Proximal point algorithms and generalized nonlinear variational problems

This work is motivated by a recent work on an extended linear proximal point algorithm (PPA) [B.S. He, X.L. Fu, and Z.K. Jiang, Proximal-point algorithm using a linear proximal term, J. Optim. Theory Appl. 141 (2009), pp. 299–319], which aims at relaxing the requirement of the linear proximal term of classical PPA. In this paper, we make further contributions along the line. First, we generalize the linear PPA-based contraction method by using a nonlinear proximal term instead of the linear one. A notable superiority over traditional PPA-like methods is that the nonlinear proximal term of the proposed method may not necessarily be a gradient of any functions. In addition, the nonlinearity of the proximal term makes the new method more flexible. To avoid solving a variational inequality subproblem exactly, we then propose an inexact version of the developed method, which may be more computationally attractive in terms of requiring lower computational cost. Finally, we gainfully employ our new methods to solve linearly constrained convex minimization and variational inequality problems.

A hybrid entropic proximal decomposition method with self-adaptive strategy for solving variational inequality problems

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.

The proximal point algorithm has many applications for convex optimization with several versions of the proximal point algorithm including generalized proximal point algorithms and accelerated proximal point algorithms that have been studied in the literature. In this paper, we propose accelerated versions of generalized proximal point algorithms to find a zero of a maximal monotone operator in Hilbert spaces. We give both weak and linear convergence results of our proposed algorithms under standard conditions. Numerical applications of our results to image recovery are given and numerical implementations show that our algorithms are effective and superior to other related accelerated proximal point algorithms in the literature.

In this paper, we study a set-valued extended generalized nonlinear mixed variational inequality problem and its generalized resolvent dynamical system. A three-step iterative algorithm is constructed for solving set-valued extended generalized nonlinear variational inequality problem. Convergence and stability analysis are also discussed. We have shown the globally exponential convergence of generalized resolvent dynamical system to a unique solution of set-valued extended generalized nonlinear mixed variational inequality problem. In support of our main result, we provide a numerical example with convergence graphs and computation tables. For illustration, a comparison of our three-step iterative algorithm with Ishikawa-type algorithm and Mann-type algorithm is shown.

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.

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.

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 focuses on some customized applications of the proximal point algorithm (PPA) to two classes of problems: the convex minimization problem with linear constraints and a generic or separable objective function, and a saddle-point problem. We treat these two classes of problems uniformly by a mixed variational inequality, and show how the application of PPA with customized metric proximal parameters can yield favorable algorithms which are able to make use of the models' structures effectively. Our customized PPA revisit turns out to unify some algorithms including some existing ones in the literature and some new ones to be proposed. From the PPA perspective, we establish the global convergence and a worst-case O(1/t) convergence rate for this series of algorithms in a unified way.

We consider the convergence rate of the proximal point algorithm (PPA) for finding a minimizer of proper lower semicontinuous convex functions. In the Hilbert space setting, Güler showed that the big-O rate of the PPA can be improved to little-o when the sequence generated by the algorithm converges strongly to a minimizer. In this paper, we establish little-o rate of the PPA in Banach spaces without requiring this assumption. Then we apply the result to give new results on the convergence rate for sequences of alternating and averaged projections.

Abstract. Based on the A maximal (m) relaxed monotonicity frame-works, the approximation solvability of a general class of variational in-clusion problems using the relaxed proximal point algorithm is explored,while generalizing most of the investigations, especially of Xu (2002) onstrong convergence of modi ed version of the relaxed proximal point al-gorithm, Eckstein and Bertsekas (1992) on weak convergence using therelaxed proximal point algorithm to the context of the Douglas-Rachfordsplitting method, and Rockafellar (1976) on weak as well as strong con-vergence results on proximal point algorithms in real Hilbert space set-tings. Furthermore, the main result has been applied to the context of theH maximal monotonicity frameworks for solving a general class of vari-ational inclusion problems. It seems the obtained results can be used togeneralize the Yosida approximation that, in turn, can be applied to rst-order evolution inclusions, and can also be applied to Douglas-Rachfordsplitting methods for nding the zero of the sum of two A maximal (m)-relaxed monotone mappings.

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

We consider variational inequality problems for set-valued vector fields on general Riemannian manifolds. The existence results of the solution, convexity of the solution set, and the convergence property of the proximal point algorithm for the variational inequality problems for set-valued mappings on Riemannian manifolds are established. Applications to convex optimization problems on Riemannian manifolds are provided.

In this paper, we propose an asymmetric proximal point algorithm for solving variational inequality problems. The algorithm is “asymmetric” in the sense that the matrix in the linear proximal term is not necessary to be a symmetric matrix, which makes the method more flexible, especially in dealing with problems with separable structures. Under some suitable conditions, we prove the global linear convergence of the algorithm. To make the method more practical, we allow the subproblem to be solved in an approximate manner and a flexible inaccuracy criterion with constant parameter is adopted. Finally, we report some preliminary numerical results.

We first introduce the notion of weak sharpness for the solution sets of variational inequality problems (in short, VIP) on Hadamard spaces. We then study the finite convergence property of sequences generated by the inexact proximal point algorithm with different error terms for solving VIP under weak sharpness of the solution set. We also give an upper bound on the number of iterations by which the sequence generated by the exact proximal point algorithm converges to a solution of VIP. An example is also given to illustrate our results.