A network formulation of market equilibrium problems and variational inequalities

This paper shows that market equilibrium problems of production may generally be modelled as equilibrium flow problems in networks and that their equilibrium conditions can be visualized as a variational inequality. This connection would allow us to transplant directly elements of the well-developed theory of equilibrium flow in networks to the theory of market equilibrium.     

Modified golden ratio algorithms for pseudomonotone equilibrium problems and variational inequalities

Applications of Mathematics - We propose a modification of the golden ratio algorithm for solving pseudomonotone equilibrium problems with a Lipschitz-type condition in Hilbert spaces. A new...     

Some uncertainty inequalities

We prove an uncertainty inequality for the Fourier transform on the Heisenberg group analogous to the classical uncertainty inequality for the Euclidean Fourier transform. Inequalities of similar form are obtained for the Hermite and Laguerre expansions.     

Some generalized vector variational inequalities and complementarity problems for multivalued mappings

In this paper, we introduce and study a new class of generalized vector variational inequalities and complementarity problems for multivalued mappings. We prove the existence of solutions for this kind of vector variational inequality and discuss the relations between the solutions of the generalized vector variational inequalities and the solutions of generalized vector complementarity problems in Hausdorff topological vector spaces. Our results extend and improve some results in this field.     

Equilibrium problems and noncoercive variational inequalities

In this paper, we are interested to establish existence results for equilibrium problems in a noncoercive framework by using techniques of recession analysis. The abstract result is then applied to find solution of noncoercive monotone and pseudomonotone variational inequalities     

Some nonlinear parabolic variational inequalities

In this paper we study initial value problems for nonlinear parabolic variational inequalities involving time-dependent subdifferentials of convex functions on a Hilbert space. We shall show the existence of a solution by a semi-discretisation method with respect to the time.     

Retraction algorithms for solving variational inequalities,pseudomonotone equilibrium problems,and fixed-point problems in Banach spaces

In this paper, using sunny generalized nonexpansive retractions which are different from the metric projection and generalized metric projection in Banach spaces, we present new extragradient and line search algorithms for finding the solution of a J-variational inequality whose constraint set is the common elements of the set of fixed points of a family of generalized nonexpansive mappings and the set of solutions of a pseudomonotone J-equilibrium problem for a J -α-inverse-strongly monotone operator in a Banach space. To prove strong convergence of generated iterates in the extragradient method, we introduce a ? _?-Lipschitz-type condition and assume that the equilibrium bifunction satisfies this condition. This condition is unnecessary when the line search method is used instead of the extragradient method. Using FMINCON optimization toolbox in MATLAB, we give some numerical examples and compare them with several existence results in literature to illustrate the usability of our results.     

Inverse problems for multi-valued quasi variational inequalities and noncoercive variational inequalities with noisy data

ABSTRACT

We study the inverse problem of identifying a variable parameter in variational and quasi-variational inequalities. We consider a quasi-variational inequality involving a multi-valued monotone map and give a new existence result. We then formulate the inverse problem as an optimization problem and prove its solvability. We also conduct a thorough study of the inverse problem of parameter identification in noncoercive variational inequalities which appear commonly in applied models. We study the inverse problem by posing optimization problems using the output least-squares and the modified output least-squares. Using regularization, penalization, and smoothing, we obtain a single-valued parameter-to-selection map and study its differentiability. We consider optimization problems using the output least-squares and the modified output least-squares for the regularized, penalized and smoothened variational inequality. We give existence results, convergence analysis, and optimality conditions. We provide applications and numerical examples to justify the proposed framework.     

Bifurcation problems for discrete variational inequalities

The buckling of a beam or a plate which is subject to obstacles is typical for the variational inequalities that are considered here. Birfurcation is known to occur from the first eigenvalue of the linearized problem. For a discretization the bifurcation point and the bifurcating branches may be obtained by solving a constrained optimization problem. An algorithm is proposed and its convergence is proved. The buckling of a clamped beam subject to point obstacles is considered in the continuous case and some numerical results for this problem are presented.     

Regularity results for evolutionary nonlinear variational and quasi-variational inequalities with applications to dynamic equilibrium problems

The aim of this paper is to obtain the continuity of solutions to time-dependent nonlinear variational and quasi-variational inequalities which express many dynamic equilibrium problems. To prove our results, we make use of Minty’s Lemma and of the notion of the Mosco’s convergence.     

Iterative methods for solving equilibrium problems, variational inequalities and fixed points of nonexpansive semigroups

In this work, strong convergence theorems by the viscosity approximation method associated with Meir–Keeler contractions are established for solving fixed point problems of a nonexpansive semigroup, a system of equilibrium problems and variational inequality problems in a real Hilbert space. Further, applications related to commutative semigroup are obtained.     

