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Resolvents and Yosida Approximations of Displacement Mappings of Isometries Set-Valued Var. Anal. (IF 1.476) Pub Date : 2021-04-12 Salihah Alwadani, Heinz H. Bauschke, Julian P. Revalski, Xianfu Wang
Maximally monotone operators are fundamental objects in modern optimization. The main classes of monotone operators are subdifferential operators and matrices with a positive semidefinite symmetric part. In this paper, we study a nice class of monotone operators: displacement mappings of isometries of finite order. We derive explicit formulas for resolvents, Yosida approximations, and (set-valued and
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On the Analysis of Variance-reduced and Randomized Projection Variants of Single Projection Schemes for Monotone Stochastic Variational Inequality Problems Set-Valued Var. Anal. (IF 1.476) Pub Date : 2021-04-08 Shisheng Cui, Uday V. Shanbhag
Classical extragradient schemes and their stochastic counterpart represent a cornerstone for resolving monotone variational inequality problems. Yet, such schemes have a per-iteration complexity of two projections onto a convex set and require two evaluations of the map, the former of which could be relatively expensive. We consider two related avenues where the per-iteration complexity is significantly
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Erratum to: New Constraint Qualifications and Optimality Conditions for Second Order Cone Programs Set-Valued Var. Anal. (IF 1.476) Pub Date : 2021-04-07 R. Andreani, E. H. Fukuda, G. Haeser, H. Ramírez, D. O. Santos, P. J. S. Silva, T. P. Silveira
In this note we show with a counter-example that all conditions proposed in Zhang and Zhang (Set-Valued Var. Anal 27:693–712 2019) are not constraint qualifications for second-order cone programming.
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Topological Approach to Mathematical Programs with Switching Constraints Set-Valued Var. Anal. (IF 1.476) Pub Date : 2021-04-07 Vladimir Shikhman
We study mathematical programs with switching constraints (for short, MPSC) from the topological perspective. Two basic theorems from Morse theory are proved. Outside the W-stationary point set, continuous deformation of lower level sets can be performed. However, when passing a W-stationary level, the topology of the lower level set changes via the attachment of a w-dimensional cell. The dimension
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Primal Superlinear Convergence of Sqp Methods in Piecewise Linear-Quadratic Composite Optimization Set-Valued Var. Anal. (IF 1.476) Pub Date : 2021-04-02 M. Ebrahim Sarabi
This paper mainly concerns with the primal superlinear convergence of the quasi-Newton sequential quadratic programming (SQP) method for piecewise linear-quadratic composite optimization problems. We show that the latter primal superlinear convergence can be justified under the noncriticality of Lagrange multipliers and a version of the Dennis-Moré condition. Furthermore, we show that if we replace
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Calmness of a Perturbed Cournot Oligopoly Game with Nonsmooth Cost Functions Set-Valued Var. Anal. (IF 1.476) Pub Date : 2021-03-23 Matthieu Maréchal
This article deals with the calmness of a solution map for a Cournot Oligopoly Game with non-smooth cost functions. The fact that the cost functions are not supposed to be differentiable allows to consider cases where some firms have different units of production, with different marginal costs. In order to obtain results concerning calmness, we use a new technique based on an outer coderivative and
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Nonsmooth Feedback Control for Multi-Agent Dynamics Set-Valued Var. Anal. (IF 1.476) Pub Date : 2021-03-20 Mira Bivas, Marc Quincampoix
We investigate a control problem with a large number of agents – a crowd. This multi-agent system is modelized by a set, each point of which is the position of an agent. The corresponding dynamical system has two-level dynamics: a microscopic one, which concerns the evolution of each agent; a macroscopic one, which describes the evolution of the whole crowd of agents. The state variable of the system
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Cone-Constrained Eigenvalue Problems: Structure of Cone Spectra Set-Valued Var. Anal. (IF 1.476) Pub Date : 2021-02-22 Alberto Seeger
