It is our immense pleasure to dedicate this special issue to celebrate the 65th birthday of Prof. Johannes Jahn who, during the last several decades, has been a mentor, an encouraging collaborator, and a dear friend to us. Prof. Johannes Jahn, an original, rigorous and brilliant mathematician, made numerous outstanding contributions to strengthen various aspects of scalar, vector and set optimization, ranging from theoretical to computation and applications. His heavily cited excellent papers and monographs contain a wealth of novel ideas that have influenced generations of researchers and will continue to do so in the future.
This special issue is comprised of 10 articles whose contribution we summarize in the following:
An interesting contribution entitled ‘Inertial forward-backward methods for solving vector optimization problems’, by R. I. Bot and S. M. Grad, proposes two forward-backward proximal point type algorithms with inertial/memory effects for computing weakly efficient solutions to a vector optimization problem posed in general spaces. The authors also present the inexact versions of the algorithms which are quite suitable for implementations. Numerical experiments, conducted on a portfolio optimization problem, show the feasibility of the developed framework.
The aim of the useful contribution entitled ‘New algorithms for discrete vector optimization based on the Graef-Younes method and cone-monotone sorting functions’ by C. Gunther and N. Popovici presents new Jahn-Graef-Younes type algorithms for solving discrete vector optimization problems. The original approach proposed by J. Jahn in 2006, computed all minimal elements of a finite set with respect to an ordering cone, using a forward iteration followed by a backward iteration. The proposed scheme involves additional sorting procedures based on scalar cone-monotone functions. The authors analyse the case where the ordering cone is polyhedral. Numerical results give evidence of the superiority of the proposed algorithm.
The objective of J. Grzybowski, D. Pallaschke, H. Przybycin and R. Urbanski in the paper entitled ‘On some consequences of Mazur–Orlicz theorem to Hahn–Banach–Lagrange theorem’ is to investigate various notions of p-convexity of function g which are sufficient for the existence of a linear functional such that
in a known result. The authors replace the sublinearity of p with convexity, the field
with Dedekind vector lattice and present
-convexity which is also a necessary condition. A result on the Mazur-Orlicz theorem is also generalized.
J. Gwinner, in his paper entitled ‘On two-coefficient identification in elliptic variational inequalities’, investigates an inverse problem of parameter identification in an elliptic variational inequalityinvolving two variable parameters. The author explores the dependence of the solution of the variational inequality on these parameters and provides a Lipschitz continuity estimate. Then the inverse problem is studied in an optimization framework. The author suggests an extension of the classical output least-squares approach, gives an existence result and provides complete convergence analysis for a finite-dimensional approximation scheme. The author also investigates the modified output least-squares approach which is based on energy functionals. Interesting examples are discussed.
T. X. D. Ha, in her interesting article entitled ‘A Hausdorff-type distance, a directional derivative of a set-valued map and applications in set optimization’, proposes, by using an ordering cone, a Hausdorff-type distance between two sets in a Banach space setting and employs it to introduce a new directional derivative for set-valued maps. Useful features of the distance and the directional derivative are presented. Focusing on the Kuriowa's set approach, which compares values of a set-valued objective map using set-order relations, the new notion of the directional derivative is then employed to present necessary and/or sufficient conditions for a variety of maximizers and minimizers of a set-valued map.
In another useful article entitled ‘Convex-cone-based comparisons of and difference evaluations for fuzzy sets’, K. Ike and T. Tanaka explore strategies for comparing two fuzzy sets and inspired by the developments in set optimization, propose eight types of fuzzy-set relations. The authors then introduce evaluation functions for fuzzy sets and under suitable conditions relate these functions to the fuzzy-set relations. A numerical procedure is presented to demonstrate the efficacy of the proposed framework.
In an exciting contribution entitled ‘An alternative theorem for set-valued maps via set relations and its application to robustness of feasible sets’, Y. Ogata, T. Tanaka, Y. Saito, G. M. Lee, and J. H. Lee present a generalized Gordan-type alternative theorem for set-valued maps to characterize various set relations without requiring any convexity assumptions. The authors then employ this result to investigate the stability of linear programming problems under modeling error. Some implications to vector optimization problems are also given.
In a useful contribution entitled ‘A systematization of convexity and quasiconvexity concepts for set-valued maps, defined by l-type and u-type preorder relations’, K. Seto, D. Kuroiwa, and N. Popovici present a broad spectrum of generalized convex/quasiconvex set-valued maps, defined by means of the l-type and u-type preorder relations. The authors single out classes of set-valued maps for which an extension of the classical characterization of convex real-valued functions by quasiconvexity of their affine perturbations can be given.
G. Wanka and O. Wilfer, in the paper entitled ‘Duality results for extended multi-facility location problems’, present duality arguments for multi-facility location problems where for each given point the sum of weighted distances to all facilities plus set-up costs is determined, and the maximal value of these sums is to be minimized. The authors develop associated dual problems, distinguishing between the cases, with and without set-up costs and present optimality conditions and use them to offer a geometrical characterization of the optimal solutions. Interesting illustrative examples are given.
In another valuable contribution entitled ‘Minimization of Gerstewitz functionals extending a scalarization by Pascoletti and Serafini’, P. Weidner investigates the minimizer sets of Gerstewitz functionals and provides conditions which ensure that such a set is nonempty and compact. She shows the interdependencies between solutions of problems with either different parameters or different feasible points and proves the impact of parameter control on the scalarization. The author notices that minimizing the Gerstewitz functional is equivalent to an optimization problem that generalizes the scalarization by Pascoletti and Serafini. The author also gives some existence results for vector optimization problems.
Finally, we give our sincerest and everlasting thanks to all the authors who have contributed to this special issue and to the collegial team of the co-editors that functioned very well together.