Solution Existence in Bifunction-Set Optimization

Existence of solutions of a generalized bifunction-set optimization problem

“…It is also shown in [28] that existence results for both Kuroiwa set optimization Problems (l-SOP) and (u-SOP) are immediate from those for Problem (u-BSOP). More importantly, the work [28] provides a new approach to the solution existence in set optimization, with tools and assumptions that are familiar in vector equilibrium problems, but are not used in all the papers [1,4,17,19,23,24] discussing the existence of solutions of Problems (l-SOP) and (u-SOP): KKMFan-Lemma [13], coercivity conditions, generalized convexity notions,... This paper aims at providing sufficient conditions for the existence of solutions of a more general setting of the bifunction-set optimization Problem (u-BSOP) of [28].…”

“…Introduction. The bifunction-set optimization problem [28] provides a general model that includes as special cases some Kuroiwa set optimization problems [23,24] and vector versions of the Ky Fan inequality [13] (called also the equilibrium problem in [6]). The optimality criteria in set optimization [23,24] are introduced by comparing values of objectives of set-valued optimization problems via set order relations defined in the image space of their objectives.…”

“…In the literature, the name "strong vector equilibrium problems" is used for several different problems. To avoid confusion, we recall the definitions of the strong vector equilibrium problems (P 1 ) and (P 2 ), that are mentioned in [28] and are used in this present paper.…”

“…[15]), under the assumption on the existence of a continuous strongly cone-monotonic function satisfying certain condition. It is shown in [28] that, if C is pointed, then existence results for both Problems (P i ), i = 1, 2, can be obtained from the existence results for the bifunction-set optimization Problem (u-BSOP) (see [28]), under assumptions quite different from the above ones. It is also shown in [28] that existence results for both Kuroiwa set optimization Problems (l-SOP) and (u-SOP) are immediate from those for Problem (u-BSOP).…”

“…It is shown in [28] that, if C is pointed, then existence results for both Problems (P i ), i = 1, 2, can be obtained from the existence results for the bifunction-set optimization Problem (u-BSOP) (see [28]), under assumptions quite different from the above ones. It is also shown in [28] that existence results for both Kuroiwa set optimization Problems (l-SOP) and (u-SOP) are immediate from those for Problem (u-BSOP). More importantly, the work [28] provides a new approach to the solution existence in set optimization, with tools and assumptions that are familiar in vector equilibrium problems, but are not used in all the papers [1,4,17,19,23,24] discussing the existence of solutions of Problems (l-SOP) and (u-SOP): KKMFan-Lemma [13], coercivity conditions, generalized convexity notions,...…”