Multivalued Variational Inequalities Research Articles

Levitin–Polyak well-posedness of split multivalued variational inequalities

ABSTRACT We introduce and study the split multivalued variational inequality problem (SMVIP) and the parametric SMVIP. We examine, in particular, Levitin–Polyak well-posedness of SMVIPs and parametric SMVIPs in Hilbert spaces. We provide several examples to illustrate our theoretical results. We also discuss several important special cases.

New inertial proximal algorithms for solving multivalued variational inequalities

New inertial proximal algorithms for solving multivalued variational inequalities

On a multivalued prescribed mean curvature problem and inclusions defined on dual spaces

This article addresses two main objectives. First, it establishes a functional analytic framework and presents existence results for a quasilinear inclusion describing a prescribed mean curvature problem with homogeneous Dirichlet boundary conditions, involving a multivalued lower order term. The formulation of the problem is done in the space of functions with bounded variation. The second objective is to introduce a general existence theory for inclusions defined on nonreflexive Banach spaces, which is specifically applicable to the aforementioned prescribed mean curvature problem. This problem can be formulated as a multivalued variational inequality in the space of functions with bounded variation, which, under suitable conditions, is equivalent to an inclusion involving a maximal monotone mapping of type (D) and a generalized pseudomonotone mapping. We prove an abstract existence theorem for inclusions of this form, under some coercivity conditions involving both the maximal monotone and the generalized pseudomonotone mappings.

An Alternated Inertial Projection Algorithm for Multi-Valued Variational Inequality and Fixed Point Problems

In this paper, we propose an alternated inertial projection algorithm for solving multi-valued variational inequality problem and fixed point problem of demi-contractive mapping. On one hand, this algorithm only requires the mapping is pseudo-monotone. On the other hand, this algorithm is combined with the alternated inertial method to accelerate the convergence speed. The global convergence of the algorithm can be obtained under mild conditions. Preliminary numerical results show that the convergence speed of our algorithm is faster than some existing algorithms.

A modified extragradient method for multi-valued variational inequality problems

In this paper, we consider a class of multi-valued variational inequality problems in which the involving function is continuous and pseudomonotone. This problem has a broad applications in equilibrium and optimization. In order to reduce the computation, we present a modified extragradient algorithm for solving multi-valued variational inequalities in which each iteration is obtained as the projection onto the intersection of the feasible set and the hyperplane containing the solution set. Moreover, we prove the sequence generated by the proposed algorithm is globally convergence with requiring the involving function being continuous and pseudomonitone on the feasible set. Finally, we give several numerical experiments to demonstrate the efficiency of the proposed algorithm.

Weak convergence of inertial proximal algorithms with self adaptive stepsize for solving multivalued variational inequalities

In this work, we introduce an inertial proximal algorithm for solving multivalued variational inequality problems in a real Hilbert space. By using self-adaptive and inertial techniques via proximal operators, we establish the weak convergence of the iteration sequences generated by these algorithms when the multivalued cost mappings associated with the problems are monotone and Lipschitz continuous. Moreover, we present the nonasymptotic O ( 1 k ) convergence rate of the proposed algorithms. We also provide some numerical examples to illustrate the accuracy and efficiency of our algorithms by comparing with other recent algorithms in the literature.

A study of multivalued variational inequalities via horizon maps and graphical convergence

Multivalued variational inequalities is an active research area that encompasses many important models in mathematics, for example in optimization, fixed point, saddle point and complementary problems. The primary objective of this work is to apply horizon maps and graphical convergence to obtain existence and stability results for these problems. We give an exhaustive list of new results for these problems in terms of horizon maps and graphical convergence. Finally, we apply these results to the pseudomonotone and maximal monotone cases, establishing new formulations of known results in terms of these tools.

An inertial Popov extragradient projection algorithm for solving multi-valued variational inequality problems

In this paper, we present an inertial Popov extragradient projection algorithm for solving multi-valued variational inequality problems in finite-dimensional Euclidean space. The global convergence of this algorithm is proved whenever the underlying mapping is Lipschitz continuous and pseudomonotone on the feasible set. Numerical experiments show that new algorithm is more efficient than algorithm of Ye [An improved projection method for solving generalized variational inequality problems. Optimization. 2018;67:1–11] whenever the underlying mapping is Lipschitz continuous. Here inertial technique can accelerate extragradient algorithm although the underlying mapping is multi-valued.

A new algorithm for solving strongly monotone equilibrium and multivalued variational inequality problems

A new algorithm for solving strongly monotone equilibrium and multivalued variational inequality problems

Multi-valued variational inequalities in unbounded domains: existence, comparison and extremal solutions

In this paper, we study multi-valued quasilinear elliptic variational inequalities of the form in all of as well as in the exterior domain , where is the p-Laplacian, K is a closed convex subset of the Beppo-Levi space with 1<p<N, or with p = N, and is the indicator functional corresponding to K with its subdifferential . The lower order multi-valued operator is generated by a multi-valued, upper semicontinuous function , and the measurable coefficient a is supposed to decay like: Our main goals are as follows. First, we provide a new approach to an existence theory for the above multi-valued variational inequalities in all for 1<p<N as well as in the exterior domain Ω for the borderline case . Second, we establish an enclosure and comparison principle based on appropriately defined sub-supersolutions, and prove the existence of extremal solutions. Third, by means of the sub-supersolution method provided here, we show that certain rather general classes of variational-hemivariational type inequalities turn out to be only subclasses of the above class of multi-valued elliptic variational inequalities. Finally, we apply the abstract theory to a multi-valued obstacle problem.

