Gordan-Type Alternative Theorems and Vector Optimization Revisited

Proper or Weak Efficiency via Saddle Point Conditions in Cone-Constrained Nonconvex Vector Optimization Problems

“…In the first subsection we consider geometric characterizations in the image space of the possibility of obtaining proper and weak efficiency by means of a linear scalarization, while in the second subsection the same characterizations are expressed in the given space by means of level sets of the objective functions. The presented results supplement those established in Sections 2.5.1 and 2.5.2 in [13].…”

“…In this paper, exploiting the image space approach [18], we analyze conditions ensuring the existence of a saddle point for the generalized Lagrangian function associated with problem (1.1) (defined in (3.7)). Such conditions are obtained by means of the notion of quasi relative interior [5,4], that recently has widely been considered in the literature [10,6,13,16,32,33,35].…”

“…Next proposition, taken from [13] after a careful reading of its proof (where the assumption A ∩ (−ri P ) = ∅ was forgotten), will play an important role in subsequent sections. The following assertions are equivalent:…”

Infinite programming and theorems of the alternative

“…Quasi-convexity or convexlikeness, together with its generalizations, have been commonly used in many problems of convex analysis, such as optimization, 6-13 minimax inequalities, 5,14-19 or theorems of the alternative, 9,[20][21][22] to mention a few. Precisely in these three areas, a not very restrictive concept of convexity, infsup-convexity (see Definition 2.1 below), arises, which generalizes the convexlikeness and that has proven to be adequate to establish very general results, in many cases sharp.…”

“…The first of them is the quasi-convexity, introduced essentially by J. von Neumman 1-4 and refers to a function: given a nonempty and convex subset C of a real vector space, a function ∶ C → R is quasi-convex on C if for any ∈ R, the corresponding sublevel set {y ∈ C ∶ f(y) ≤ } is convex. The second concept, independent of quasi-convexity, is due to K. Fan ( 5, p. 42 ) and involves a family of functions: if X and Λ are nonempty sets, a family {f } ∈ Λ of real-valued functions defined on X is convexlike on X provided that for each 0 ≤ t ≤ 1 and x 1 , x 2 ∈ X, there exists x 0 ∈ X in such a way thatQuasi-convexity or convexlikeness, together with its generalizations, have been commonly used in many problems of convex analysis, such as optimization, 6-13 minimax inequalities, 5,14-19 or theorems of the alternative, 9,[20][21][22] to mention a few. Precisely in these three areas, a not very restrictive concept of convexity, infsup-convexity (see Definition 2.1 below), arises, which generalizes the convexlikeness and that has proven to be adequate to establish very general results, in many cases sharp.…”