Versions of Ekeland’s variational principle involving set perturbations

Variational principles in set optimization with domination structures and application to changing jobs

“…Inspired by Ha's work, Qiu [54] obtained an improvement of Ha's result by relaxing the lower boundedness condition imposed on objective maps. It was further extended in [35,55]. However, there has, to the best of our knowledge, no extensions/generalizations to set optimization with ordering structure.…”

A pre-order principle and set-valued Ekeland variational principle

“…The key to the proof of the general pre-order principle is to distinguish two different points by scalarizations. From the pre-order principle we obtained a very general set-valued Ekeland variational principle (briefly, denoted by EVP), which implies most of the known set-valued EVPs and their improvements, for example, Ha's EVP in [7], Qiu's EVP in [17], Bednarczuk and Zagrodny's EVP in [3], Gutiérrez, Jiménez and Novo's EVPs in [6], Tammer and Zȃlinescu's EVPs in [22], Flores-Bazán, Gutiérrez and Novo's EVPs in [4], Liu and Ng's EVPs in [13], Qiu's EVPs in [18], Khanh and Quy's EVPs in [11] and Bao and Mordukhovich's EVPs in [1,2]. However, it could not imply Khanh and Quy's EVPs in [10], where the perturbations contain weak τ -functions.…”

Ekeland Variational Principle for Generalized Vector Equilibrium Problems with Equivalences and Applications

“…An Ekeland variational principle [1] (also [2,3]) appeared first as an existence result of approximate minimizer for a lower semicontinuous and bounded below function on complete metric spaces. It subsequently developed an important tool of many subjects, such as in nonlinear analysis (e.g., [4]), optimization theory (e.g., [5][6][7][8][9][10][11]), game theory (e.g., [12]), dynamical systems (e.g., [13]), and others (e.g., [14,15]). By reason of the fact that equilibrium problems contain many problems as their special cases, such as optimization problems, fixed-point problems, variational inequality problems, complementary problems, and Nash equilibrium problems (see [16]), a new direction of research on variational principle for equilibrium problems has arisen.…”

“…The variational principle for equilibrium problems (e.g., [17]) and for vector equilibrium problems (e.g., [18][19][20][21][22][23]) and/or their applications or equivalent results were discussed. In terms of set-valued objective mappings, variational principle for vector optimization problems was first introduced by Chen and Huang [5] and was reported in many literatures (e.g., see [7,[9][10][11]) in the sequel. In 2009, Zeng and Li [24] discussed Ekeland variational principle for vector equilibrium problems with setvalued objective mappings (generalized vector equilibrium problems).…”