problems with an altered feasible set are proved. The considered problems depend on certain parameters when they are applied to scalarization. We examine the related parameter control. Assumptions are given under which the considered problems are equivalent to the minimization of special Minkowski functionals and norms.

In this chapter, vector optimization problems in linear spaces are studied. Here the notion of vector optimization is used in a very general way. A vector optimization problem with a variable domination structure is equivalent to the problem of finding nondominated solutions in a set related to some arbitrary binary relation. Each decision problem can be shown to be of this type. We introduce the basic concept for variable domination structures and connect it to decision problems. Optimal decisions are investigated as optima w.r.t. (with regard to) the preference relation, and the results can straightforwardly be transferred to optima w.r.t. arbitrary binary relations. It will be proved that—even for variable domination structures—efficient points can be helpful in finding the best solutions. Later on, we investigate the case where the domination structure can be described by a single set, namely the domination set of the vector optimization problem. We define efficiency w.r.t. reference sets which are not necessarily domination sets and give a thorough motivation for this approach. Properties of such efficient point sets are investigated. We study efficient and weakly efficient solutions of vector optimization problems, including surrogates for the weakly efficient point set and problems with uncertainties or perturbations. Scalarization results are proved, with an emphasis on translation invariant functions and implications for norms. Beside this, some basic properties of efficient and weakly efficient solutions are examined, especially their existence. The results imply statements for properly efficient solutions and related subsets of the efficient point set.