Variational principles for variational inequalities

Auchmuty, Giles

doi:10.1080/01630568908816335

Cited by 93 publications

(46 citation statements)

References 3 publications

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“…This paper is devoted to the study of a new merit function for proximal point problems and associated error bounds. It also discusses the relation between this merit function and the regularized gap function [1,13] in the context of variational inequalities. Finally, these theoretical results are applied to devise inexact proximal-based algorithms for the variational inequality problem which employ constructive approximation criteria.…”

Section: Introductionmentioning

confidence: 99%

“…Clearly, (2) can be related to (1) by the choice of C = H, F = T . Conversely, (1) can be reformulated as the inclusion problem (2) if we take T = F + N C , where N C is the normal operator for the set C. This operator T is monotone, and it is also maximal because C, the domain of N C , intersects the interior of the domain of F (which we assume here to be H) [26].…”

Section: Introductionmentioning

confidence: 99%

“…This method consists of solving a sequence of regularized subproblems. Specifically, if applied to (1), the method consists of resolving subproblems of the form…”

Section: Introductionmentioning

confidence: 99%

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Error bounds for proximal point subproblems and associated inexact proximal point algorithms

Solodov

Svaiter

2000

Math. Program.

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Abstract. We study various error measures for approximate solution of proximal point regularizations of the variational inequality problem, and of the closely related problem of finding a zero of a maximal monotone operator. A new merit function is proposed for proximal point subproblems associated with the latter. This merit function is based on Burachik-Iusem-Svaiter's concept of ε-enlargement of a maximal monotone operator. For variational inequalities, we establish a precise relationship between the regularized gap function, which is a natural error measure in this context, and our new merit function. Some error bounds are derived using both merit functions for the corresponding formulations of the proximal subproblem. We further use the regularized gap function to devise a new inexact proximal point algorithm for solving monotone variational inequalities. This inexact proximal point method preserves all the desirable global and local convergence properties of the classical exact/inexact method, while providing a constructive error tolerance criterion, suitable for further practical applications. The use of other tolerance rules is also discussed.Key words. variational inequalities -proximal point subproblems -error bound -(enlargement of) maximal monotone operator -regularized gap function

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Error bounds for proximal point subproblems and associated inexact proximal point algorithms

Solodov

Svaiter

2000

Math. Program.

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“…Regularized gap functions were introduced in [1] and [8] (see also [7, section 10.2.1] for a discussion of gap functions). We have that, for given x and ω, a pointζ ∈ Q is a solution of (2.10), i.e.,ζ ∈ S(x, ω), iff γ(x,ζ, ω) = 0.…”

Section: Theorem 1 Let G : R N × ω → R Be a Random Lower Semicontinuomentioning

confidence: 99%

Stochastic Programming with Equilibrium Constraints

Shapiro

2006

J Optim Theory Appl

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“…Facchinei and Kanzow (1997) improved the result in (Yamashita and Fukushima 1995) and gave a necessary and sufficient condition for the stationary point of the implicit Lagrangian to be a solution of NCP. extended the implicit Lagrangian to the variational inequality problem (VIP) and showed that the implicit Lagrangian can be represented as the difference of two regularized gap functions proposed by Fukushima (1992) and Auchmuty (1989) independently. Yamashita, Taji and Fukushima (1997) extended the results of and studied various properties of the D-gap function g αβ := f α (x) − f β (x) (1.5) where α and β are arbitrary positive parameters with α < β, and f α is the regularized gap function defined by f α (x) := αF (x), x − y α (x) − 1 2 x − y α (x) 2 (1.6) with y α (x) = X (x − αF (x)) and X (·) being the projection operator onto the constraint set X of VIP.…”

Section: Introductionmentioning

confidence: 99%

Vector-Valued Implicit Lagrangian for Symmetric Cone Complementarity Problems

Kong

Tunçel

Xiu

2009

Asia Pac. J. Oper. Res.

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The implicit Lagrangian was first proposed by Mangasarian and Solodov as a smooth merit function for the nonnegative orthant complementarity problem. It has attracted much attention in the past ten years because of its utility in reformulating complementarity problems as unconstrained minimization problems. In this paper, exploiting the Jordan-algebraic structure, we extend it to the vector-valued implicit Lagrangian for symmetric cone complementary problem (SCCP), and show that it is a continuously differentiable complementarity function for SCCP and whose Jacobian is strongly semismooth. As an application, we develop the real-valued implicit Lagrangian and the corresponding smooth merit function for SCCP, and give a necessary and sufficient condition for the stationary point of the merit function to be a solution of SCCP. Finally, we show that this merit function can provide a global error bound for SCCP with the uniform Cartesian P-property.

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Variational principles for variational inequalities

Cited by 93 publications

References 3 publications

Error bounds for proximal point subproblems and associated inexact proximal point algorithms

Error bounds for proximal point subproblems and associated inexact proximal point algorithms

Stochastic Programming with Equilibrium Constraints

Vector-Valued Implicit Lagrangian for Symmetric Cone Complementarity Problems

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