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Invariant Closed Sets with Respect to Differential Inclusions with Time-Dependent Maximal Monotone Operators Set-Valued Var. Anal. (IF 1.6) Pub Date : 2024-03-12 Dalila Azzam-Laouir, Karima Dib
The main purpose of the present paper is the characterization, in the finite dimensional setting, of weak and strong invariance of closed sets with respect to a differential inclusion governed by time-dependent maximal monotone operators and multi-valued perturbation, by the use of the corresponding Hamiltonians.
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Conjugation-Based Approach to the $\varepsilon $ -Subdifferential of Convex Suprema Set-Valued Var. Anal. (IF 1.6) Pub Date : 2024-03-12 Rafael Correa, Abderrahim Hantoute, Marco A. López
We provide new characterizations of the \(\varepsilon \)-subdifferential of the supremum of an arbitrary family of convex functions. The resulting formulas only involve approximate subdifferentials of adequate convex combinations of the data functions. Families of convex functions with a concavity-like property are introduced and their relationship with affine models is studied. The role of the lower
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Probability Functions Generated by Set-Valued Mappings: A Study of First Order Information Set-Valued Var. Anal. (IF 1.6) Pub Date : 2024-02-29 Wim van Ackooij, Pedro Pérez-Aros, Claudia Soto
Probability functions appear in constraints of many optimization problems in practice and have become quite popular. Understanding their first-order properties has proven useful, not only theoretically but also in implementable algorithms, giving rise to competitive algorithms in several situations. Probability functions are built up from a random vector belonging to some parameter-dependent subset
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Evolution Integro-Differential Inclusions Set-Valued Var. Anal. (IF 1.6) Pub Date : 2024-02-09 Abderrahim Bouach, Tahar Haddad, Lionel Thibault
The aim of the present paper is to state and discuss the well-posedness of a new evolution inclusion governed by the subdifferential of a function \(\varphi \) perturbed both by a Carathéodory mapping and by an integral forcing term. The integrand of the forcing term depends on two time-variables. We prove the existence and uniqueness of local and global solution, assuming that \(\varphi \) is primal
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Representative Functions, Variational Convergence and Almost Convexity Set-Valued Var. Anal. (IF 1.6) Pub Date : 2024-02-07 A. Eberhard, R. Wenczel
We develop a new epi-convergence based on the use of bounded convergent nets on the product topology of the strong topology on the primal space and weak star topology on the dual space of a general real Banach space. We study the propagation of the associated variational convergences through conjugation of convex functions defined on this product space. These results are then applied to the problem
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Strong Solutions and Mild Solutions for Sturm-Liouville Differential Inclusions Set-Valued Var. Anal. (IF 1.6) Pub Date : 2024-01-30 Tiziana Cardinali, Giulia Duricchi
Existence results for a Cauchy problem driven by a semilinear differential Sturm-Liouville inclusion are achived by proving, in a preliminary way, an existence theorem for a suitable integral inclusion. In order to obtain this proposition we use a recent fixed point theorem that allows us to work with the weak topology and the De Blasi measure of weak noncompactness. So we avoid requests of compactness
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BV Sweeping Process Involving Prox-Regular Sets and a Composed Perturbation Set-Valued Var. Anal. (IF 1.6) Pub Date : 2024-01-25 Alexander Tolstonogov
In this paper we study the existence and properties of solutions for a discontinuous sweeping process involving prox-regular sets in a Hilbert spaces. The variation of the moving set is controlled by a positive Radon measure and the perturbation is the sum of two multivalued mappings. The values of the first one are closed, bounded, not necessarily convex sets. It is measurable in the time variable
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Approximations of Quasi-Equilibria and Nash Quasi-Equilibria in Terms of Variational Convergence Set-Valued Var. Anal. (IF 1.6) Pub Date : 2023-12-08 Huynh Thi Hong Diem, Phan Quoc Khanh
Various types of variational convergence of functions and bifunctions are applied to study global approximations of a quasi-equilibrium problem and a Nash quasi-game (generalized noncooperative game), two typical and important optimization models. We prove that when the data of approximating problems of a problem under consideration converge in the sense of suitable types of epi-, epi/hypo-, lop-convergence
