Kuhn–Tucker optimality conditions; Lagrangian duality: Basics; Variational inequalities IMPLICIT LAGRANGIAN

Solodov, M. V.

doi:10.1007/0-306-48332-7_206

Cited by 6 publications

(5 citation statements)

References 20 publications

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“…In the case when F (·) is strongly monotone and/or the feasible set C has some special structure (e.g., C is the nonnegative orthant or, more generally, a box), there exist many efficient methods that can be used to solve those special cases of VI(F, C) (see [4,5,14,18,19,20,22,23,34,36,2,16,29,15,28,27,31]). In some of those methods, F (·) is further assumed to be differentiable, or Lipschitz continuous, or affine.…”

Section: Introductionmentioning

confidence: 99%

A New Projection Method for Variational Inequality Problems

Solodov

Svaiter

1999

SIAM J. Control Optim.

475

265

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Abstract. We propose a new projection algorithm for solving the variational inequality problem, where the underlying function is continuous and satisfies a certain generalized monotonicity assumption (e.g., it can be pseudomonotone). The method is simple and admits a nice geometric interpretation. It consists of two steps. First, we construct an appropriate hyperplane which strictly separates the current iterate from the solutions of the problem. This procedure requires a single projection onto the feasible set and employs an Armijo-type linesearch along a feasible direction. Then the next iterate is obtained as the projection of the current iterate onto the intersection of the feasible set with the halfspace containing the solution set. Thus, in contrast with most other projection-type methods, only two projection operations per iteration are needed. The method is shown to be globally convergent to a solution of the variational inequality problem under minimal assumptions. Preliminary computational experience is also reported.

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Section: Introductionmentioning

confidence: 99%

A New Projection Method for Variational Inequality Problems

Solodov

Svaiter

1999

SIAM J. Control Optim.

475

265

View full text Add to dashboard Cite

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“…For the future, we can expect further improvement in the accuracy of the experimental (g − 2)/2 measurement. As far as the SM prediction is concerned, we may anticipate low energy data for a variety of e + e − channels, produced via initial state radiation, at the φ-factory DAΦNE [33] and at the B-factories, BaBar and Belle, see, for example, [34]. For instance, by detecting the π + π − γ channel, it may be possible to measure e + e − → π + π − as low as about √ s = 400 MeV.…”

Section: Discussionmentioning

confidence: 99%

The SM prediction of g − 2 of the muon

Hagiwara¹,

Martin²,

Nomura³

et al. 2003

Proceedings of the 31st International Conference on High Energy Physics Ichep 2002

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We calculate (g − 2)/2 of the muon, by improving the determination of the hadronic vacuum polarisation contribution, a had,LO µ , and its uncertainties. The different e + e − data sets for each exclusive (and the inclusive) channel are combined in order to obtain the optimum estimate of the cross sections and their uncertainties. QCD sum rules are evaluated in order to resolve an apparent discrepancy between the inclusive data and the sum of the exclusive channels. We conclude a had,LO µ = (683.1 ± 5.9 exp ± 2.0 rad ) × 10 −10 which, when combined with the other contributions to (g − 2)/2, is about 3σ below the present world average measurement.

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“…In the complementarity literature, the idea of reformulation proved to be one of the most productive; see Mangasarian (1976), Mangasarian and Solodov (1993), Kanzow (1994), Facchinei andSoares (1997), andSolodov (2001), the collection (Fukushima and Qi 1999), and references therein. In the present context, of relevance are equation-based reformulations.…”

Section: Ifx Is a Local Solution Of Problem (13) (14) Thenmentioning

confidence: 99%

The Theory of 2-Regularity for Mappings with Lipschitzian Derivatives and its Applications to Optimality Conditions

Izmailov

Solodov

2002

Mathematics of OR

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We study local structure of a nonlinear mapping near points where standard regularity and/or smoothness assumptions need not be satisfied. We introduce a new concept of 2-regularity (a certain kind of second-order regularity) for a once differentiable mapping whose derivative is Lipschitz continuous. Under this 2-regularity condition, we obtain the representation theorem and the covering theorem (i.e., stability with respect to "right-hand side" perturbations) under assumptions that are weaker than those previously employed in the literature for results of this type. These results are further used to derive a constructive description of the tangent cone to a set defined by (2-regular) equality constraints and optimality conditions for related optimization problems. The class of mappings introduced and studied in the paper appears to be a convenient tool for treating complementarity structures by means of an appropriate equation-based reformulation. Optimality conditions for mathematical programs with (equivalently reformulated) complementarity constraints are also discussed.1. Introduction. In this paper, we consider the problem of local approximation of a nonlinear mapping near a given point in the absence of standard combinations of smoothness and regularity assumptions. We further study consequences of this approximation, in particular those relevant for constructive description of tangent directions to level surfaces, and optimality conditions for problems with feasible regions defined by irregular mappings.Let X and Y be Banach spaces, and consider a pointx ∈ X, its neighborhood V in X, and a mapping F V → Y . In what follows, X Y stands for the space of continuous linear operators from X to Y . For a linear operator ∈ X Y , ker = x ∈ X x = 0 is its null space, im = y ∈ Y y = x for some x ∈ X is its image space, and * Y * → X * is the linear operator adjoint to , where X * and Y * denote the dual spaces of X and Y , respectively.It is well known that if F is (first-order) regular atx, i.e.,

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Kuhn–Tucker optimality conditions; Lagrangian duality: Basics; Variational inequalities IMPLICIT LAGRANGIAN

Cited by 6 publications

References 20 publications

A New Projection Method for Variational Inequality Problems

A New Projection Method for Variational Inequality Problems

The SM prediction of g − 2 of the muon

The Theory of 2-Regularity for Mappings with Lipschitzian Derivatives and its Applications to Optimality Conditions

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