Publications

Multiobjective optimization problems with a variable ordering structure, instead of a partial ordering, have recently gained interest due to several applications. In the previous years, a basic theory has been developed for such problems. The binary relations of a variable ordering structure are defined by a cone-valued map that associates, with each element of the linear space R m, a pointed convex cone of dominated or preferred directions. The difficulty in the study of the variable ordering structures arises from the fact that the binary relations 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.

The Stochastic Dynamic Distance Optimal Partitioning problem (SDDP problem) is a complex Operations Research problem. The SDDP problem is based on an industrial problem, which contains an optimal conversion of machines. Partitions of integers as states of these stochastic dynamic programming problems involves combinatorial aspects of SDDP problems. Under the assumption of identical "basic costs" (in other words of "unit distances") and independent and identically distributed requirements we will show (in many cases) by means of combinatorial ideas that decisions for feasible states with least square sums of their parts are optimal solutions. Corresponding Markov kernels are called Partitions-Requirements-Matrices (PRMs). Optimal decisions of such problems can be used as approximate solutions of corresponding SDDP problems, in which the basic costs differ only slightly from each other or as starting decisions if corresponding SDDP problems are solved by iterative methods, such as the Howard algorithm.

In this paper, an erratum is provided to the article "On the set-semidefinite representation of nonconvex quadratic programs over arbitrary feasible sets", published in Optim Lett, 2012. Due to precise observation of the first author, it has been found that the proof of Lemma 9 has a nontrivial gap, and consequently the main result (Theorem 10) is incorrect. In this erratum, we prove that Corollary 14 is still correct in the original setting while to fix the proof of Theorem 10 we need additional assumptions. We provide a list of different commonly used assumptions making this theorem to be true, and a new version of this theorem, which is now Theorem 17.

Our main concern in this article are concepts of nondominatedness w.r.t. a variable ordering structure introduced by Yu [P.L. Yu, Cone convexity, cone extreme points, and nondominated solutions in decision problems with multiobjectives, J. Optim. Theory Appl. 14 (1974), pp. 319-377]. Our studies are motivated by some recent applications e.g. in medical image registration. Restricting ourselves to the case when the values of a cone-valued map defining the ordering structure are Bishop-Phelps cones, we obtain for the first time scalarizing functionals for nondominated elements, Fermat rule, Lagrange multiplier rule and duality results for a single- or set-valued vector optimization problem with a variable ordering structure.