A method for solving variational inequalities

Translated from Kibernetika i Sistemnyi Analiz, No. 6, pp. 48–55, November–December, 1992.     

A hybrid LQP-based method for structured variational inequalities

This paper proposes a hybrid LQP-based method (LQP, logarithmic-quadratic proximal) to solve a class of structured variational inequalities. In this method, an intermediate point is produced by solving a nonlinear equation system based on the LQP method; a descent direction is constructed using this iterate and the new iterate is obtained by a convex combination of the previous point and the one generated by a projection-type method along this descent direction. Global convergence of the new method is proved under mild assumptions. Preliminary numerical results for traffic equilibrium problems verify the computational preferences of the new method.     

The generalized set-valued strongly nonlinear implicit variational inequalities

In this paper, we introduce and study a new class of generalized set-valued strongly nonlinear implicit variational inequalities in Hilbert spaces. We construct the algorithm for solving this kind of generalized set-valued strongly nonlinear implicit variational inequality problem by using the auxiliary principle technique of Glowinski, Lions, and Tremolieres, prove the existence of solutions for this class of generalized set-valued strongly nonlinear implicit variational inequalities and the convergence of iterative sequences generated by the algorithm.     

A new alternating direction method for solving variational inequalities

We introduce a modified alternating direction method for structured monotone variational inequalities by performing an additional projection step at each iteration and another optimal step length is employed to reach substantial progress in each iteration. This method only needs functional values for given variables in the solution process and avoids the task of estimating the co-coercive modulus. All the computing process are easily implemented and the global convergence is also presented under mild assumptions. Some preliminary computational results are given.     

A multi-grid method for variational inequalities in contact problems

The convergence of the method is proved and it is shown that the objective function corresponding to the quadratic programming problem is monotonically decreasing. The results of numerical tests for an elasticity contact problem are presented.     

An iterative algorithm for elliptic variational inequalities of the second kind

In this paper, we propose an iterative algorithm for a simplified friction problem which is formulated as an elliptic variational inequality of the second kind. We approximate the simplified friction problem by a discrete system with the finite element method. Based on the use of the linearized technique and by constructing a particular function, we put forward the new algorithm to get the discrete solution. This algorithm is attractive due to its simple proof of convergence and easy implementation. A linear equation is solved in each iteration. Numerical results confirm that our algorithm is efficient and mesh independent.     

Descent method for monotone mixed variational inequalities

Abstract We consider mixed variational inequalities involving a non-strictly monotone, differentiable cost mapping and a convex nondifferentiable function. We apply the Tikhonov–Browder regularization technique to these problems. We use uniformly monotone auxiliary functions for constructing regularized problems and apply the gap function approach for the perturbed uniformly monotone variational inequalities. Then we propose a combined regularization and descent method for initial monotone problems and establish convergence of its iteration sequence. Keywords. Variational inequalities, nonsmooth functions, descent methods     

Iterative methods for solving extended general mixed variational inequalities

In this paper, we introduce and consider a new class of mixed variational inequalities involving four operators, which are called extended general mixed variational inequalities. Using the resolvent operator technique, we establish the equivalence between the extended general mixed variational inequalities and fixed point problems as well as resolvent equations. We use this alternative equivalent formulation to suggest and analyze some iterative methods for solving general mixed variational inequalities. We study the convergence criteria for the suggested iterative methods under suitable conditions. Our methods of proof are very simple as compared with other techniques. The results proved in this paper may be viewed as refinements and important generalizations of the previous known results.     

