Gap functions and error bounds for weak vector variational inequalities

In this article, by using the image space analysis, a gap function for weak vector variational inequalities is obtained. Its lower semicontinuity is also discussed. Then, these results are applied to obtain the error bounds for weak vector variational inequalities. These bounds provide effective estimated distances between a feasible point and the solution set of the weak vector variational inequalities.     

Gap functions and global error bounds for set-valued variational inequalities

The set-valued variational inequality problem is very useful in economics theory and nonsmooth optimization. In this paper, we introduce some gap functions for set-valued variational inequality problems under suitable assumptions. By using these gap functions we derive global error bounds for the solution of the set-valued variational inequality problems. Our results not only generalize the previously known results for classical variational inequalities from single-valued case to set-valued, but also present a way to construct gap functions and derive global error bounds for set-valued variational inequality problems.     

Gap functions and error bounds for generalized vector variational inequalities

This paper deals with generalized vector variational inequalities. Without any scalarization approach, the gap functions and their regularized versions for generalized vector variational inequalities are first obtained. Then, in the absence of the projection operator method, some error bounds for generalized vector variational inequalities are established in terms of these regularized gap functions. Further, the results obtained in this paper are more simpler from the computational view.     

Merit functions and error bounds for generalized variational inequalities

We consider the generalized variational inequality and construct certain merit functions associated with this problem. In particular, those merit functions are everywhere nonnegative and their zero-sets are precisely solutions of the variational inequality. We further use those functions to obtain error bounds, i.e., upper estimates for the distance to solutions of the problem.     

Gap functions for vector variational inequalities

In this article, we intend to study several scalar-valued gap functions for Stampacchia and Minty-type vector variational inequalities. We first introduce gap functions based on a scalarization technique and then develop a gap function without any scalarizing parameter. We then develop its regularized version and under mild conditions develop an error bound for vector variational inequalities with strongly monotone data. Further, we introduce the notion of a partial gap function which satisfies all, but one of the properties of the usual gap function. However, the partial gap function is convex and we provide upper and lower estimates of its directional derivative.     

Merit functions and error bounds for constrained mixed set-valued variational inequalities via generalized f-projection operators

In this paper, we introduce and investigate a constrained mixed set-valued variational inequality (MSVI) in Hilbert spaces. We prove the solution set of the constrained MSVI is a singleton under strict monotonicity. We also propose four merit functions for the constrained MSVI, that is, the natural residual, gap function, regularized gap function and D-gap function. We further use these functions to obtain error bounds, i.e. upper estimates for the distance to solutions of the constrained MSVI under strong monotonicity and Lipschitz continuity. The approach exploited in this paper is based on the generalized f-projection operator due to Wu and Huang, but not the well-known proximal mapping.     

Gap functions for constrained vector variational inequalities with applications

In this paper, the image space analysis is applied to investigate scalar-valued gap functions and their applications for a (parametric)-constrained vector variational inequality. Firstly, using a non-linear regular weak separation function in image space, a gap function of a constrained vector variational inequality is obtained without any assumptions. Then, as an application of the gap function, two error bounds for the constrained vector variational inequality are derived by means of the gap function under some mild assumptions. Further, a parametric gap function of a parametric constrained vector variational inequality is presented. As an application of the parametric gap function, a sufficient condition for the continuity of the solution map of the parametric constrained vector variational inequality is established within the continuity and strict convexity of the parametric gap function. These assumptions do not include any information on the solution set of the parametric constrained vector variational inequality.     

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.     

Generalized solutions of quasi variational inequalities

This paper deals with multivalued quasi variational inequalities with pseudo-monotone and monotone maps. The primary objective of this work is to show that the notion of generalized solutions can be employed to investigate multivalued pseudo-monotone quasi variational inequalities. It is a well-known fact that a quasi variational inequality can conveniently be posed as a fixed point problem through the so-called variational selection. For pseudo-monotone maps, the associated variational selection is a nonconvex map, and the fixed point theorems can only be applied under restrictive assumptions on the data of quasi variational inequalities. On the other hand, the generalized solutions are defined by posing a minimization problem which can be solved by a variant of classical Weierstrass theorem. It turns out that far less restrictive assumptions on the data are needed in this case. To emphasis on the strong difference between a classical solution and a generalized solution, we also give a new existence theorem for quasi variational inequalities with monotone maps. The main existence result is proved under a milder coercivity condition. We also relax a few other conditions from the monotone map. Due to its flexibility, it seems that the notion of generalized solutions can be employed to study quasi variational inequalities for other classes of maps as well.     

