Proper solutions of vector optimization problems

First and second-order optimality conditions without differentiability in multivalued vector optimization

“…We focus on optimality conditions, not comparing them. To have relatively comprehensible presentations and comparisons of various kinds of solutions, the reader is referred to, e.g., [20][21][22][23]. We will need the following properties of infeasible directions in the sequel.…”

Multifunction Optimization Problems Involving Parameters: Necessary Optimality Conditions

“…LUU then (x 0 ; f 0 ) is said to be a (global) weak minimum of F on A. If there is a neighborhood N of x 0 such that there holds (7) or (8) with F ðAÞ replaced by F ðA \ NÞ, then (x 0 ; f 0 ) is called a local minimum or a local weak minimum, respectively, of F . Since U is a set with no structure, we adopt for U the trivial topology consisting of two sets 6 0 and U, on considering local minima of problems (1)-(3) or (4)- (6).…”

“…Stimulated by the book by Ioffe and Tihomirov [5], where scalar optimization with parameters was investigated. We examined vector optimization with parameters in [7][8][9][10][11]. Recently we considered multifunction optimization with parameters dealing with a special regular case in [12].…”

Continuity properties of solution maps of parametric lexicographic equilibrium problems

“…For application purposes, to avoid anomalous mimimal points, one of the possible approaches is to consider bilevel models, i.e., to find Pareto minima under constraints which are in turn solution sets of other optimization-related problems, see, e.g., [6]. Another more direct way is, besides the Pareto minimum, using other stronger notions of minimality such as proper minima (for various kinds of properness see, e.g., [11,18,22]), strict (or firm) minima (see, e.g., recent papers [10,13]), strong (or ideal) minima (e.g., [8,12]), etc.x ∈ A ⊆ X is called a strong minimum of A if (A − x) ⊆ C. But, usually for a convex cone C and a set A, there hardly is a strong minimum. Fortunately, if C is the lexicographic cone C lex (defined in Sect.…”