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On Diffeologies for Power Sets and Measures Set-Valued Var. Anal. (IF 1.6) Pub Date : 2023-11-21 Alireza Ahmadi, Jean-Pierre Magnot
We consider a differential geometric setting on power sets and Borel algebras. Our chosen framework is based on diffeologies, and we make a link between the various diffeological structures that we propose, having in mind set-valued maps, relations, set-valued gradients, differentiable measures, and shape analysis. This work intends to establish rigorous properties on sample diffeologies that seem
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Regularity of Sets Under a Reformulation in a Product Space with Reduced Dimension Set-Valued Var. Anal. (IF 1.6) Pub Date : 2023-11-20 Rubén Campoy
Different notions on regularity of sets and of collection of sets play an important role in the analysis of the convergence of projection algorithms in nonconvex scenarios. While some projection algorithms can be applied to feasibility problems defined by finitely many sets, some other require the use of a product space reformulation to construct equivalent problems with two sets. In this work we analyze
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A Strong Law of Large Numbers for Random Monotone Operators Set-Valued Var. Anal. (IF 1.6) Pub Date : 2023-11-06 Adil Salim
Random monotone operators are stochastic versions of maximal monotone operators which play an important role in stochastic nonsmooth optimization. Several stochastic nonsmooth optimization algorithms have been shown to converge to a zero of a mean operator defined as the expectation, in the sense of the Aumann integral, of a random monotone operator. In this note, we prove a strong law of large numbers
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Nonlinear Forward-Backward Splitting with Momentum Correction Set-Valued Var. Anal. (IF 1.6) Pub Date : 2023-11-03 Martin Morin, Sebastian Banert, Pontus Giselsson
The nonlinear, or warped, resolvent recently explored by Giselsson and Bùi-Combettes has been used to model a large set of existing and new monotone inclusion algorithms. To establish convergent algorithms based on these resolvents, corrective projection steps are utilized in both works. We present a different way of ensuring convergence by means of a nonlinear momentum term, which in many cases leads
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Projectional Coderivatives and Calculus Rules Set-Valued Var. Anal. (IF 1.6) Pub Date : 2023-10-30 Wenfang Yao, Kaiwen Meng, Minghua Li, Xiaoqi Yang
This paper is devoted to the study of a newly introduced tool, projectional coderivatives, and the corresponding calculus rules in finite dimensional spaces. We show that when the restricted set has some nice properties, more specifically, it is a smooth manifold, the projectional coderivative can be refined as a fixed-point expression. We will also improve the generalized Mordukhovich criterion to
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Caristi-Type Conditions in Constraint Minimisation of Mappings in Metric and Partially Ordered Spaces Set-Valued Var. Anal. (IF 1.6) Pub Date : 2023-10-24 Evgeny Zhukovskiy, Evgenii Burlakov, Ivan Malkov
We consider the problem of finding minima of mappings defined on metric and partially ordered spaces subject to constraints in the form of inclusions (and as a consequence in the form of equalities and/or inequalities). We introduce analogues of Caristi-type inequality proposed in the studies on minimisation of non-convex functionals in metric spaces. Statements on attainment of minima of non-convex
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Viscous Approximations of Non-Convex Sweeping Processes in the Space of Regulated Functions Set-Valued Var. Anal. (IF 1.6) Pub Date : 2023-09-19 Pavel Krejčí, Giselle A. Monteiro, Vincenzo Recupero
Vanishing viscosity approximations are considered here for discontinuous sweeping processes with non-convex constraints. It is shown that they are well-posed for sufficiently small viscosity parameters, and that their solutions converge pointwise, as the viscosity parameter tends to zero, to the left-continuous solution to the sweeping process in the Kurzweil integral setting. The convergence is uniform
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Singular Value Analysis of Linear Maps Under Conic Constraints Set-Valued Var. Anal. (IF 1.6) Pub Date : 2023-09-19 Alberto Seeger, David Sossa
We have recently introduced and studied the concept of singular value of a rectangular matrix relative to a pair of closed convex cones. Such cones act as constraint sets for the left-singular and right-singular vectors. This work extends the theory of cone-constrained singular value problems from rectangular matrices to linear maps between Euclidean spaces.
