A new ABB theorem in normed vector spaces

Abstract

We extend the Arrow, Barankin and Blackwell (ABB) theorem for Henig efficient points for nonconvex sets in normed vector spaces. The novelty of our result is especially represented by the fact that we do not assume compactness of the set; in fact it can be an unbounded asymptotically compact set. Our result subsumes several generalizations of this important theorem.

Extrapolation algorithm for affine-convex feasibility problems

“…(v) S i = {x ∈ H | f i (x) ≤ 0}, where f i : H → R is a continuous convex function which is bounded on bounded sets and, for every n ∈ N, T i,pn is a subgradient projector onto S i (see (2.4) In practice, item (i) corresponds to the case when it is relatively easy to compute the best approximation to x from S i (see [3,20,30] for examples); item (ii) corresponds to the monotone inclusion problem 0 ∈ A i x, which arises in many applied mathematics problems [24,52,56]; item (iii) corresponds to equilibrium problems [8,25,39,46]; item (iv) corresponds to (firmly) nonexpansive fixed point problems [36,37] (recall that T is firmly nonexpansive if and only if T = 2T − Id is nonexpansive [37, Theorem 12.1], while Fix T = Fix T ); finally, item (v) corresponds to the inequality f i (x) ≤ 0, which arises in convex inequality systems [21,55] (note that, if dim H < +∞, then bounded sets are relatively compact and thus the boundedness condition on the function is always satisfied).…”

Vector Equilibrium Problems on Dense Sets

“…These vector equilibrium problems, much like their scalar counterpart, offer a unified framework for treating vector optimization, vector variational inequalities or cone saddle point problems, to name just a few [1,3,2,4,11,12].…”

Krasnoselski–mann Iteration for Hierarchical Fixed Points and Equilibrium Problem

“…Blum and Oettli [1,19] show that, in the case of a single equilibrium problem, the formulation (1.6) covers monotone inclusion problems, saddlepoint problems, VIPs, minimization problems, Nash equilibria in noncooperative games, vector equilibrium problems and certain fixed point problems (see [8]). It is also worth remarking that, in the case of VIP (1.1), the induced bifunction G(x, y) := (I − V )x, y − x satisfies the following condition.…”