There is a rich literature devoted to the eigenvalue analysis of variational inequalities. Of special interest is the case in which the constraint set of the variational inequality is a closed convex cone. The set of eigenvalues of a matrix A relative to a closed convex cone K is called the K-spectrum of A. Cardinality and topological results for cone spectra depend on the kind of matrices and cones
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Perturbation Techniques for Convergence Analysis of Proximal Gradient Method and Other First-Order Algorithms via Variational Analysis Set-Valued Var. Anal. (IF 1.476) Pub Date : 2021-01-28 Xiangfeng Wang, Jane J. Ye, Xiaoming Yuan, Shangzhi Zeng, Jin Zhang
Wedevelopnew perturbation techniques for conducting convergence analysis of various first-order algorithms for a class of nonsmooth optimization problems. We consider the iteration scheme of an algorithm to construct a perturbed stationary point set-valued map, and define the perturbing parameter by the difference of two consecutive iterates. Then, we show that the calmness condition of the induced
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On the Quantitative Solution Stability of Parameterized Set-Valued Inclusions Set-Valued Var. Anal. (IF 1.476) Pub Date : 2021-01-09 A. Uderzo
The subject of the present paper are stability properties of the solution set to set-valued inclusions. The latter are problems emerging in robust optimization and mathematical economics, which can not be cast in traditional generalized equations. The analysis here reported focuses on several quantitative forms of semicontinuity for set-valued mappings, widely investigated in variational analysis,
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Sufficient Conditions for Metric Subregularity of Constraint Systems with Applications to Disjunctive and Ortho-Disjunctive Programs Set-Valued Var. Anal. (IF 1.476) Pub Date : 2021-01-05 Matúš Benko, Michal Červinka, Tim Hoheisel
This paper is devoted to the study of the metric subregularity constraint qualification for general optimization problems, with the emphasis on the nonconvex setting. We elaborate on notions of directional pseudo- and quasi-normality, recently introduced by Bai et al., which combine the standard approach via pseudo- and quasi-normality with modern tools of directional variational analysis. We focus
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Bregman Forward-Backward Operator Splitting Set-Valued Var. Anal. (IF 1.476) Pub Date : 2020-11-28 Minh N. Bùi, Patrick L. Combettes
We establish the convergence of the forward-backward splitting algorithm based on Bregman distances for the sum of two monotone operators in reflexive Banach spaces. Even in Euclidean spaces, the convergence of this algorithm has so far been proved only in the case of minimization problems. The proposed framework features Bregman distances that vary over the iterations and a novel assumption on the
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Ergodic Approach to Robust Optimization and Infinite Programming Problems Set-Valued Var. Anal. (IF 1.476) Pub Date : 2020-11-11 Pedro Pérez-Aros
In this work, we show the consistency of an approach for solving robust optimization problems using sequences of sub-problems generated by ergodic measure preserving transformations. The main result of this paper is that the minimizers and the optimal value of the sub-problems converge, in some sense, to the minimizers and the optimal value of the initial problem, respectively. Our result particularly
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Constrained Lipschitzian Error Bounds and Noncritical Solutions of Constrained Equations Set-Valued Var. Anal. (IF 1.476) Pub Date : 2020-11-09 A. Fischer, A. F. Izmailov, M. Jelitte
For many years, local Lipschitzian error bounds for systems of equations have been successfully used for the design and analysis of Newton-type methods. There are characterizations of those error bounds by means of first-order derivatives like a recent result by Izmailov, Kurennoy, and Solodov on critical solutions of nonlinear equations. We aim at extending this result in two directions which shall
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Second-Order Lagrange Multiplier Rules in Multiobjective Optimal Control of Semilinear Parabolic Equations Set-Valued Var. Anal. (IF 1.476) Pub Date : 2020-10-31 Tuan Nguyen Dinh