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Nonlocal Error Bounds for Piecewise Affine Functions Set-Valued Var. Anal. (IF 1.6) Pub Date : 2023-09-14 M. V. Dolgopolik
The paper is devoted to a detailed analysis of nonlocal error bounds for nonconvex piecewise affine functions. We both improve some existing results on error bounds for such functions and present completely new necessary and/or sufficient conditions for a piecewise affine function to have an error bound on various types of bounded and unbounded sets. In particular, we show that any piecewise affine
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Viability and Filippov-type Lemma for Stieltjes Differential Inclusions Set-Valued Var. Anal. (IF 1.6) Pub Date : 2023-09-06 Bianca Satco, George Smyrlis
We prove a viability result for differential inclusions involving the Stieltjes derivative with respect to a left-continuous non-decreasing function with time dependent state constraints. A tangential condition using a generalized notion of the contingent derivative is imposed. Classical viability results (for usual differential inclusions) are thus generalized and, at the same time, the gate to new
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A Note on the Hausdorff Distance Between Norm Balls and Their Linear Maps Set-Valued Var. Anal. (IF 1.6) Pub Date : 2023-09-01 Shadi Haddad, Abhishek Halder
We consider the problem of computing the (two-sided) Hausdorff distance between the unit \(\ell _{p_{1}}\) and \(\ell _{p_{2}}\) norm balls in finite dimensional Euclidean space for \(1 \leq p_{1} < p_{2} \leq \infty \), and derive a closed-form formula for the same. We also derive a closed-form formula for the Hausdorff distance between the \(k_{1}\) and \(k_{2}\) unit \(D\)-norm balls, which are
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The Homogenization Cone: Polar Cone and Projection Set-Valued Var. Anal. (IF 1.6) Pub Date : 2023-08-14 Heinz H. Bauschke, Theo Bendit, Hansen Wang
Let C be a closed convex subset of a real Hilbert space containing the origin, and assume that K is the homogenization cone of C, i.e., the smallest closed convex cone containing \(C\times \{1\}\). Homogenization cones play an important role in optimization for the construction of examples and counterexamples. A famous examples is the second-order/Lorentz/“ice cream” cone which is the homogenization
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Random Lift of Set Valued Maps and Applications to Multiagent Dynamics Set-Valued Var. Anal. (IF 1.6) Pub Date : 2023-08-11 Rossana Capuani, Antonio Marigonda, Michele Ricciardi
We introduce an abstract framework for the study of general mean field games and mean field control problems. Given a multiagent system, its macroscopic description is provided by a time-depending probability measure, where at every instant of time the measure of a set represents the fraction of (microscopic) agents contained in it. The trajectories available to each of the microscopic agents are affected
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Variational Analysis of Norm Cones in Finite Dimensional Euclidean Spaces Set-Valued Var. Anal. (IF 1.6) Pub Date : 2023-08-10 Haoyang Liu, Jia Wu, Liwei Zhang
A norm cone in a finite dimensional Euclidean space is the epigraph of a norm. Many important practical optimization problems are formulated as norm conic optimization problems, a typical example is the second-order conic optimization problem. This paper is devoted to the study of variational analysis of norm cones. For a general norm cone, formulas for the tangent cone, normal cone and second-order
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Relaxed Constant Positive Linear Dependence Constraint Qualification for Disjunctive Systems Set-Valued Var. Anal. (IF 1.6) Pub Date : 2023-08-08 Mengwei Xu, Jane J. Ye
The disjunctive system is a system involving a disjunctive set which is the union of finitely many polyhedral convex sets. In this paper, we introduce a notion of the relaxed constant positive linear dependence constraint qualification (RCPLD) for the disjunctive system. For a disjunctive system, our notion is weaker than the one we introduced for a more general system recently (J. Glob. Optim. 2020)
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Impulsive Input-to-State Stabilization of an Ensemble Set-Valued Var. Anal. (IF 1.6) Pub Date : 2023-08-03 Ivan Atamas, Sergey Dashkovskiy, Vitalii Slynko
