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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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A Projection Algorithm for Non-Monotone Variational Inequalities Set-Valued Var. Anal. (IF 1.476) Pub Date : 2019-07-25 Regina S. Burachik; R. Díaz Millán
We introduce a projection-type algorithm for solving the variational inequality problem for point-to-set operators, and establish its convergence properties. Namely, we assume that the operator of the variational inequality is continuous in the point-to-set sense, i.e., inner- and outer-semicontinuous. Under the assumption that the dual solution set is not empty, we prove that our method converges
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Stability Analysis for Parameterized Variational Systems with Implicit Constraints Set-Valued Var. Anal. (IF 1.476) Pub Date : 2019-07-03 Matúš Benko; Helmut Gfrerer; Jiří V. Outrata
In the paper we provide new conditions ensuring the isolated calmness property and the Aubin property of parameterized variational systems with constraints depending, apart from the parameter, also on the solution itself. Such systems include, e.g., quasi-variational inequalities and implicit complementarity problems. Concerning the Aubin property, possible restrictions imposed on the parameter are
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Duality for the Robust Sum of Functions Set-Valued Var. Anal. (IF 1.476) Pub Date : 2019-06-20 N. Dinh; M. A. Goberna; M. Volle
In this paper we associate with an infinite family of real extended functions defined on a locally convex space a sum, called robust sum, which is always well-defined. We also associate with that family of functions a dual pair of problems formed by the unconstrained minimization of its robust sum and the so-called optimistic dual. For such a dual pair, we characterize weak duality, zero duality gap
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Lyapunov Stability of Differential Inclusions with Lipschitz Cusco Perturbations of Maximal Monotone Operators Set-Valued Var. Anal. (IF 1.476) Pub Date : 2019-06-10 Samir Adly; Abderrahim Hantoute; Bao Tran Nguyen
We give new criteria for weak and strong invariant closed sets for differential inclusions in \(\mathbb {R}^{n}\), and which are simultaneously governed by Lipschitz Cusco mapping and by maximal monotone operators. Correspondingly, we provide different characterizations for the associated strong Lyapunov functions and pairs. The resulting conditions only depend on the data of the system, while the
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Viability Constraints for Computing Solutions to the Lighthill-Whitham-Richards Model Involving Partial Autonomous Vehicle Flow Set-Valued Var. Anal. (IF 1.476) Pub Date : 2019-06-05 Tian Lei; Christian G. Claudel
This article proposes a new algorithm for computing solutions to mixed flow problems involving the classical Lighthill-Whitham-Richards (LWR) flow model. We show that the behavior of autonomous vehicles can be described by viability constrained solutions to an alternate Hamilton-Jacobi formulation of the same model, for appropriate upper constraint functions. We also provide the physical interpretation
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Qualitative Properties of the Solution Set for Time-Delayed Discontinuous Dynamics Set-Valued Var. Anal. (IF 1.476) Pub Date : 2019-04-18 Norma Ortiz-Robinson; Vinicio Ríos
This article presents a survey of several properties of the set of solutions for a differential inclusion involving a time-delayed component and with right-hand side parametrized by either an upper semicontinuous or lower semicontinuous multifunction. Our results include: existence of solutions, compactness and contractibility of the solution and reachable sets in the upper semicontinuous case, precompactness
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On Inexact Relative-Error Hybrid Proximal Extragradient, Forward-Backward and Tseng’s Modified Forward-Backward Methods with Inertial Effects Set-Valued Var. Anal. (IF 1.476) Pub Date : 2019-03-15 M. Marques Alves; Raul T. Marcavillaca
For solving monotone inclusion problems, we propose an inertial under-relaxed version of the relative-error hybrid proximal extragradient method. We study the asymptotic convergence of the method, as well as its nonasymptotic global convergence rates in terms of iteration complexity. We analyze the new method under more flexible assumptions than existing ones, both on the extrapolation and on the relative-error
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Topological Properties of the Space of Convex Minimal Usco Maps Set-Valued Var. Anal. (IF 1.476) Pub Date : 2019-03-08 Ľubica Holá; Branislav Novotný
We investigate the space of convex minimal usco maps from a Tychonoff space to the space of real numbers. Its elements are set-valued maps that are important e.g. in the study of subdifferentials of convex functions. We show that if the underlying space is normal, convex minimal usco maps can be approximated in the Vietoris topology by continuous functions. Using the strong Choquet game we prove complete
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Generalized Derivatives for the Solution Operator of the Obstacle Problem Set-Valued Var. Anal. (IF 1.476) Pub Date : 2019-02-07 Anne-Therese Rauls; Gerd Wachsmuth
We characterize generalized derivatives of the solution operator of the obstacle problem. This precise characterization requires the usage of the theory of so-called capacitary measures and the associated solution operators of relaxed Dirichlet problems. The generalized derivatives can be used to obtain a novel necessary optimality condition for the optimal control of the obstacle problem with control
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