We consider multiobjective optimal control problems for semilinear parabolic systems subject to pointwise state constraints, integral state-control constraints and pointwise state-control constraints. In addition, the data of the problems need not be twice Fréchet differentiable. Employing the second-order directional derivative (in the sense of Demyanov-Pevnyi) for the involved functions, we establish
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Metric Regularity and Lyusternik-Graves Theorem via Approximate Fixed Points of Set-Valued Maps in Noncomplete Metric Spaces Set-Valued Var. Anal. (IF 1.476) Pub Date : 2020-10-29 Mohamed Ait Mansour, Mohamed Amin Bahraoui, Adham El Bekkali
This paper considers global metric regularity and approximate fixed points of set-valued mappings. We establish a very general Theorem extending to noncomplete metric spaces a recent result by A.L. Dontchev and R.T. Rockafellar on sharp estimates of the distance from a point to the set of exact fixed points of composition set-valued mappings. In this way, we find again the famous Nadler’s Theorem,
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The ABC of DC Programming Set-Valued Var. Anal. (IF 1.476) Pub Date : 2020-10-28 Welington de Oliveira
A function is called DC if it is expressible as the difference of two convex functions. In this work, we present a short tutorial on difference-of-convex optimization surveying and highlighting some important facts about DC functions, optimality conditions, and recent algorithms. The manuscript, accessible to a wide range of readers familiar with the convex analysis machinery, builds upon three pillars
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Set-Convergence and Its Application: A Tutorial Set-Valued Var. Anal. (IF 1.476) Pub Date : 2020-10-23 Johannes O. Royset
Optimization problems, generalized equations, and a multitude of other variational problems invariably lead to the analysis of sets and set-valued mappings as well as their approximations. We review the central concept of set-convergence and explain its role in defining a notion of proximity between sets, especially for epigraphs of functions and graphs of set-valued mappings. The development leads
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The Obstacle Problem at Zero for the Fractional p -Laplacian Set-Valued Var. Anal. (IF 1.476) Pub Date : 2020-10-22 Silvia Frassu, Eugénio M. Rocha, Vasile Staicu
In this paper we establish a multiplicity result for a class of unilateral, nonlinear, nonlocal problems with nonsmooth potential (variational-hemivariational inequalities), using the degree map of multivalued perturbations of fractional nonlinear operators of monotone type, the fact that the degree at a local minimizer of the corresponding Euler functional is equal one, and controlling the degree
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Nonsmoothness in Machine Learning: Specific Structure, Proximal Identification, and Applications Set-Valued Var. Anal. (IF 1.476) Pub Date : 2020-10-19 Franck Iutzeler, Jérôme Malick
Nonsmoothness is often a curse for optimization; but it is sometimes a blessing, in particular for applications in machine learning. In this paper, we present the specific structure of nonsmooth optimization problems appearing in machine learning and illustrate how to leverage this structure in practice, for compression, acceleration, or dimension reduction. We pay a special attention to the presentation
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State-Constrained Control-Affine Parabolic Problems I: First and Second Order Necessary Optimality Conditions Set-Valued Var. Anal. (IF 1.476) Pub Date : 2020-10-17 M. Soledad Aronna, J. Frédéric Bonnans, Axel Kröner
In this paper we consider an optimal control problem governed by a semilinear heat equation with bilinear control-state terms and subject to control and state constraints. The state constraints are of integral type, the integral being with respect to the space variable. The control is multidimensional. The cost functional is of a tracking type and contains a linear term in the control variables. We
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Continuous Newton-like Inertial Dynamics for Monotone Inclusions Set-Valued Var. Anal. (IF 1.476) Pub Date : 2020-10-15 Hedy Attouch, Szilárd Csaba László