We consider an ensemble of trajectories generated by a linear differential equation subjected to disturbance and parameterized by the initial state. The scalar output of the system is the volume comprised by the states of the whole ensemble. Already the unperturbed dynamics is assumed to be unstable. In order to stabilize the system with unknown inputs in the ISS sense we design impulsive control actions
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On Successive Approximations for Compact-Valued Nonexpansive Mappings Set-Valued Var. Anal. (IF 1.6) Pub Date : 2023-07-18 Emir Medjic
We show that for a given initial point the typical, in the sense of Baire category, nonexpansive compact valued mapping F has the following properties: there is a unique sequence of successive approximations and this sequence converges to a fixed point of F. In the case of separable Banach spaces we show that for the typical mapping there is a residual set of initial points that have a unique trajectory
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A Control Space Ensuring the Strong Convergence of Continuous Approximation for a Controlled Sweeping Process Set-Valued Var. Anal. (IF 1.6) Pub Date : 2023-07-11 Chadi Nour, Vera Zeidan
A controlled sweeping process with prox-regular set, \(W^{1,2}\)-controls, and separable endpoints constraints is considered in this paper. Existence of optimal solutions is established and local optimality conditions are derived via strong converging continuous approximations, whose state entirely resides in the interior of the prox-regular set. Consequently, subdifferentials smaller than the standard
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Resolvent and Proximal Compositions Set-Valued Var. Anal. (IF 1.6) Pub Date : 2023-07-10 Patrick L. Combettes
We introduce the resolvent composition, a monotonicity-preserving operation between a linear operator and a set-valued operator, as well as the proximal composition, a convexity-preserving operation between a linear operator and a function. The two operations are linked by the fact that, under mild assumptions, the subdifferential of the proximal composition of a convex function is the resolvent composition
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Characteristic curves for Set-Valued Hamilton-Jacobi Equations Set-Valued Var. Anal. (IF 1.6) Pub Date : 2023-07-06 D. Visetti
The method of characteristics is extended to set-valued Hamilton-Jacobi equations. This problem arises from a calculus of variations’ problem with a multicriteria Lagrangian function: through an embedding into a set-valued framework, a set-valued Hamilton-Jacobi equation is derived, where the Hamiltonian function is the Fenchel conjugate of the Lagrangian function. In this paper a method of characteristics
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The Radius of Metric Regularity Revisited Set-Valued Var. Anal. (IF 1.6) Pub Date : 2023-07-01 Helmut Gfrerer, Alexander Y. Kruger
The paper extends the radius of metric regularity theorem by Dontchev, Lewis and Rockafellar (2003) by providing an exact formula for the radius with respect to Lipschitz continuous perturbations in general Asplund spaces, thus, answering affirmatively an open question raised twenty years ago by Ioffe. In the non-Asplund case, we give a natural upper bound for the radius complementing the conventional
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Conservative Parametric Optimality and the Ridge Method for Tame Min-Max Problems Set-Valued Var. Anal. (IF 1.6) Pub Date : 2023-06-22 Edouard Pauwels
We study the ridge method for min-max problems, and investigate its convergence without any convexity, differentiability or qualification assumption. The central issue is to determine whether the “parametric optimality formula” provides a conservative gradient, a notion of generalized derivative well suited for optimization. The answer to this question is positive in a semi-algebraic, and more generally
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Metric Regularity for Set-Valued Maps in Fréchet-Montel Spaces. Implicit Mapping Theorem Set-Valued Var. Anal. (IF 1.6) Pub Date : 2023-05-30 M. Ivanov, M. Quincampoix, N. Zlateva
This paper is devoted to new derivative criterion of metric regularity for set-valued map from a Fréchet-Montel space to a Fréchet space. Such type of criteria have been well studied in Banach spaces. Our work extends recent studies motivated by Nash-Moser-Ekeland approach. As a consequence of our criterion, an implicit mapping result is also given in the paper.
