A Generalized Proximal Point Algorithm for the Variational Inequality Problem in a Hilbert Space

Burachik, Regina S.; Iusem, Alfredo N.

doi:10.1137/s1052623495286302

Cited by 129 publications

(67 citation statements)

References 14 publications

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“…(See also [58,59], where extensions of (1.4) to nondifferentiable functions were studied.) Extensions to Hilbert [18,20,61] and Banach [1,8,21,23,24,25,26,27,55,56,75] spaces have also been considered more recently. In the present paper, we adopt the following definition for Bregman distances.…”

Section: Introductionmentioning

confidence: 99%

Bregman Monotone Optimization Algorithms

Bauschke

Borwein

Combettes

2003

SIAM J. Control Optim.

246

216

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Abstract.A broad class of optimization algorithms based on Bregman distances in Banach spaces is unified around the notion of Bregman monotonicity. A systematic investigation of this notion leads to a simplified analysis of numerous algorithms and to the development of a new class of parallel block-iterative surrogate Bregman projection schemes. Another key contribution is the introduction of a class of operators that is shown to be intrinsically tied to the notion of Bregman monotonicity and to include the operators commonly found in Bregman optimization methods. Special emphasis is placed on the viability of the algorithms and the importance of Legendre functions in this regard. Various applications are discussed.

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Section: Introductionmentioning

confidence: 99%

Bregman Monotone Optimization Algorithms

Bauschke

Borwein

Combettes

2003

SIAM J. Control Optim.

246

216

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“…For a given distance D, a coercivity property (namely surjectivity of ∇ 1 D(•, y) for y fixed) is required (see, for instance [8,Proposition 3]). The result we need to ensure well-definedness of our scheme, which we state below, is [3, Proposition 3.1], which establishes the desired surjectivity in our particular setting.…”

Section: Basic Assumptions and Propertiesmentioning

confidence: 99%

“…Examples of these regularizing functionals are the Bregman distances (see, e.g. [1,8,13,14,20,25]), ϕ-divergences ( [26,5,15,18,19,27,28]) and log-quadratic regularizations ( [3,4]). Being interior point methods, it is a basic assumption that the topological interior of C is nonempty.…”

Section: Introductionmentioning

confidence: 99%

An inexact interior point proximal method for the variational inequality problem

Burachik¹,

Lopes²,

Silva³

2009

Comput.appl.math.

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Abstract.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.

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“…To find such a root, we then apply generalized proximal algorithms based on Bregman functions [6,7,9,12,17,18,33]. A number of recent papers [5,6,8] have stressed the ability of proximal terms arising from appropriately-formulated Bregman functions to act like barrier functions, giving rise to "interior point" proximal methods for variational inequality problems. Such methods are derived by applying Bregman proximal methods to a primal formulation of (1) or (2).…”

Section: This Paper Concerns the Solution Of The Nonlinear Complementmentioning

confidence: 99%

“…[30]). This primal-dual method combines the best theoretical features of primal methods in the spirit of [5,6,8] with the best features of the new dual method. Some preliminary computational results on the MCPLIB [14] suite of test problems are given in Section 4.…”

Section: This Paper Concerns the Solution Of The Nonlinear Complementmentioning

confidence: 99%

Smooth methods of multipliers for complementarity problems

Eckstein¹,

Ferris²

1999

Math. Program.

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Abstract. This paper describes several methods for solving nonlinear complementarity problems. A general duality framework for pairs of monotone operators is developed and then applied to the monotone complementarity problem, obtaining primal, dual, and primal-dual formulations. We derive Bregman-function-based generalized proximal algorithms for each of these formulations, generating three classes of complementarity algorithms. The primal class is well-known. The dual class is new and constitutes a general collection of methods of multipliers, or augmented Lagrangian methods, for complementarity problems. In a special case, it corresponds to a class of variational inequality algorithms proposed by Gabay. By appropriate choice of Bregman function, the augmented Lagrangian subproblem in these methods can be made continuously differentiable. The primal-dual class of methods is entirely new and combines the best theoretical features of the primal and dual methods. Some preliminary computation shows that this class of algorithms is effective at solving many of the standard complementarity test problems.

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A Generalized Proximal Point Algorithm for the Variational Inequality Problem in a Hilbert Space

Cited by 129 publications

References 14 publications

Bregman Monotone Optimization Algorithms

Bregman Monotone Optimization Algorithms

An inexact interior point proximal method for the variational inequality problem

Smooth methods of multipliers for complementarity problems

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