In a Hilbert framework ℌ, we study the convergence properties of a Newton-like inertial dynamical system governed by a general maximally monotone operator A ℌ: → 2ℌ. When A is equal to the subdifferential of a convex lower semicontinuous proper function, the dynamic corresponds to the introduction of the Hessian-driven damping in the continuous version of the accelerated gradient method of Nesterov
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Measure Differential Inclusions: Existence Results and Minimum Problems Set-Valued Var. Anal. (IF 1.476) Pub Date : 2020-10-10 Luisa Di Piazza, Valeria Marraffa, Bianca Satco
We focus on a very general problem in the theory of dynamic systems, namely that of studying measure differential inclusions with varying measures. The multifunction on the right hand side has compact non-necessarily convex values in a real Euclidean space and satisfies bounded variation hypotheses with respect to the Pompeiu excess (and not to the Hausdorff-Pompeiu distance, as usually in literature)
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The Stationary Point Set Map in General Parametric Optimization Problems Set-Valued Var. Anal. (IF 1.476) Pub Date : 2020-10-09 D. T. K. Huyen, J.-C. Yao, N. D. Yen
The present paper shows how the linear independence constraint qualification (LICQ) can be combined with some conditions put on the first-order and second-order derivatives of the objective function and the constraint functions to ensure the Robinson stability and the Lipschitz-like property of the stationary point set map of a general C2-smooth parametric constrained optimization problem. So, a part
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A Discussion on Variational Analysis in Derivative-Free Optimization Set-Valued Var. Anal. (IF 1.476) Pub Date : 2020-09-25 Warren Hare
Variational Analysis studies mathematical objects under small variations. With regards to optimization, these objects are typified by representations of first-order or second-order information (gradients, subgradients, Hessians, etc). On the other hand, Derivative-Free Optimization studies algorithms for continuous optimization that do not use first-order information. As such, researchers might conclude
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On Existence of Solutions of Parametrized Generalized Equations Set-Valued Var. Anal. (IF 1.476) Pub Date : 2020-09-22 Asen L. Dontchev
In this paper we study existence of solutions to the generalized equation 0 ∈ f(p,x) + F(x), where f is a function, F is a set-valued mapping, and p is a parameter. Conditions are given, in terms of metric regularity of F, local convex-valuedness of F− 1, and partial calmness of f with respect to x uniformly in p, for the property that, for any p near the reference value, the generalized equation has
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Error Bounds for Approximate Solutions of Abstract Inequality Systems and Infinite Systems of Inequalities on Banach Spaces Set-Valued Var. Anal. (IF 1.476) Pub Date : 2020-09-11 Jinhua Wang, Mingwu Ye, Sy-Ming Guu, Chong Li
Using the result of the error estimate of the simple extended Newton method established in the present paper for solving abstract inequality systems, we study the error bound property of approximate solutions of abstract inequality systems on Banach spaces with the involved function F being Fréchet differentiable and its derivative \(F^{\prime }\) satisfying the center-Lipschitz condition (not necessarily
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A Discussion of Probability Functions and Constraints from a Variational Perspective Set-Valued Var. Anal. (IF 1.476) Pub Date : 2020-09-05 Wim van Ackooij
Probability constraints are a popular modelling mechanism in applications. They help to model feasible decisions when the latter are taken prior to observing uncertainty and both decisions and uncertainty are involved in a constraint structure of an optimization problem. The popularity of this paradigm is attested by a vast literature using probability constraints. In this work we try to provide, with
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The Gradient Projection Algorithm for Smooth Sets and Functions in Nonconvex Case Set-Valued Var. Anal. (IF 1.476) Pub Date : 2020-08-01 Maxim V. Balashov
