The Proximal Point Method for Nonmonotone Variational Inequalities (original) (raw)
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A proximal point method for a class of monotone equilibrium problems with linear constraints
Operational Research, 2015
It is well known that the equilibrium problems which often arise in engineering, economics and management applications, provide a unified framework for variational inequality, complementarity problem, optimization problem, saddle point problem and fixed point problem. In this paper, a proximal point method is proposed for solving a class of monotone equilibrium problems with linear constraints (MEP). The updates of all variables of the proximal point method are given in closed form. An auxiliary equilibrium problem is introduced for MEP via its saddle point problem. Further, we present some characterizations for solution of the auxiliary equilibrium problem and fixed point of corresponding resolvent operator. Thirdly, a proximal point method for MEP is suggested by fixed point technique. The asymptotic behavior of the proposed algorithm is established under some mild assumptions. Finally, some numerical examples are reported to show the feasibility of the proposed algorithm.
Inexact Proximal Point Methods for Variational Inequality Problems
SIAM Journal on Optimization, 2010
We present a new family of proximal point methods for solving monotone variational inequalities. Our algorithm has a relative error tolerance criterion in solving the proximal subproblems. Our convergence analysis covers a wide family of regularization functions, including double regularizations recently introduced by Silva, Eckstein, and Humes, Jr. [SIAM J. Optim., 12 (2001), pp. 238-261] and the Bregman distance induced by h(x) = n i=1 x i log x i . We do not use in our analysis the assumption of paramonotonicity, which is standard in proving convergence of Bregman-based proximal methods.
Mathematics
In this paper, we propose two modified two-step proximal methods that are formed through the proximal-like mapping and inertial effect for solving two classes of equilibrium problems. A weak convergence theorem for the first method and the strong convergence result of the second method are well established based on the mild condition on a bifunction. Such methods have the advantage of not involving any line search procedure or any knowledge of the Lipschitz-type constants of the bifunction. One practical reason is that the stepsize involving in these methods is updated based on some previous iterations or uses a stepsize sequence that is non-summable. We consider the well-known Nash–Cournot equilibrium models to support our well-established convergence results and see the advantage of the proposed methods over other well-known methods.
An inexact interior point proximal method for the variational inequality problem
Computational & Applied Mathematics, 2009
We propose an infeasible interior proximal method for solving variational inequality problems with maximal monotone operators and linear constraints. The interior proximal method proposed by Auslender, Teboulle and Ben-Tiba [3] is a proximal method using a distance-like barrier function and it has a global convergence property under mild assumptions. However, this method is applicable only to problems whose feasible region has nonempty interior. The algorithm we propose is applicable to problems whose feasible region may have empty interior. Moreover, a new kind of inexact scheme is used. We present a full convergence analysis for our algorithm. Mathematical subject classification: 90C51, 65K10, 47J20, 49J40, 49J52, 49J53.
On New Proximal Point Methods for Solving the Variational Inequalities
Journal of Applied Mathematics, 2012
It is well known that the variational inequalities are equivalent to the fixed point problem. We use this alternative equivalent formulation to suggest and analyze some new proximal point methods for solving the variational inequalities. These new methods include the explicit, the implicit, and the extragradient methods as special cases. The convergence analysis of the new methods is considered under some suitable conditions. Results proved in this paper may stimulate further research in this direction.
Theory and Algorithms of Variational Inequality and Equilibrium Problems, and Their Applications
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The variational inequality problem is a general problem formulation that encompasses many mathematical problems, among others, including nonlinear equations, optimization problems, complementarity problems, and fixed point problems. Variational inequality is developed as a tool for the study of certain classes of partial deferential equations, economic equilibrium problems, and the pricing model of the option.
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A proximal point method for solving mixed variational inequalities is suggested and analyzed by using the auxiliary principle technique. It is shown that the convergence of the proposed method requires only the pseudomonotonicity of the operator, which is a weaker condition than monotonicity. As special cases, we obtain various known and new results for solving variational inequalities and related problems. Our proof of convergence is very simple as compared with other methods.
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Projection-proximal methods for general variational inequalities
Journal of Mathematical Analysis and Applications, 2006
In this paper, we consider and analyze some new projection-proximal methods for solving general variational inequalities. The modified methods converge for pseudomonotone operators which is a weaker condition than monotonicity. The proposed methods include several new and known methods as special cases. Our results can be considered as a novel and important extension of the previously known results. Since the general variational inequalities include the quasi-variational inequalities and implicit complementarity problems as special cases, results proved in this paper continue to hold for these problems.