Optimality Conditions for Vector Equilibrium Problems with Applications

Abstract

We use asymptotic analysis for studying noncoercive pseudomonotone equilibrium problems and vector equilibrium problems. We introduce suitable notions of asymptotic functions, which provide sufficient conditions for the set of solutions of these problems to be nonempty and compact under quasiconvexity of the objective function. We characterize the efficient and weakly efficient solution set for the nonconvex vector equilibrium problem via scalarization. A sufficient condition for the closedness of the image of a nonempty, closed and convex set via a quasiconvex vector-valued function is given. Finally, applications to the quadratic fractional programming problem are also presented.

Solving Mixed Variational Inequalities Beyond Convexity

“…This conflict is restricted to nonconvex functions, since if h is convex (3.1) has a solution on any compact and convex set K . Existence results for the nonconvex and noncoercive setting can be found in [19][20][21], in which this conflict appears in a theoretical assumption called mixed variational inequality property (see [42,Definition 3.1]).…”

“…Note also that the generalized monotonicity condition (A3) can be further weakened to [27,Equation (71)] without losing the convergence of the algorithm proposed below. (ii) Results guaranteeing existence of solutions to mixed variational inequalities beyond convexity may be found in [19][20][21].…”

“…The first motivation behind this study comes from [19,20], where existence results for mixed variational inequalities involving quasiconvex functions were provided, hence the question of numerically solving such problems arose naturally. As far as we know in the literature one can find only iterative methods for solving mixed variational inequalities without convexity assumptions where the involved functions are taken continuous in works like [32][33][34], however these algorithms are either implicitly formulated or merely conceptual schemes, no implementations of them being known to us.…”

Approximate Benson efficient solutions for set-valued equilibrium problems

“…It is widely used in investment decision-making, quantitative economy, optimal control, and engineering technology. Because of the universality and unity of the problems involved and the profundity of solving them, vector equilibrium has become a hot issue in the field of nonlinear analysis and operational research [1][2][3][4][5][6]. In Banach spaces, Feng et al [1] established Kuhn-Tucker-like conditions for weakly efficient solutions of vector equilibrium problems with constraints by using the Gerstewitz's functional, and obtained sufficient conditions of weakly efficient solutions under the assumption of generalized invexity.…”

“…Under the assumption of cone-convexity, Gong [4] obtained necessary and sufficient optimality conditions for several efficient solutions to constrained vector equilibrium problems. By using asymptotic analysis, Iusem et al [5] studied vector equilibrium problems and noncoercive pseudomonotone equilibrium problems.…”