We consider the problem of minimization for a function with Lipschitz continuous gradient on a proximally smooth and smooth manifold in a finite dimensional Euclidean space. We consider the Lezanski-Polyak-Lojasiewicz (LPL) conditions in this problem of constrained optimization. We prove that the gradient projection algorithm for the problem converges with a linear rate when the LPL condition holds
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On the Positive Definiteness of Limiting Coderivative for Set-Valued Mappings Set-Valued Var. Anal. (IF 1.476) Pub Date : 2020-07-30 T. T. A. Nghia, D. T. Pham, T. T. T. Tran
This paper concerns the interconnection between the positive definiteness of limiting coderivative and the local strong maximal monotonicity of set-valued mappings suspected in Mordukhovich and Nghia (SIAM J. Optim. 26, 1032–1059, 2016, Conjecture 3.6). We disprove the conjecture by a counterexample and provide some special classes at which it is true. However, the positive definiteness of limiting
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On the FPV Property, Monotone Operator Structure and the Monotone Polar of Representable Monotone Sets Set-Valued Var. Anal. (IF 1.476) Pub Date : 2020-07-20 A. Eberhard, R. Wenczel
We study the pointwise partial ordering of representative functions for a monotone operator and in particular we focus on the bigger conjugate representative functions that represent a fixed initial (non-maximal) monotone operator. The first problem considered is that of constructing a new representative function from a given member of this class when wanting to add an additional monotonically related
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Continuous Dynamics Related to Monotone Inclusions and Non-Smooth Optimization Problems Set-Valued Var. Anal. (IF 1.476) Pub Date : 2020-07-13 Ernö Robert Csetnek
The aim of this survey is to present the main important techniques and tools from variational analysis used for first and second order dynamical systems of implicit type for solving monotone inclusions and non-smooth optimization problems. The differential equations are expressed by means of the resolvent (in case of a maximally monotone set valued operator) or the proximal operator for non-smooth
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Upper Semicontinuity of the Solution Map to a Parametric Elliptic Optimal Control Problem Set-Valued Var. Anal. (IF 1.476) Pub Date : 2020-07-10 N. H. Son, N. B. Giang
This paper studies solution stability of a parametric optimal control problem governed by semilinear elliptic equations and nonconvex objective function with mixed pointwise constrains in which the controls act both in the domain and on the boundary. We give sufficient conditions under which the solution map is upper semicontinuous and continuous in parameters.
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Transversality Properties: Primal Sufficient Conditions Set-Valued Var. Anal. (IF 1.476) Pub Date : 2020-06-09 Nguyen Duy Cuong, Alexander Y. Kruger
The paper studies ‘good arrangements’ (transversality properties) of collections of sets in a normed vector space near a given point in their intersection. We target primal (metric and slope) characterizations of transversality properties in the nonlinear setting. The Hölder case is given a special attention. Our main objective is not formally extending our earlier results from the Hölder to a more
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Regularization of Brézis pseudomonotone variational inequalities Set-Valued Var. Anal. (IF 1.476) Pub Date : 2020-06-02 M. Bianchi, G. Kassay, R. Pini
In this paper we prove the existence of solutions of regularized set-valued variational inequalities involving Brézis pseudomonotone operators in reflexive and locally uniformly convex Banach spaces. By taking advantage of this result, we approximate a general set-valued variational inequality with suitable regularized set-valued variational inequalities, and we show that their solutions weakly converge
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On a Time and State Dependent Maximal Monotone Operator Coupled with a Sweeping Process with Perturbations Set-Valued Var. Anal. (IF 1.476) Pub Date : 2020-05-29 Dalila Azzam-Laouir, Messaouda Benguessoum, Charles Castaing
In this paper, we state, in separable Hilbert spaces, the existence of absolutely continuous solutions for a couple of evolution problems governed by time and state dependent maximal monotone operator and closed convex sweeping process, with perturbations.
