We establish formulas for computing/estimating the regular and Mordukhovich coderivatives of implicit multifunctions defined by generalized equations in Asplund spaces. These formulas are applied to obtain conditions ...
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We establish formulas for computing/estimating the regular and Mordukhovich coderivatives of implicit multifunctions defined by generalized equations in Asplund spaces. These formulas are applied to obtain conditions for solution stability of parametric variational systems over perturbed smooth-boundary constraint sets.
The Lipschitz-like property and the metric regularity in the sense of Robinson of the solution map of a parametric linear constraint system are investigated thoroughly by means of normal coderivative, the Mordukhovich...
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The Lipschitz-like property and the metric regularity in the sense of Robinson of the solution map of a parametric linear constraint system are investigated thoroughly by means of normal coderivative, the Mordukhovich criterion, and a related theorem due to Levy and Mordukhovich [Math. Program., 99 (2004), pp. 311-327]. Among other things, the obtained results yield uniform local error bounds and traditional local error bounds for the linear complementarity problem and the general affine variational inequality problem, as well as verifiable sufficient conditions for the Lipschitz-like property of the solution map of the linear complementarity problem and a class of affine variational inequalities, where all components of the problem data are subject to perturbations.
Sufficient conditions for a norm-to-weak* continuous mapping f : X -> X* being monotone or submonotone are established by its Frechet and normal coderivatives, where X is an Asplund space with its dual space X*. Un...
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Sufficient conditions for a norm-to-weak* continuous mapping f : X -> X* being monotone or submonotone are established by its Frechet and normal coderivatives, where X is an Asplund space with its dual space X*. Under some additional assumptions, they are also necessary conditions. Among other things, we obtain a criterion for the monotonicity of continuous mappings which extends the following classical result: a differentiable mapping F : R-n -> R-n is monotone if and only if for each x is an element of R-n the Jacobian matrix del F(x) is positive semi-definite;see [22, Proposition 12.3]. As a by-product, sufficient conditions for a function being convex or approximately convex are given.
In this paper, error bounds for gamma-paraconvex multifunctions are considered. Characterizations of a gamma-paraconvex multifunction are given. In terms of normal cone and coderivative, some results on the existence ...
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In this paper, error bounds for gamma-paraconvex multifunctions are considered. Characterizations of a gamma-paraconvex multifunction are given. In terms of normal cone and coderivative, some results on the existence of error bounds are presented.
A class of evolution quasi-static systems which leads, after a suitable time discretization, to recursive non-linear programs, is considered and optimal control or identification problems governed by such systems are ...
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A class of evolution quasi-static systems which leads, after a suitable time discretization, to recursive non-linear programs, is considered and optimal control or identification problems governed by such systems are investigated. The resulting problem is an evolutionary Mathematical Programs with Equilibrium Constraints. A subgradient information of the (in general nonsmooth) composite objective function is evaluated and the problem is solved by the implicit programming approach. The abstract theory is illustrated on an identification problem governed by delamination of a unilateral adhesive contact of elastic bodies discretized by finite-element method in space and a semiimplicit formula in time. Being motivated by practical tasks, an identification problem of the fracture toughness and of the elasticity moduli of the adhesive is computationally implemented and tested numerically on a two-dimensional example. Other applications including frictional contacts or bulk damage, plasticity or phase transformations are outlined.
In this paper, we develop a geometric approach to convex subdifferential calculus in finite dimensions with employing some ideas of modern variational analysis. This approach allows us to obtain natural and rather eas...
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In this paper, we develop a geometric approach to convex subdifferential calculus in finite dimensions with employing some ideas of modern variational analysis. This approach allows us to obtain natural and rather easy proofs of basic results of convex subdifferential calculus in full generality and also derive new results of convex analysis concerning optimal value/marginal functions, normals to inverse images of sets under set-valued mappings, calculus rules for coderivatives of single-valued and set-valued mappings, and calculating coderivatives of solution maps to parameterized generalized equations governed by set-valued mappings with convex graphs.
The aim of this paper is to investigate the convergence properties for Mordukhovich's coderivative of the solution map of the sample average approximation (SAA) problem for a parametric stochastic variational ineq...
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The aim of this paper is to investigate the convergence properties for Mordukhovich's coderivative of the solution map of the sample average approximation (SAA) problem for a parametric stochastic variational inequality with equality and inequality constraints. The notion of integrated deviation is introduced to characterize the outer limit of a sequence of sets. It is demonstrated that, under suitable conditions, both the cosmic deviation and the integrated deviation between the coderivative of the solution mapping to SAA problem and that of the solution mapping to the parametric stochastic variational inequality converge almost surely to zero as the sample size tends to infinity. Moreover, the exponential convergence rate of coderivatives of the solution maps to the SAA parametric stochastic variational inequality is established. The results are used to develop sufficient conditions for the consistency of the Lipschitz-like property of the solution map of SAA problem and the consistency of stationary points of the SAA estimator for a stochastic bilevel program.
The purpose of this paper is to investigate coderivatives of the gap function involving the Minty vector variational inequality. First, we discuss the regular coderivative, the normal coderivative, and the mixed coder...
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The purpose of this paper is to investigate coderivatives of the gap function involving the Minty vector variational inequality. First, we discuss the regular coderivative, the normal coderivative, and the mixed coderivative of a class of set-valued maps. Then, by using the relationships between the coderivatives of a set-valued map and its efficient points set-valued map, we obtain the coderivatives of the gap function for the Minty vector variational inequality.
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