Variants of the Ekeland variational principle for approximate proper solutions of vector equilibrium problems

Ekeland variational principles for vector equilibrium problems with solid ordering cones

“…Next, a recent lower semicontinuity concept for vector functions mapping to a finite dimensional space is recalled (see [15,Definition 3]). It was applied to the particular case when the convex cone H is solid and b ∈ intH (see, for instance, [15, Theorem 3]).…”

“…(2) Let q ∈ intD. Since intD ⊂ −sq + intD, for all s > 0, a stronger condition is C ∩ (sq − intD) = / 0, for some s > 0, which is equivalent to inf c∈C ϕ q D (c) > 0 (see part (a) of [15,Remark 6]).…”

“…Nowadays, these variational principles are a growing research line (see [1,4,6,10,14,15,19,20,21]) since they can be formulated as vector equilibrium problems, which encompass several really important problems, like vector optimization problems, vector variational inequalities and vector complementary problems (see [2,3,7,11,17] and the references therein). This work is motivated by the recent Ekeland variational principles in [15], where they are derived by considering the so-called Gerstewitz scalarization functional and a kind of proper approximate solutions of vector equilibrium problems. These solutions are just weak approximate solutions with respect to certain dilating cones.…”

Error bound analysis for vector equilibrium problems with partial order provided by a polyhedral cone

“…From a computational point of view, they showed that the results in [15,16] based on the ordering cone generated by some matrix are attractive. Using the partial order considered in [16], Hai et al [17] continued to investigating vector equilibrium problems with a polyhedral ordering cone. In [17], variants of the Ekeland variational principle for a type of approximate proper solutions of those vector equilibrium problems were also provided.…”

“…Using the partial order considered in [16], Hai et al [17] continued to investigating vector equilibrium problems with a polyhedral ordering cone. In [17], variants of the Ekeland variational principle for a type of approximate proper solutions of those vector equilibrium problems were also provided. However, to the best of our knowledge, up to now, there is no paper devoted to error bounds for vector equilibrium problems and vector network equilibrium problems, whose final space is finite dimensional partially ordered by a polyhedral cone generated by some matrix.…”