Vector Equilibrium Problems Research Articles

Weak set-equilibrium problems: existence and stability conditions

We formulate four types of set-equilibrium problems based on some well-known set-order relations. Then, we study the existence of solutions for these problems by imposing standard generalized convexity and Lipschitz conditions. A comparison with some widely studied vector equilibrium problems is performed. Finally, we are interested in an issue concerning the stability of solutions of some related approximate set-optimization problems.

Multilevel interpolations for the generalized Nikishin system on a tree graph

We study a multilevel interpolation problem for a system of Markov functions defined by a tree graph. The normality of all indices has been proven. Asymptotic properties are studied in terms of the vector equilibrium problem for logarithmic potential.

Efficiency Conditions for Nonsmooth Vector Equilibrium Problems with Constraints via Generalized Subdifferentials

Efficiency Conditions for Nonsmooth Vector Equilibrium Problems with Constraints via Generalized Subdifferentials

Directional Compactness, Approximations and Efficiency Conditions for Nonsmooth Vector Equilibrium Problems with Constraints

Directional Compactness, Approximations and Efficiency Conditions for Nonsmooth Vector Equilibrium Problems with Constraints

Vector Equilibrium Problems—A Unified Approach and Applications

We present existing results and properties for the solutions of some vector equilibrium problems with set-valued functions in the case of a vector space ordered by a cone with some “interiority” properties. Some applications concerning the existence of equilibrium for abstract economies and vector optimization problems are given.

Strict Vector Equilibrium Problems of Multi-Product Supply–Demand Networks with Capacity Constraints and Uncertain Demands

This paper considers a multi-product, multi-criteria supply–demand network equilibrium model with capacity constraints and uncertain demands. Strict network equilibrium principles are proposed both in the case of a single criterion and multi-criteria, respectively. Based on a single criterion, it proves that strict network equilibrium flows are equivalent to vector variational inequalities, and the existence of strict network equilibrium flows is derived by virtue of the Fan–Browder fixed point theorem. Based on multi-criteria, the scalarization of strict network equilibrium flows is given by using Gerstewitz’s function without any convexity assumptions. Meanwhile, the necessary and sufficient conditions of strict network equilibrium flows are derived in terms of vector variational inequalities. Finally, an example is given to illustrate the application of the derived theoretical results.

Higher-order KKT optimality conditions through contingent derivatives for constrained nonsmooth vector equilibrium problems

In this paper, we deal with some higher-order optimality conditions for local strict efficient solutions to a nonsmooth vector equilibrium problem with set, cone and equality constraints. For this aim, the concept of m−stable and m−steady functions (m≥2 and integer) for single-valued functions and some constraint qualifications of higher order in terms of contingent derivatives are proposed accordingly. We analyze the sum calculus rule of mth-order adjacent set, mth-order interior set, asymptotic mth-order tangent cone and asymptotic mth-order adjacent cone. Subsequently, we employ the obtained calculus rules to treat KKT necessary and sufficient optimality conditions of higher order in terms of contingent derivatives for the mth-order (local) strict efficient solutions to such problem. Simultaneously, we employ these rules to study KKT higher-order optimality conditions for such efficient solutions for a nonsmooth vector optimization problem with constraints. Some illustrative examples are provided to demonstrate the main results of the literature.

Continuity of the solution set mappings to parametric unified weak vector equilibrium problems via free-disposal sets

In this paper, the lower and upper semicontinuity of the solution mappings for parametric unified weak vector equilibrium problem (PUWVEP) are established, via freedisposal set and nonlinear scalarization function. Moreover, example is given to illustrate the obtained results. The results improve the corresponding ones in the literature [12-14].

Optimality and scalarization of approximate solutions for vector equilibrium problems via Michel-Penot subdifferential

Optimality and scalarization of approximate solutions for vector equilibrium problems via Michel-Penot subdifferential

D-gap functions and error bounds for vector equilibrium problems

D-gap functions and error bounds for vector equilibrium problems

A New Existence Result of Equilibria for Vector Equilibrium Problems

In this paper, we deal with vector equilibrium problems. We prove the nonemptiness of the solution set for this type of problems in the sequential compactness case and in the absence of convexity and lower semicontinuity assumptions. Some examples are presented and an existence result for countable systems of vector equilibrium problems is stated.

Equilibrium problems when the equilibrium condition is missing

Given a nonempty convex subset X of a topological vector space and a real bifunction f defined on X times X, the associated equilibrium problem consists in finding a point x_0 in X such that f(x_0, y) ge 0, for all y in X. A standard condition in equilibrium problems is that the values of f to be nonnegative on the diagonal of X times X. In this paper, we deal with equilibrium problems in which this condition is missing. For this purpose, we will need to consider, besides the function f, another one g: X times X rightarrow mathbb {R}, the two bifunctions being linked by a certain compatibility condition. Applications to variational inequality problems, quasiequilibrium problems and vector equilibrium problems are given.