Gap functions and error bounds for nonsmooth convex vector optimization problem

Abstract

In this article, our main aim is to develop gap functions and error bounds for a (non-smooth) convex vector optimization problem. We show that by focusing on convexity we are able to quite efficiently compute the gap functions and try to gain insight about the structure of set of weak Pareto minimizers by viewing its graph. We will discuss several properties of gap functions and develop error bounds when the data are strongly convex. We also compare our results with some recent results on weak vector variational inequalities with set-valued maps, and also argue as to why we focus on the convex case.     

Lagrange multipliers, (exact) regularization and error bounds for monotone variational inequalities

We examine two central regularization strategies for monotone variational inequalities, the first a direct regularization of the operative monotone mapping, and the second via regularization of the associated dual gap function. A key link in the relationship between the solution sets to these various regularized problems is the idea of exact regularization, which, in turn, is fundamentally associated with the existence of Lagrange multipliers for the regularized variational inequality. A regularization is said to be exact if a solution to the regularized problem is a solution to the unregularized problem for all parameters beyond a certain value. The Lagrange multipliers corresponding to a particular regularization of a variational inequality, on the other hand, are defined via the dual gap function. Our analysis suggests various conceptual, iteratively regularized numerical schemes, for which we provide error bounds, and hence stopping criteria, under the additional assumption that the solution set to the unregularized problem is what we call weakly sharp of order greater than one.     

On a class of quasi variational inequalities

In this paper, we introduce and study a new class of quasi variational inequalities. Using essentially the projection technique and its variant forms, we establish the equivalence between generalized nonlinear quasi variational inequalities and the fixed point problems. This equivalence is then used to suggest and analyze a number of new iterative algorithms. These new results include the corresponding known results for generalized quasi variational inequalities as special cases.     

Optimal control of multivalued quasi variational inequalities

Optimal control of various variational problems has been an area of active research. On the other hand, in recent years many important models in mechanics and economics have been formulated as multi-valued quasi variational inequalities. The primary objective of this work is to study optimal control of the general nonlinear problems of this type. Under suitable conditions, we ensure the existence of an optimal control for a quasi variational inequality with multivalued pseudo-monotone maps. Convergence behavior of the control is studied when the data for the state quasi variational inequality is contaminated by some noise. Some possible applications are discussed.     

Generalized set-valued mixed nonlinear quasi variational inequalities

In this paper we introduce and study a number of new classes of quasi variational inequalities. Using essentially the projection technique and its variant forms we prove that the generalized set-valued mixed quasivariational inequalities are equivalent to the fixed point problem and the Wiener-Hopf equations (normal maps). This equivalence enables us to suggest a number of iterative algorithms for solving the generalized variational inequalities. As a special case of the generalized set-valued mixed quasi variational inequalities, we obtain a class of quasi variational inequalities studied by Siddiqi, Husain and Kazmi [35], but there are several inaccuracies in their formulation of the problem, the statement and the proofs of their results. We have removed these inaccuracies. The correct formulation of their results can be obtained as special cases from our main results.     

Gradient bounds for solutions of degenerate variational inequalities

We note here that a technique of Oleinik for obtaining derivative estimates for degenerate second-order elliptic equations applies even to certain degenerate variational inequalities.Supported in part by NSF Grant 77-01952; Alfred P. Sloan Fellow 1979–81.     

Nonconvex functions and variational inequalities

In this paper, we study some properties of a class of nonconvex functions, called semipreinvex functions, which includes the classes of preinvex functions and arc-connected convex functions. It is shown that the minimum of an arcwise directionally differentiable semi-invex functions on a semi-invex set can be characterized by a class of variational inequalities, known as variational-like inequalities. We use the auxiliary principle technique to prove the existence of a solution of a variational-like inequality and suggest a novel iterative algorithm.     

Merit functions for general variational inequalities

In this paper, we consider some classes of merit functions for general variational inequalities. Using these functions, we obtain error bounds for the solution of general variational inequalities under some mild conditions. Since the general variational inequalities include variational inequalities, quasivariational inequalities and complementarity problems as special cases, results proved in this paper hold for these problems. In this respect, results obtained in this paper represent a refinement of previously known results for classical variational inequalities.     

An iterative scheme for a class of quasi variational inequalities

An iterative scheme is given to obtain the approximate solution of a class of quasi variational inequalities. It is shown that the approximate solution obtained by the iterative scheme converges strongly in the Hilbert space to the exact solution. As a special case, we obtain the corresponding iterative scheme for variational inequalities.