Convergence of the solution method for variational inequalities

Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 176–180, May–June, 1994     

Nonlocal quasi-Newton method for solving convex variational inequalities

Conclusion In the optimization problem [f ₀(x)│h_i(x)<-0,i=1,…,l] relaxation of the functionf ₀(x)+Nh⁺(x) does not produce, as we know [6, 7], α_k=1 in Newton's method with the auxiliary problem (5), (6), whereF(x)=f _0′(x). For this reason, Newton type methods based on relaxation off ₀(x)+Nh⁺(x) are not superlinearly convergent (so-called Maratos effect). The results of this article indicate that if (F(x)=f _0′(x), then replacement of the initial optimization problem with a larger equivalent problem (7) eliminates the Maratos effect in the proposed quasi-Newton method. This result is mainly of theoretical interest, because Newton type optimization methods in the space of the variablesx ∈R ⁿ are less complex. However to the best of our knowledge, the difficulties with nonlocal convergence arising in these methods (choice of parameters, etc.) have not been fully resolved [10, 11]. The discussion of these difficulties and comparison with the proposed method fall outside the scope of the present article, which focuses on solution of variational inequalities (1), (2) for the general caseF′(x)≠F′ ^T(x). Translated from Kibernetika i Sistemnyi Analiz, No. 6, pp. 78–91, November–December, 1994.     

Theoretically supported scalable BETI method for variational inequalities

Summary The Boundary Element Tearing and Interconnecting (BETI) methods were recently introduced as boundary element counterparts of the well established Finite Element Tearing and Interconnecting (FETI) methods. Here we combine the BETI method preconditioned by the projector to the “natural coarse grid” with recently proposed optimal algorithms for the solution of bound and equality constrained quadratic programming problems in order to develop a theoretically supported scalable solver for elliptic multidomain boundary variational inequalities such as those describing the equilibrium of a system of bodies in mutual contact. The key observation is that the “natural coarse grid” defines a subspace that contains the solution, so that the preconditioning affects also the non-linear steps. The results are validated by numerical experiments.      

A recurrent neural network for solving a class of general variational inequalities.

This paper presents a recurrent neural-network model for solving a special class of general variational inequalities (GVIs), which includes classical VIs as special cases. It is proved that the proposed neural network (NN) for solving this class of GVIs can be globally convergent, globally asymptotically stable, and globally exponentially stable under different conditions. The proposed NN can be viewed as a modified version of the general projection NN existing in the literature. Several numerical examples are provided to demonstrate the effectiveness and performance of the proposed NN.     

Penalty method for variational inequalities with multivalued mappings. Part I

The regularization method of variational inequalities is generalized by means of penalty operators to a class of variational inequalities with multivalued mappings. Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 57–69, July–August, 2000.     

A continuous method model for solving general variational inequality

Based on the projection operator, this paper presents a continuous method model for solving general variational inequality problems (VIPs) with bound constraints. A main feature of the proposed model is that it does not involve any form of matrix information in analysing its convergence properties. Under some reasonable assumptions, the convergence results of the proposed method model are established. Numerical results on some problems show that the proposed approach is efficient and can be applied to solve large scale VIPs with bound constraints.     

A general projection neural network for solving monotone variational inequalities and related optimization problems

Recently, a projection neural network for solving monotone variational inequalities and constrained optimization problems was developed. In this paper, we propose a general projection neural network for solving a wider class of variational inequalities and related optimization problems. In addition to its simple structure and low complexity, the proposed neural network includes existing neural networks for optimization, such as the projection neural network, the primal-dual neural network, and the dual neural network, as special cases. Under various mild conditions, the proposed general projection neural network is shown to be globally convergent, globally asymptotically stable, and globally exponentially stable. Furthermore, several improved stability criteria on two special cases of the general projection neural network are obtained under weaker conditions. Simulation results demonstrate the effectiveness and characteristics of the proposed neural network.     

A self-adaptive projection and contraction method for monotone symmetric linear variational inequalities

In this paper, we present a self-adaptive projection and contraction (SAPC) method for solving symmetric linear variational inequalities. Preliminary numerical tests show that the proposed method is efficient and effective and depends only slightly on its initial parameter. The global convergence of the new method is also addressed.     

A cascadic multigrid algorithm for variational inequalities

When classical multigrid methods are applied to discretizations of variational inequalities, several complications are frequently encountered mainly due to the lack of simple feasible restriction operators. These difficulties vanish in the application of the cascadic version of the multigrid method which in this sense yields greater advantages than in the linear case. Furthermore, a cg-method is proposed as smoother and as solver on coarse meshes. The efficiency of the new algorithm is elucidated by test calculations for an obstacle problem and for a Signorini problem.