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Coincidence and Fixed Points of Set-Valued Mappings Via Regularity in Metric Spaces Set-Valued Var. Anal. (IF 1.6) Pub Date : 2023-05-29 Nguyen Huu Tron
In this paper, we construct a common iterative scheme that allows to unify two important results established recently by Ioffe and Ait Mansour, Bahraoui, El Bekkali, respectively. Our results rely on a weaker concept of metric regularity, called orbital regularity. Some applications are given to approximate and/or exact coincidence double fixed point problems as well as to the perturbation stability
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Solving monotone inclusions involving the sum of three maximally monotone operators and a cocoercive operator with applications Set-Valued Var. Anal. (IF 1.6) Pub Date : 2023-04-28 Chunxiang Zong, Yuchao Tang, Guofeng Zhang
In this paper, we propose two four-operator splitting algorithms for approaching the set of zeros of the sum of three maximally monotone operators and a cocoercive operator. Our methods do not rely on the traditional product space technique. The sequences of the proposed algorithms are proven to be convergent under mild conditions. In applications of interest to us, we employ the proposed algorithms
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On the Weak Second-order Optimality Condition for Nonlinear Semidefinite and Second-order Cone Programming Set-Valued Var. Anal. (IF 1.6) Pub Date : 2023-04-26 Ellen H. Fukuda, Gabriel Haeser, Leonardo M. Mito
Second-order necessary optimality conditions for nonlinear conic programming problems that depend on a single Lagrange multiplier are usually built under nondegeneracy and strict complementarity. In this paper we establish a condition of such type for two classes of nonlinear conic problems, namely semidefinite and second-order cone programming, assuming Robinson’s constraint qualification and a weak
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The Intrinsic Core and Minimal Faces of Convex Sets in General Vector Spaces Set-Valued Var. Anal. (IF 1.6) Pub Date : 2023-04-05 R. Díaz Millán, Vera Roshchina
Intrinsic core generalises the finite-dimensional notion of the relative interior to arbitrary (real) vector spaces. Our main goal is to provide a self-contained overview of the key results pertaining to the intrinsic core and to elucidate the relations between intrinsic core and facial structure of convex sets in this general context. We gather several equivalent definitions of the intrinsic core
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A Mean Value Theorem for Tangentially Convex Functions Set-Valued Var. Anal. (IF 1.6) Pub Date : 2023-03-27 Juan Enrique Martínez-Legaz
The main result is an equality type mean value theorem for tangentially convex functions in terms of tangential subdifferentials, which generalizes the classical one for differentiable functions, as well as Wegge theorem for convex functions. The new mean value theorem is then applied, analogously to what is done in the classical case, to characterize, in the tangentially convex context, Lipschitz
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Michael Selections and Castaing Representations with càdlàg Functions Set-Valued Var. Anal. (IF 1.6) Pub Date : 2023-03-14 Ari-Pekka Perkkiö, Erick Treviño-Aguilar
It follows from Michael’s selection theory that a closed convex nonempty-valued mapping from the Sorgenfrey line to a euclidean space is inner semicontinuous if and only if the mapping can be represented as the image closure of right-continuous selections of the mapping. This article gives necessary and sufficient conditions for the representation to hold for càdlàg selections, i.e., for selections
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A Radius of Robust Feasibility for Uncertain Farthest Voronoi Cells Set-Valued Var. Anal. (IF 1.6) Pub Date : 2023-03-10 Andrea B. Ridolfi, Virginia N. Vera de Serio
Given an arbitrary non-empty set of sites, T in \(\mathbb {R}^{n},\) and a specific site s ∈ T, the farthest Voronoi cell of s is the set consisting of all points in \(\mathbb {R}^{n}\) that are farther from s, with respect to the Euclidean norm, than from any other site in T. In this paper, we propose a new concept of a radius of robust feasibility for farthest Voronoi cells in the ball uncertainty
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Duality for Sets of Strong Slater Points Set-Valued Var. Anal. (IF 1.6) Pub Date : 2023-03-07 Margarita M. L. Rodríguez, José Vicente-Pérez