Revisiting the construction of gap functions for variational inequalities and equilibrium problems via conjugate duality

Based on conjugate duality we construct several gap functions for general variational inequalities and equilibrium problems, in the formulation of which a so-called perturbation function is used. These functions are written with the help of the Fenchel-Moreau conjugate of the functions involved. In case we are working in the convex setting and a regularity condition is fulfilled, these functions become gap functions. The techniques used are the ones considered in [Altangerel L., Bo? R.I., Wanka G., On gap functions for equilibrium problems via Fenchel duality, Pac. J. Optim., 2006, 2(3), 667–678] and [Altangerel L., Bo? R.I., Wanka G., On the construction of gap functions for variational inequalities via conjugate duality, Asia-Pac. J. Oper. Res., 2007, 24(3), 353–371]. By particularizing the perturbation function we rediscover several gap functions from the literature. We also characterize the solutions of various variational inequalities and equilibrium problems by means of the properties of the convex subdifferential. In case no regularity condition is fulfilled, we deliver also necessary and sufficient sequential characterizations for these solutions. Several examples are illustrating the theoretical aspects.     

Some open questions about variational inequalities

We discuss some open questions especially concerning the coincidence set and the curve of separation in two-dimensional variational inequalities.     

Some aspects of elliptic variational inequalities

In this paper we study an existence and the approximation of the solution of the elliptic variational inequality from an abstract axiomatic point of view. We discuss convergence results and give an error estimate for the difference of the two solutions in an appropriate norm. Also, we present some computational results by using fixed point method.     

Strong convergence for maximal monotone operators, relatively quasi-nonexpansive mappings, variational inequalities and equilibrium problems

In this paper, we introduce a new hybrid iterative scheme for finding a common element of the set of zeroes of a maximal monotone operator, the set of fixed points of a relatively quasi-nonexpansive mapping, the sets of solutions of an equilibrium problem and the variational inequality problem in Banach spaces. As applications, we apply our results to obtain strong convergence theorems for a maximal monotone operator and quasi-nonexpansive mappings in Hilbert spaces and we consider a problem of finding a minimizer of a convex function.     

Brunn-Minkowski inequalities for variational functionals and related problems

Recently, several inequalities of Brunn-Minkowski type have been proved for well-known functionals in the Calculus of Variations, e.g. the first eigenvalue of the Laplacian, the Newton capacity, the torsional rigidity and generalizations of these examples. In this paper, we add new contributions to this topic: in particular, we establish equality conditions in the case of the first eigenvalue of the Laplacian and of the torsional rigidity, and we prove a Brunn-Minkowski inequality for another class of variational functionals. Moreover, we describe the links between Brunn-Minkowski type inequalities and the resolution of Minkowski type problems.     

Monotone generalized variational inequalities and generalized complementarity problems

Some existence results for generalized variational inequalities and generalized complementarity problems involving quasimonotone and pseudomonotone set-valued mappings in reflexive Banach spaces are proved. In particular, some known results for nonlinear variational inequalities and complementarity problems in finite-dimensional and infinite-dimensional Hilbert spaces are generalized to quasimonotone and pseudomonotone set-valued mappings and reflexive Banach spaces. Application to a class of generalized nonlinear complementarity problems studied as mathematical models for mechanical problems is given.The research of the first author was supported by the National Natural Science Foundation of P. R. China and by the Ethel Raybould Fellowship, University of Queensland, St. Lucia, Brisbane, Australia.     

Jumping problems for quasilinear elliptic variational inequalities

We investigate the existence and multiplicity of solutions for a certain class of quasilinear variational inequalities, whenever between the obstacle and the behavior at +￥+\infty of the nonlinearity there is a situation of jumping type.     

Reinforcement problems for elliptic equations and variational inequalities

Summary Consider the Dirichlet problem for an elliptic equation in a domain , with coefficients having discontinuity on a surface . Suppose divides into _{1 2}(₂ the inner core), the thickness of ₁ is of order of magnitude , and the modulus of ellipticity in ₁ is of order magnitude ₁. The asymptotic behavior of the solution is studied as 0, ₁ 0, provided lim (/₁) exists. Other questions of this type are studied both for elliptic equations and for elliptic variational inequalities.The second author is partially supported by National Science Foundation Grant 7406375 A01. The third author is partially supported by National Science Foundation Grant MC575-21416 A01.     

On variational inequalities for auction market problems

We give an equivalent variational inequality formulation for a general class of equilibrium problems based upon auction decision rules. We show that a general relaxation iterative process with conditional gradient extrapolation ensures convergence to a solution under rather mild assumptions.