Well-posedness for bilevel vector equilibrium problems with variable domination structures

Abstract In this article, well-posedness for two types of bilevel vector equilibrium problems with variable domination structures are introduced and studied. With the help of cosmically upper continuity or Hausdorff upper semi-continuity for variable domination structures, sufficient and necessary conditions are given for such problems to be Levitin-Polyak (LP) well-posed and LP well-posedness in the generalized sense. As variable domination structure is a valid generalization of fixed one, the main results obtained in this article extend and develop some recent works in the literature.

Existence and stability of generalized weakly-mixed vector equilibrium problems

Existence and stability of generalized weakly-mixed vector equilibrium problems

Electrostatic Partners and Zeros of Orthogonal and Multiple Orthogonal Polynomials

For a given polynomial P with simple zeros, and a given semiclassical weight w, we present a construction that yields a linear second-order differential equation (ODE), and in consequence, an electrostatic model for zeros of P. The coefficients of this ODE are written in terms of a dual polynomial that we call the electrostatic partner of P. This construction is absolutely general and can be carried out for any polynomial with simple zeros and any semiclassical weight on the complex plane. An additional assumption of quasi-orthogonality of P with respect to w allows us to give more precise bounds on the degree of the electrostatic partner. In the case of orthogonal and quasi-orthogonal polynomials, we recover some of the known results and generalize others. Additionally, for the Hermite–Padé or multiple orthogonal polynomials of type II, this approach yields a system of linear second-order differential equations, from which we derive an electrostatic interpretation of their zeros in terms of a vector equilibrium. More detailed results are obtained in the special cases of Angelesco, Nikishin, and generalized Nikishin systems. We also discuss the discrete-to-continuous transition of these models in the asymptotic regime, as the number of zeros tends to infinity, into the known vector equilibrium problems. Finally, we discuss how the system of obtained second-order ODEs yields a third-order differential equation for these polynomials, well described in the literature. We finish the paper by presenting several illustrative examples.

Necessary Optimality Conditions for Second-Order Local Strict Efficiency for Constrained Nonsmooth Vector Equilibrium Problems

Necessary Optimality Conditions for Second-Order Local Strict Efficiency for Constrained Nonsmooth Vector Equilibrium Problems

Iterative methods for vector equilibrium and fixed point problems in Hilbert spaces

In this paper, algorithms for finding common solutions of strong vector equilibrium and fixed point problems of multivalued mappings are considered. First, a Minty vector equilibrium problem is introduced and the relationship between the Minty vector equilibrium problem and the strong equilibrium problem is discussed. Then, by applying the Minty vector equilibrium problem, projection iterative methods are proposed and some convergence results are established in Hilbert spaces. The main results obtained in this paper develop and improve some recent works in this field.

Second-Order Optimality Conditions for Strict Pareto Minima and Weak Efficiency for Nonsmooth Constrained Vector Equilibrium Problems

In this article, some types of lower and upper second-order strictly pseudoconvexity are provided for establishing sufficient conditions for the second-order strict local Pareto minima of nonsmooth vector equilibrium problem with set, inequality and equality constraints. Based on the notion of Gerstewitz mappings, some Kuhn-Tucker-type multiplier rules for the strict local Pareto minima of such problem are obtained. We also construct the second-order constraint qualification in terms of first- and second-order directional derivatives of the (CQ) and (CQ1) types. Using this constraint qualifications, some second-order primal and dual necessary optimality conditions in terms of second-order upper and lower Dini directional derivatives for such minima are derived. Under suitable assumptions on the lower and upper strictly pseudoconvexity of order two of objective and constraint functions, second-order necessary optimality conditions become sufficient optimality conditions to such problem. Some illustrative examples are also given for our findings.

Ekeland variational principles for vector equilibrium problems

This work concerns Ekeland variational principles for scalar and vector cyclically antimonotone bifunctions on complete metric spaces. The scalar results work for extended bifunctions and they are obtained by a generalized version of the Dancs–Hegedüs–Medvegyev's fixed point theorem. As a result, weaker lower-semicontinuity assumptions have been considered, that generalize the concept of strictly decreasingly lower-semicontinuous real-valued function. The vector results are derived from the previous ones by a scalarization approach and are based on new notions of cyclical antimonotonicity, lower boundedness and strictly decreasingly lower-semicontinuity for vector bifunctions. Several results in the literature are improved since they are stated by weaker assumptions.

Regularization of vector equilibrium problems

Regularization of vector equilibrium problems