The strong Slater condition plays a significant role in the stability analysis of linear semi-infinite inequality systems. This piece of work studies the set of strong Slater points, whose non-emptiness guarantees the fullfilment of the strong Slater condition. Given a linear inequality system, we firstly establish some basic properties of the set of strong Slater points. Then, we derive dual characterizations
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Optimality Conditions for Mathematical Programs with Orthogonality Type Constraints Set-Valued Var. Anal. (IF 1.6) Pub Date : 2023-03-03 S. Lämmel, V. Shikhman
We consider the class of mathematical programs with orthogonality type constraints. Orthogonality type constraints appear by reformulating the sparsity constraint via auxiliary binary variables and relaxing them afterwards. For mathematical programs with orthogonality type constraints a necessary optimality condition in terms of T-stationarity is stated. The justification of T-stationarity is threefold
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Necessary Conditions in Infinite-Horizon Control Problems that Need no Asymptotic Assumptions Set-Valued Var. Anal. (IF 1.6) Pub Date : 2023-02-23 Dmitry Khlopin
We consider an infinite-horizon optimal control problem with an asymptotic terminal constraint. For the weakly overtaking criterion and the overtaking criterion, necessary boundary conditions on co-state arcs are deduced, these conditions need no assumptions about the asymptotic behavior of the motion, co-state arc, cost functional, and its derivatives. In the absence of an asymptotic terminal constraint
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Dynamics of Nonautomous Impulsive Multivalued Processes Set-Valued Var. Anal. (IF 1.6) Pub Date : 2023-02-10 Tomás Caraballo, José M. Uzal
In this paper we study the asymptotic behaviour of multivalued processes which are under the influence of impulsive action. We provide conditions to guarantee the existence of a pullback attractor and we illustrate the results with several examples.
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Non-Linear Operators and Differentiability of Lipschitz Functions Set-Valued Var. Anal. (IF 1.6) Pub Date : 2023-02-07 Mohammed Bachir, Sebastián Tapia-García
In this work we provide a characterization of distinct types of (linear and non-linear) maps between Banach spaces in terms of the differentiability of certain class of Lipschitz functions. Our results are stated in an abstract bornological and non-linear framework. Restricted to the linear case, we can apply our results to compact, weakly-compact, limited and completely continuous linear operators
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Fast Continuous Dynamics Inside the Graph of Maximally Monotone Operators Set-Valued Var. Anal. (IF 1.6) Pub Date : 2023-02-03 Paul-Emile Maingé, André Weng-Law
In a Hilbert framework, we introduce a new class of fast continuous dissipative dynamical systems for approximating zeroes of an arbitrary maximally monotone operator. This system originates from some change of variable operated in a continuous Nesterov-like model that is driven by the Yosida regularization of the operator and that involves an asymptotic vanishing damping. The proposed model (based
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Sensitivity Analysis of Stochastic Constraint and Variational Systems via Generalized Differentiation Set-Valued Var. Anal. (IF 1.6) Pub Date : 2023-02-02 Boris S. Mordukhovich, Pedro Pérez-Aros
This paper conducts sensitivity analysis of random constraint and variational systems related to stochastic optimization and variational inequalities. We establish efficient conditions for well-posedness, in the sense of robust Lipschitzian stability and/or metric regularity, of such systems by employing and developing coderivative characterizations of well-posedness properties for random multifunctions
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Sequential Constant Rank Constraint Qualifications for Nonlinear Semidefinite Programming with Algorithmic Applications Set-Valued Var. Anal. (IF 1.6) Pub Date : 2023-01-27 Roberto Andreani, Gabriel Haeser, Leonardo M. Mito, Héctor Ramírez
