Local maximizers of generalized convex vector-valued functions

Any local maximizer of an explicitly quasiconvex real-valued function is actually a global minimizer, if it belongs to the intrinsic core of the function's domain. In this paper we show that similar properties hold for componentwise explicitly quasiconvex vector-valued functions, with respect to the concepts of ideal, strong and weak optimality. We illustrate these results in the particular framework of linear fractional multicriteria optimization problems.Any local maximizer of an explicitly quasiconvex real-valued function is actually a global minimizer, if it belongs to the intrinsic core of the function's domain. In this paper we show that similar properties hold for componentwise explicitly quasiconvex vector-valued functions, with respect to the concepts of ideal, strong and weak optimality. We illustrate these results in the particular framework of linear fractional multicriteria optimization problems

Simple Problems: The Simplicial Gluing Structure of Pareto Sets and Pareto Fronts

Quite a few studies on real-world applications of multi-objective optimization reported that their Pareto sets and Pareto fronts form a topological simplex. Such a class of problems was recently named the simple problems, and their Pareto set and Pareto front were observed to have a gluing structure similar to the faces of a simplex. This paper gives a theoretical justification for that observation by proving the gluing structure of the Pareto sets/fronts of subproblems of a simple problem. The simplicity of standard benchmark problems is studied.Comment: 10 pages, accepted at GECCO'17 as a poster paper (2 pages

Arcwise cone-quasiconvex multicriteria optimization

The role played by the generalized convexity in optimization is nowadays well recognized. In this paper we introduce a new concept of quasiconvexity for vector functions, which is shown to have some interesting applications in multicriteria optimization, especially as regards the Pareto reducibility and the strong contractibility of the efficient outcome set. The notion of Pareto reducibility, introduced by Popovici in [8], allows one to reduce the complexity of a multicriteria problem by considering new problems obtained from the original one by selecting a certain number of criteria. More precisely, a multicriteria optimization problem is said to be Pareto reducible if its weakly efficient solutions actually are efficient solutions for the problem itself or for a subproblem obtained from it by selecting certain criteria. In [9] some results concerning the Pareto reducibility and the contractibility of efficient sets have been established for multicriteria optimization problems involving lexicographic quasiconvex objective functions. Our paper aims to extend these result