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A Reflected Forward-Backward Splitting Method for Monotone Inclusions Involving Lipschitzian Operators Set-Valued Var. Anal. (IF 1.476) Pub Date : 2020-03-30 Volkan Cevher, Bằng Công Vũ
In this paper, we propose a novel splitting method for finding a zero point of the sum of two monotone operators where one of them is Lipschizian. The weak convergence the method is proved in real Hilbert spaces. Applying the proposed method to composite monotone inclusions involving parallel sums yields a new primal-dual splitting which is different from the existing methods. Connections to existing
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New inertial factors of the Krasnosel’skiı̆-Mann iteration Set-Valued Var. Anal. (IF 1.476) Pub Date : 2020-03-26 Yunda Dong
We consider inertial iterative schemes for approximating a fixed point of any given non-expansive operator in real Hilbert spaces. We provide new conditions on the inertial factors that ensure weak convergence and depend only on the iteration coefficients. For the special case of the Douglas-Rachford splitting, the conditions boil down to a sufficiently small upper bound on the sequence of inertial
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A Global Linear and Local Superlinear (Quadratic) Inexact Non-Interior Continuation Method for Variational Inequalities Over General Closed Convex Sets Set-Valued Var. Anal. (IF 1.476) Pub Date : 2020-03-18 Le Thi Khanh Hien, Chek Beng Chua
We use the concept of barrier-based smoothing approximations to extend the non-interior continuation method, which was proposed by B. Chen and N. Xiu for nonlinear complementarity problems based on Chen-Mangasarian smoothing functions, to an inexact non-interior continuation method for variational inequalities over general closed convex sets. Newton equations involved in the method are solved inexactly
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Regularity of the Minimum Time and of Viscosity Solutions of Degenerate Eikonal Equations via Generalized Lie Brackets Set-Valued Var. Anal. (IF 1.476) Pub Date : 2020-03-14 Martino Bardi, Ermal Feleqi, Pierpaolo Soravia
In this paper we relax the current regularity theory for the eikonal equation by using the recent theory of set-valued iterated Lie brackets. We give sufficient conditions for small time local attainability of general, symmetric, nonlinear systems, which have as a consequence the Hölder regularity of the minimum time function in optimal control. We then apply such result to prove Hölder continuity
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A Constant Rank Constraint Qualification in Continuous-Time Nonlinear Programming Set-Valued Var. Anal. (IF 1.476) Pub Date : 2020-03-02 Moisés Rodrigues Cirilo do Monte, Valeriano Antunes de Oliveira
The paper addresses continuous-time nonlinear programming problems with equality and inequality constraints. First and second order necessary optimality conditions are obtained under a constant rank type constraint qualification. The first order necessary conditions are of Karush-Kuhn-Tucker type.
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BV Continuous Solutions of an Evolution Inclusion with Maximal Monotone Operator and Nonconvex-Valued Perturbation. Existence Theorem Set-Valued Var. Anal. (IF 1.476) Pub Date : 2020-03-02 A. A. Tolstonogov
An evolution inclusion with the right-hand side containing a time-dependent maximal monotone operator and a multivalued mapping with closed nonconvex values is studied in a separable Hilbert space. The dependence of the maximal monotone operator on time is described with the help of the distance between maximal monotone operators in the sense of Vladimirov. This distance as a function of time has bounded
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Categorizing with Catastrophic Radii in Numerical Minimization Set-Valued Var. Anal. (IF 1.476) Pub Date : 2020-03-02 Adam B. Levy
We introduce and develop a notion of “catastrophic radii” to identify where a minimization method may require an arbitrarily large number of steps to approximate a minimizer of an objective function, and we use this notion to categorize the performance of method/objective combinations. In order to investigate the different categories, we explore simple examples where explicit formulas can be used,
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An Inverse Mapping Theorem in Fréchet-Montel Spaces Set-Valued Var. Anal. (IF 1.476) Pub Date : 2020-02-17 Radek Cibulka, Marián Fabian, Tomáš Roubal
Influenced by a recent note by M. Ivanov and N. Zlateva, we prove a statement in the style of Nash-Moser-Ekeland theorem for mappings from a Fréchet-Montel space with values in any Fréchet space (not necessarily standard). The mapping under consideration is supposed to be continuous and directionally differentiable (in particular Gateaux differentiable) with the derivative having a right inverse. We
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Deep Neural Network Structures Solving Variational Inequalities Set-Valued Var. Anal. (IF 1.476) Pub Date : 2020-02-13 Patrick L. Combettes, Jean-Christophe Pesquet
Motivated by structures that appear in deep neural networks, we investigate nonlinear composite models alternating proximity and affine operators defined on different spaces. We first show that a wide range of activation operators used in neural networks are actually proximity operators. We then establish conditions for the averagedness of the proposed composite constructs and investigate their asymptotic
