Convergence of the projection and contraction methods for solving bilevel variational inequality problems

This paper is concerned with the mixed linear complementarity problem and the role it and its variants play in the stability analysis of the nonlinear complementarity problem and the Karush-Kuhn-Tucker system of a variational inequality problem. Under a nonsingular assumption, the mixed linear complementarity problem can be converted to the standard problem; in this case, the rich theory of the latter can be directly applied to the former. In this work, we employ degree theory to derive some sufficient conditions for the existence of a solution to the mixed linear complementarity problem in the absence of the nonsingularity property. Next, we extend this existence theory to the mixed nonlinear complementarity problem and establish a main stability result under a certain degree-theoretic assumption concerning the linearized problem. We then specialize this stability result and its consequences to the parametric variational inequality problem under the assumption of a unique set of multipliers. Finally, we consider the latter problem with the uniqueness assumption of the multipliers replaced by a convexity assumption and obtain stability results under some weak second-order conditions. In addition to the new existence results for the mixed linear complementarity problem, the main contributions of this paper in the stability category are the following: a resolution to a conjecture concerning the local solvability of a parametric variational inequality, the use of the generalized linear complementarity problem as a tool to broaden the second-order conditions, the characterization of the solution stability of the linear complementarity problem and the affine variational inequality problem in terms of the solution isolatedness under some weak hypotheses, and various stability theorems under some weak second-order conditions.

We consider a new class of multiplier interior point methods for solving variational inequality problems with maximal monotone operators and explicit convex constraint inequalities. Developing a simple Lagrangian duality scheme which is combined with the recent logarithmic-quadratic proximal (LQP) theory introduced by the authors, we derive three algorithms for solving the variational inequality (VI) problem. This provides a natural extension of the methods of multipliers used in convex optimization and leads to smooth interior point multiplier algorithms. We prove primal, dual, and primal-dual convergence under very mild assumptions, eliminating all the usual assumptions used until now in the literature for related algorithms. Applications to complementarity problems are also discussed.

In this article, a generalized variational inequality problem in the setting of Hadamard spaces is introduced and analyzed. For approximating a solution of the problem when the underlined mapping is monotone, an adaptive algorithm that requires the computation of only one proximal operator at each iteration is proposed. The work gives a step towards investigating monotone variational inequality problems in geodesic settings.

<abstract><p>The theory of variational inequalities is an important tool in physics, engineering, finance, and optimization theory. The projection algorithm and its variants are useful tools for determining the approximate solution to the variational inequality problem. This paper introduces three distinct extragradient algorithms for dealing with variational inequality problems involving quasi-monotone and semistrictly quasi-monotone operators in infinite-dimensional real Hilbert spaces. This problem is a general mathematical model that incorporates a set of applied mathematical models as an example, such as equilibrium models, optimization problems, fixed point problems, saddle point problems, and Nash equilibrium point problems. The proposed algorithms employ both fixed and variable stepsize rules that are iteratively transformed based on previous iterations. These algorithms are based on the fact that no prior knowledge of the Lipschitz constant or any line-search framework is required. To demonstrate the convergence of the proposed algorithms, some simple conditions are used. Numerous experiments have been conducted to highlight the numerical capabilities of algorithms.</p></abstract>

Hierarchical variational inequality problem and split common fixed point of averaged operators

The alternating direction method is an attractive method for solving large-scale variational inequality problems whenever the subproblems can be solved efficiently. However, the subproblems are still variational inequality problems, which are as structurally difficult to solve as the original one. To overcome this disadvantage, in this paper we propose a new alternating direction method for solving a class of nonlinear monotone variational inequality problems. In each iteration the method just makes an orthogonal projection to a simple set and some function evaluations. We report some preliminary computational results to illustrate the efficiency of the method.

Abstract. In this paper, we consider and study a new class of general-ized vector quasi-variational type inequalities and obtain some existencetheorems for both under compact and noncompact assumptions in topo-logical vector spaces without using monotonicity. For the noncompactcase, we use the concept of escaping sequences. 1. IntroductionThe vector variational inequality problem was initiated by Giannessi [11] in nite dimensional Euclidean spaces with applications. Later on many authors[6, 7, 12, 15, 24] generalized vector variational inequalities in abstract spacesin several ways. The vector variational-like inequality is one of the generalizedforms of vector variational inequalities [11]. In 1989, Parida et al [19] stud-ied the existence of solutions for variational-like inequalities in R n space andhave shown a relationship between variational like inequalities problems andconvex programming as well as with complementarity problems. The vectorvariational like inequality and generalized vector variational like inequality arepowerful tools to study non-convex vector optimization problems and convexand nondi erentiable vector optimization problems respectively, see [10, 18, 21].In 1973, Bensoussan and Lions [3] introduced the quasi-variational inequalityproblem (QVIP). Since then, many generalizations of the QVIP have beenappeared in the literature (see, for example, [1, 2]). In the last decade, becauseof applications in the optimization problems, mathematical programming andequilibrium problems, the QVIP has been intensively studied by many authors[5, 8, 16, 20, 22, 23, 25].In this paper, we consider the generalized vector quasivariational type in-equality problem (GVQVTIP) and obtain some existence theorems for solutions

This paper deals with a variational inequality problem over a solution set of another variational inequality problem, which essentially is called the double-hierarchical constrained optimization problem. Inertial-based algorithms are devised for this problem and solution theorems of strong convergence are established. Simulation results on numerical examples are given to illustrate the behaviour of our proposed algorithms compared with some existing ones.