We present new constraint qualification conditions for nonlinear semidefinite programming that extend some of the constant rank-type conditions from nonlinear programming. As an application of these conditions, we provide a unified global convergence proof of a class of algorithms to stationary points without assuming neither uniqueness of the Lagrange multiplier nor boundedness of the Lagrange multipliers
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Limiting Behaviour of the Generalized Simplex Gradient as the Number of Points Tends to Infinity on a Fixed Shape in IRn Set-Valued Var. Anal. (IF 1.6) Pub Date : 2023-01-20 Warren Hare, Gabriel Jarry-Bolduc, Chayne Planiden
This work investigates the asymptotic behaviour of the gradient approximation method called the generalized simplex gradient (GSG). This method has an error bound that at first glance seems to tend to infinity as the number of sample points increases, but with some careful construction, we show that this is not the case. For functions in finite dimensions, we present two new error bounds ad infinitum
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Multi-dimensional Path-dependent Forward-backward Stochastic Variational Inequalities Set-Valued Var. Anal. (IF 1.6) Pub Date : 2023-01-20 Ning Ning, Jing Wu
In this article, we consider a system of stochastic variational inequalities (SVIs) in the differential form. The system has a d-dimensional forward SVI X that depends on its path and carries a subdifferential operator, and a n-dimensional backward SVI coupled with X through the path of X and has another subdifferential operator. This system extends all classical stochastic processes to SVIs with general
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The Boosted DC Algorithm for Linearly Constrained DC Programming Set-Valued Var. Anal. (IF 1.6) Pub Date : 2022-12-21 F. J. Aragón-Artacho, R. Campoy, P. T. Vuong
The Boosted Difference of Convex functions Algorithm (BDCA) has been recently introduced to accelerate the performance of the classical Difference of Convex functions Algorithm (DCA). This acceleration is achieved thanks to an extrapolation step from the point computed by DCA via a line search procedure. In this work, we propose an extension of BDCA that can be applied to difference of convex functions
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On the Application of the SCD Semismooth* Newton Method to Variational Inequalities of the Second Kind Set-Valued Var. Anal. (IF 1.6) Pub Date : 2022-11-28 Helmut Gfrerer, Jiří V. Outrata, Jan Valdman
The paper starts with a description of SCD (subspace containing derivative) mappings and the SCD Newton method for the solution of general inclusions. This method is then applied to a class of variational inequalities of the second kind. As a result, one obtains an implementable algorithm which exhibits locally superlinear convergence. Thereafter we suggest several globally convergent hybrid algorithms
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Generalized Differentiation of a Class of Normal Cone Operators and Sensitivity of Optimal Control Problems Set-Valued Var. Anal. (IF 1.6) Pub Date : 2022-11-28 N. T. Toan, L. Q. Thuy
This paper studies generalized differentiation for a class of normal cone operators and sensitivity of solution to a class of parametric discrete optimal control problems. We first establish the Fréchet and Mordukhovich coderivatives of a class of normal cone operators in Asplund spaces. We then use the obtained results to compute and estimate the Fréchet coderivative of the solution map to parametric
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About Error Bounds in Metrizable Topological Vector Spaces Set-Valued Var. Anal. (IF 1.6) Pub Date : 2022-09-23 Malek Abbasi, Michel Théra
This paper aims to present some sufficient criteria under which a given function between two spaces that are either topological vector spaces whose topologies are generated by metrics or metrizable subsets of some topological vector spaces, satisfies the error bound property. Then, we discuss the Hoffman estimation and obtain some results for the estimate of the distance to the set of solutions to
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Forward-partial inverse-half-forward splitting algorithm for solving monotone inclusions Set-Valued Var. Anal. (IF 1.6) Pub Date : 2022-08-31 Luis Briceño-Arias, Jinjian Chen, Fernando Roldán, Yuchao Tang
In this paper we provide a splitting algorithm for solving coupled monotone inclusions in a real Hilbert space involving the sum of a normal cone to a vector subspace, a maximally monotone, a monotone-Lipschitzian, and a cocoercive operator. The proposed method takes advantage of the intrinsic properties of each operator and generalizes the method of partial inverses and the forward-backward-half forward