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Some New Characterizations of Intrinsic Transversality in Hilbert Spaces Set-Valued Var. Anal. (IF 1.476) Pub Date : 2020-02-12 Nguyen Hieu Thao, Hoa T. Bui, Nguyen Duy Cuong, Michel Verhaegen
Motivated by a number of questions concerning transversality-type properties of pairs of sets recently raised by Ioffe and Kruger, this paper reports several new characterizations of the intrinsic transversality property in Hilbert spaces. New results in terms of normal vectors clarify the picture of intrinsic transversality, its variants and sufficient conditions for subtransversality, and unify several
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Finite Alternation Theorems and a Constructive Approach to Piecewise Polynomial Approximation in Chebyshev Norm Set-Valued Var. Anal. (IF 1.476) Pub Date : 2020-02-07 Jean-Pierre Crouzeix, Nadezda Sukhorukova, Julien Ugon
One of the purposes in this paper is to provide a better understanding of the alternance property which occurs in Chebyshev polynomial approximation and continuous piecewise polynomial approximation problems. In the first part of this paper, we prove that alternating sequences of any continuous function are finite in any given segment and then propose an original approach to obtain new proofs of the
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Set-Valued Means Set-Valued Var. Anal. (IF 1.476) Pub Date : 2020-02-03 Kazimierz Nikodem
The notion of set-valued means is introduced. Set-valued counterparts of the arithmetic, quasi-arithmetic and Lagrangian means are investigated and various properties of them are presented.
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Evolution Inclusions Governed by Time-Dependent Maximal Monotone Operators with a Full Domain Set-Valued Var. Anal. (IF 1.476) Pub Date : 2020-02-01 Emilio Vilches, Bao Tran Nguyen
In this paper, we study the existence of solutions for evolution inclusions governed by time-dependent maximal monotone operators with a full domain. Without assumptions concerning time-regularity on the time-dependent maximal monotone operators, and by using the Moreau-Yosida regularization technique, we establish the existence of solutions in Hilbert spaces. The theoretical result is applied to prove
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Epi/Hypo-Convergence of Bifunctions on General Domains and Approximations of Quasi-Variational Problems Set-Valued Var. Anal. (IF 1.476) Pub Date : 2020-01-30 H. T. H. Diem, P. Q. Khanh
Epi/hypo-convergence is extended to the case of bifunctions defined on general domains. Its basic characterizations are established. Variational properties such as those about saddle points, weak saddle points, minsup-points, sup-projections, etc, of bifunctions are shown to be preserved for their epi/hypo-limits (possibly under some additional assumptions). Approximations of quasi-equilibrium problems
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Immobile Indices and CQ-Free Optimality Criteria for Linear Copositive Programming Problems Set-Valued Var. Anal. (IF 1.476) Pub Date : 2020-01-27 O. I. Kostyukova, T. V. Tchemisova, O. S. Dudina
We consider problems of linear copositive programming where feasible sets consist of vectors for which the quadratic forms induced by the corresponding linear matrix combinations are nonnegative over the nonnegative orthant. Given a linear copositive problem, we define immobile indices of its constraints and a normalized immobile index set. We prove that the normalized immobile index set is either
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On Linear Transformations of Intersections Set-Valued Var. Anal. (IF 1.476) Pub Date : 2020-01-21 Alexey Kushnir, Shuo Liu
For any linear transformation and two convex closed sets, we provide necessary and sufficient conditions for the transformation of the intersection of the sets to coincide with the intersection of their images. We also identify conditions for non-convex closed sets, continuous transformations, and multiple sets. We demonstrate the usefulness of our results via an application to the economics literature
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On a Class of Lur’e Dynamical Systems with State-Dependent Set-Valued Feedback Set-Valued Var. Anal. (IF 1.476) Pub Date : 2020-01-14 Ba Khiet Le
Using a new implicit discretization scheme, we study in this paper the existence and uniqueness of strong solutions for a class of Lur’e dynamical systems where the set-valued feedback depends on both the time and the state. This work is a generalization of Tanwani et al. (SIAM J. Control Opti. 56(2), 751–781, 2018) where the time-dependent set-valued feedback is considered to acquire weak solutions
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Directional Metric Pseudo Subregularity of Set-valued Mappings: a General Model Set-Valued Var. Anal. (IF 1.476) Pub Date : 2019-11-25 Huynh Van Ngai, Nguyen Huu Tron, Nguyen Van Vu, Michel Théra
This paper investigates a new general pseudo subregularity model which unifies some important nonlinear (sub)regularity models studied recently in the literature. Some slope and abstract coderivative characterizations are established.