The purpose of this paper is to study variational inequality problem over the solution set of multiple-set split monotone variational inclusion problem. We propose an iterative algorithm with inertial method for finding an approximate solution of this problem in real Hilbert spaces. Strong convergence of the sequence of iterates generated from the proposed method is obtained under some mild assumptions. The iterative scheme does not require prior knowledge of operator norm. Also we present some applications of our main result to solve the bilevel programming problem, the bilevel monotone variational inequalities, the split minimization problem, the multiple-set split feasibility problem and the multiple set split variational inequality problem.

General convergence analysis for two-step projection methods and applications to variational problems

In this paper we develop a framework for the study of financial equilibrium in the case of sectors in the economy, each of which is faced with two objectives/criteria in his portfolio selection decision making. In particular, we first present the bicriteria decision model of an individual financial sector, who seeks an optimal portfolio composition, given that the wishes to minimize his risk and to maximize his return. We utilize a value function approach to reformulate a sector's bicriteria optimization problem as a single optimization problem and argue that constant weight value functions may not adequately reveal a sector's preference over the return and the risk. Hence, we introduce state-dependent weights for the modeling of a sector's decision-making problem. We, subsequently, provide qualitative properties of the value function. We state the economic system conditions governing the instrument prices, define the financial equilibrium conditions, and show that they can be formulated as a variational inequality problem. Finally, qualitative properties of existence and uniqueness of the equilibrium are obtained. This work is the first to establish the connections among bicriteria problems (in the general setting of value functions), financial equilibrium problems, and variational inequality problems.

We propose a projection algorithm for solving the variational inequality problem, where the underlying function is continuous and satisfies a certain generalized monotonicity assumption (for example, it can be pseudomonotone). The method is simple and admits a nice geometric interpretation. It consists of two steps. First, we construct an appropriate hyperplane which strictly separates the current iterate from the solutions of the problem. This procedure requires a single projection onto the feasible set and employs an Armijo-type line-search along a feasible direction. Then the next iterate is obtained as the projection of the current iterate onto the intersection of the feasible set with the halfspace containing the solution set. Thus, in contrast with most other projection-type methods, only two projection operations per iteration are needed. The method is shown to be globally convergent to a solution of the variational inequality problem under minimal assumptions.

In the paper, we consider bilevel problems: variational inequality problems over the set of solutions of the equilibrium problem. Finding normal Nash equilibrium is an example of such a problem. To solve these problems, an iterative algorithm is proposed that combines the ideas of the two-stage proximal method, adaptability, and iterative regularization. In contrast to the previously used rules for choosing the step size, the proposed algorithm does not calculate bifunction values at additional points and does not require knowledge of information on bifunction’s Lipschitz constants and operator’s Lipschitz and strong monotonicity constants. For monotone bifunctions of Lipschitz type and strongly monotone Lipschitz operators, the theorem on strong convergence of sequences generated by the algorithm is proved. The proposed algorithm is shown to be applicable to monotone bilevel variational inequalities in Hilbert spaces.

In this paper, we propose a globally convergent BFGS method to solve Variational Inequality Problems (VIPs). In fact, a globalization technique on the basis of the hyperplane projection method is applied to the BFGS method. The technique, which is independent of any merit function, is applicable for pseudo-monotone problems. The proposed method applies the BFGS direction and tries to reduce the distance of iterates to the solution set. This property, called Fejer monotonicity of iterates with respect to the solution set, is the basis of the convergence analysis. The method applied to pseudo-monotone VIP is globally convergent in the sense that subproblems always have unique solutions, and the sequence of iterates converges to a solution to the problem without any regularity assumption. Finally, some numerical simulations are included to evaluate the efficiency of the proposed algorithm.

We propose and study an alternating inertial algorithm for approximating a solution to the variational inequality problem on an Hadamard manifold. The proposed method combines the inertial technique and the Tseng extragradient method with self-adaptive step sizes. We establish some convergence results for our iterative procedure when the variational inequality problem is governed by pseudomonotone vector fields. We present the results of some numerical experiments which show the competitive advantage of our algorithm over earlier methods in the literature.