Strong well-posedness for a class of split variational inequalities with set-valued maps

In this paper, we present a generalization of well-posedness for a system of split multi-valued variational inequalities with set-valued maps and establish a metric characterization for them. Moreover, we show that the strong well-posedness is equivalent to the existence and uniqueness of solution for a split multi-valued variational inequality../files/site1/files/72/11Abstract.pdf

A subgradient proximal method for solving a class of monotone multivalued variational inequality problems

It is well known that the algorithms with using a proximal operator can be not convergent for monotone variational inequality problems in the general case. Malitsky (Optim. Methods Softw. 33 (1) 140–164, ??) proposed a proximal extrapolated gradient algorithm ensuring convergence for the problems, where the constraints are a finite-dimensional vector space. Based on this proximal extrapolated gradient techniques, we propose a new subgradient proximal iteration method for solving monotone multivalued variational inequality problems with the closed convex constraint. At each iteration, two strongly convex subprograms are required to solve separately by using proximal operators. Then, the algorithm is convergent for monotone and Lipschitz continuous cost mapping. We also use the proposed algorithm to solve a jointly constrained Cournot-Nash equilibirum model. Some numerical experiment and comparison results for convex nonlinear programming confirm efficiency of the proposed modification.

New Inertial Projection Methods for Solving Multivalued Variational Inequality Problems Beyond Monotonicity

In this paper, we present two new inertial projection-type methods for solving multivalued variational inequality problems in finite-dimensional spaces. We establish the convergence of the sequence generated by these methods when the multivalued mapping associated with the problem is only required to be locally bounded without any monotonicity assumption. Furthermore, the inertial techniques that we employ in this paper are quite different from the ones used in most papers. Moreover, based on the weaker assumptions on the inertial factor in our methods, we derive several special cases of our methods. Finally, we present some experimental results to illustrate the profits that we gain by introducing the inertial extrapolation steps.

Strong well-posedness of a system of split variational inequalities

In this paper, we consider a generalization of well-posedness for a class of split multi-valued variational inequalities and establish some metric characterizations for them. Moreover, we show that the existence and uniqueness of solution for a split multi-valued variational inequality is equivalent to the strong well-posedness.

Halpern projection methods for solving pseudomonotone multivalued variational inequalities in Hilbert spaces

In this paper, we introduce new approximate projection and proximal algorithms for solving multivalued variational inequalities involving pseudomonotone and Lipschitz continuous multivalued cost mappings in a real Hilbert space. The first proposed algorithm combines the approximate projection method with the Halpern iteration technique. The second one is an extension of the Halpern projection method to variational inequalities by using proximal operators. The strongly convergent theorems are established under standard assumptions imposed on cost mappings. Finally we introduce a new and interesting example to the multivalued cost mapping, and show its pseudomontone and Lipschitz continuous properties. We also present some numerical experiments to illustrate the behavior of the proposed algorithms.

Projection-type methods with alternating inertial steps for solving multivalued variational inequalities beyond monotonicity

Projection-type methods with alternating inertial steps for solving multivalued variational inequalities beyond monotonicity

Modified Projection Methods for Solving Multi-valued Variational Inequality without Monotonicity

In this paper, we propose two new projection-type algorithms for solving the multi-valued variational inequality in finite dimensional spaces. We prove the convergence of the sequences generated by the proposed projection-type algorithms without any monotonicity. Moreover, we provide some numerical experiments to illustrate the efficiency of the proposed projection-type algorithms.

A new algorithm for solving multi-valued variational inequality problems

In this paper, we present a new algorithm for solving multi-valued variational inequality problems, which combines the subgradient extragradient algorithm with inertial algorithm. We prove that the algorithm is globally convergent when the multi-valued mapping is continuous and pseudomonotone with nonempty compact convex values. And the convergence rate of this algorithm is Q-linear convergence.

Extremal solutions of multi-valued variational inequalities in plane exterior domains

Let Ω=R2∖B(0,1)‾ be the exterior of the closed unit disc in the plane. In this paper we prove existence and enclosure results of multi-valued variational inequalities in Ω of the form: Find u∈K and η∈F(u) such that〈−Δu,v−u〉≥〈aη,v−u〉,∀v∈K, where K is a closed convex subset of the Hilbert space X=D1,20(Ω) which is the completion of Cc∞(Ω) with respect to the ‖∇⋅‖2,Ω-norm. The lower order multi-valued operator F is generated by an upper semicontinuous multi-valued function f:R→2R∖{∅}, and the (single-valued) coefficient a:Ω→R+ is supposed to decay like |x|−2−α with α>0. Unlike in the situation of higher-dimensional exterior domain, that is RN∖B(0,1)‾ with N≥3, the borderline case N=2 considered here requires new tools for its treatment and results in a qualitatively different behaviour of its solutions. We establish a sub-supersolution principle for the above multi-valued variational inequality and prove the existence of extremal solutions. Moreover, we are going to show that classes of generalized variational-hemivariational inequalities turn out to be merely special cases of the above multi-valued variational inequality.

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

ABSTRACTWe 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.