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Bartle-Graves Theorem Revisited Set-Valued Var. Anal. (IF 1.476) Pub Date : 2019-11-21 Asen L. Dontchev
We take a fresh look at the Bartle-Graves theorem pointing out the main differences with the standard implicit function theorem. We then present a set-valued version of this theorem which generalizes some recent results. Applications to variational inequalities and differential inclusions are also given.
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The Radius of Metric Subregularity Set-Valued Var. Anal. (IF 1.476) Pub Date : 2019-11-19 Asen L. Dontchev, Helmut Gfrerer, Alexander Y. Kruger, Jiří V. Outrata
There is a basic paradigm, called here the radius of well-posedness, which quantifies the “distance” from a given well-posed problem to the set of ill-posed problems of the same kind. In variational analysis, well-posedness is often understood as a regularity property, which is usually employed to measure the effect of perturbations and approximations of a problem on its solutions. In this paper we
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Correction to: Optimal Control Involving Sweeping Processes Set-Valued Var. Anal. (IF 1.476) Pub Date : 2019-09-09 M. d. R. de Pinho,M. M. A. Ferreira,G. V. Smirnov
We would like to make corrections to a result, Lemma 2, in the above paper.
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Metric Regularity of Quasidifferentiable Mappings and Optimality Conditions for Nonsmooth Mathematical Programming Problems Set-Valued Var. Anal. (IF 1.476) Pub Date : 2019-09-04 M. V. Dolgopolik
This article is devoted to the analysis of necessary and/or sufficient conditions for metric regularity in terms of Demyanov-Rubinov-Polyakova quasidifferentials. We obtain new necessary and sufficient conditions for the local metric regularity of a multifunction in terms of quasidifferentials of the distance function to this multifunction. We also propose a new MFCQ-type constraint qualification for
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New Sharp Necessary Optimality Conditions for Mathematical Programs with Equilibrium Constraints Set-Valued Var. Anal. (IF 1.476) Pub Date : 2019-08-29 Helmut Gfrerer, Jane J. Ye
In this paper, we study the mathematical program with equilibrium constraints formulated as a mathematical program with a parametric generalized equation involving the regular normal cone. We derive a new necessary optimality condition which is sharper than the usual M-stationary condition and is applicable even when no constraint qualifications hold for the corresponding mathematical program with
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The U$\mathcal {U}$ -Lagrangian, Fast Track, and Partial Smoothness of a Prox-regular Function Set-Valued Var. Anal. (IF 1.476) Pub Date : 2019-08-27 Shuai Liu, Andrew Eberhard, Yousong Luo
When restricted to a subspace, a nonsmooth function can be differentiable. It is known that for a nonsmooth convex function and a point, the Euclidean space can be decomposed into two subspaces: \(\mathcal {U}\), over which a special Lagrangian can be defined and has nice smooth properties and \(\mathcal {V}\), the orthogonal complement subspace of \(\mathcal {U}\). In this paper we generalize the
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