Duality assertions in vector optimization w.r.t. relatively solid convex cones in real linear spaces

We derive duality assertions for vector optimization problems in real linear spaces based on a scalarization using recent results concerning the concept of relative solidness for convex cones (i.e., convex cones with nonempty intrinsic cores). In our paper, we consider an abstract vector optimization problem with generalized inequality constraints and investigate Lagrangian type duality assertions … Read more

A Fixed Point Approach with a New Solution Concept for Set-valued Optimization

We present a fixed point approach to find the whole solution set of a set-valued optimization problem though a parametric problem, in which the height of the level set of the objective function is regarded as the parameter. First, the solution concept based on the vector approach is considered in this method. Then, we propose … Read more

Vector Optimization w.r.t. Relatively Solid Convex Cones in Real Linear Spaces

In vector optimization, it is of increasing interest to study problems where the image space (a real linear space) is preordered by a not necessarily solid (and not necessarily pointed) convex cone. It is well-known that there are many examples where the ordering cone of the image space has an empty (topological / algebraic) interior, … Read more

A Steepest Descent Method for Set Optimization Problems with Set-Valued Mappings of Finite Cardinality

In this paper, we study a first-order solution method for a particular class of set optimization problems where the solution concept is given by the set approach. We consider the case in which the set-valued objective mapping is identified by a finite number of continuously differentiable selections. The corresponding set optimization problem is then equivalent … Read more

The Fermat Rule for Set Optimization Problems with Lipschitzian Set-Valued Mappings

n this paper, we consider set optimization problems with respect to the set approach. Specifically, we deal with the lower less and the upper less set relations. First, we derive properties of convexity and Lipschitzianity of suitable scalarizing functionals, under the same assumption on the set-valued objective mapping. We then obtain upper estimates of the … Read more

On the intrinsic core of convex cones in real linear spaces

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 they are useful geometric objects for describing optimality conditions. As important applications (especially, in the fields of optimal control with PDE constraints, … Read more

Subdifferentials and SNC property of scalarization functionals with uniform level sets and applications

This paper deals with necessary conditions for minimal solutions of constrained and unconstrained optimization problems with respect to general domination sets by using a well-known nonlinear scalarization functional with uniform level sets (called Gerstewitz’ functional in the literature). The primary objective of this work is to establish revised formulas for basic and singular subdifferentials of … Read more

A Unified Characterization of Nonlinear Scalarizing Functionals in Optimization

Over the years, several classes of scalarization techniques in optimization have been introduced and employed in deriving separation theorems, optimality conditions and algorithms. In this paper, we study the relationships between some of those classes in the sense of inclusion. We focus on three types of scalarizing functionals (by Hiriart-Urruty, Drummond and Svaiter, Gerstewitz) and … Read more

On generalized-convex constrained multi-objective optimization

In this paper, we consider multi-objective optimization problems involving not necessarily convex constraints and componentwise generalized-convex (e.g., semi-strictly quasi-convex, quasi-convex, or explicitly quasi-convex) vector-valued objective functions that are acting between a real linear topological pre-image space and a finite dimensional image space. For these multi-objective optimization problems, we show that the set of (strictly, weakly) … Read more

A new algorithm for solving planar multiobjective location problems involving the Manhattan norm

This paper is devoted to the study of unconstrained planar multiobjective location problems, where distances between points are defined by means of the Manhattan norm. By identifying all nonessential objectives, we develop an effective algorithm for generating the whole set of efficient solutions. We prove the correctness of this algorithm and present some computational results, … Read more