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Generalized Differentiation and Duality in Infinite Dimensions under Polyhedral Convexity Set-Valued Var. Anal. (IF 1.6) Pub Date : 2022-08-08 D. V. Cuong, B. S. Mordukhovich, N. M. Nam, G. Sandine
This paper addresses the study and applications of polyhedral duality in locally convex topological vector (LCTV) spaces. We first revisit the classical Rockafellar’s proper separation theorem for two convex sets one of which is polyhedral and then present its LCTV extension replacing the relative interior by its quasi-relative interior counterpart. Then we apply this result to derive enhanced calculus
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Sums of Squares Polynomial Program Reformulations for Adjustable Robust Linear Optimization Problems with Separable Polynomial Decision Rules Set-Valued Var. Anal. (IF 1.6) Pub Date : 2022-07-28 V. Jeyakumar, J. H. Lee, G. M. Lee, G. Li, D. Woolnough
We show that adjustable robust linear programs with affinely adjustable box data uncertainties under separable polynomial decision rules admit exact sums of squares (SOS) polynomial reformulations. These problems share the same optimal values and admit a one-to-one correspondence between the optimal solutions. A sum of squares representation of non-negativity of a separable non-convex polynomial over
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Convex Robust Sum Optimization Problems with Conic and Set Constraints: Duality and Optimality Conditions Revisited Set-Valued Var. Anal. (IF 1.6) Pub Date : 2022-07-22 Nguyen Dinh, Dang Hai Long, Michel Volle
This paper deals with optimization problems consisting in the minimization of a robust sum of infinitely many functions under a conic and a set constraints. Through suitable perturbation functions, the problem is embedded into a family of linearly perturbed problems which have an associated qualifying set which is contained in the Cartesian product of the dual space to the decision space by the real
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Stability and Sensitivity of Uncertain Linear Programs Set-Valued Var. Anal. (IF 1.6) Pub Date : 2022-07-19 María J. Cánovas, Juan Parra
The present paper deals with uncertain linear optimization problems where the objective function coefficient vector belongs to a compact convex uncertainty set and the feasible set is described by a linear semi-infinite inequality system (finitely many variables and possibly infinitely many constrainsts), whose coefficients are also uncertain. Perturbations of both, the objective coefficient vector
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On Subdifferentials Via a Generalized Conjugation Scheme: An Application to DC Problems and Optimality Conditions Set-Valued Var. Anal. (IF 1.6) Pub Date : 2022-06-16 M.D. Fajardo, J. Vidal
This paper studies properties of a subdifferential defined using a generalized conjugation scheme. We relate this subdifferential together with the domain of an appropriate conjugate function and the ε-directional derivative. In addition, we also present necessary conditions for ε-optimality and global optimality in optimization problems involving the difference of two convex functions. These conditions
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On the Nonemptiness and Boundedness of Solution Sets of Weakly Homogeneous Optimization Problems Set-Valued Var. Anal. (IF 1.6) Pub Date : 2022-06-07 Vu Trung Hieu
In this paper, we introduce a new class of optimization problems whose objective functions are weakly homogeneous relative to the constraint sets. By using the normalization argument in asymptotic analysis, we prove two criteria for the nonemptiness and boundedness of the solution set of a weakly homogeneous optimization problem. Moreover, we discuss the existence and stability of the solution sets
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Suns, Moons, and $$\mathring{B}$$ B ˚ -complete Sets in Asymmetric Spaces Set-Valued Var. Anal. (IF 1.6) Pub Date : 2022-05-26 Alexey R. Alimov, Igor’ G. Tsar’kov
Classical concepts and problems of geometric approximation theory are considered in normed and asymmetric spaces. Relations between strict suns, sets with outer radially continuous (ORL continuous) metric projection, unimodal sets, \(\mathring{B}\)-complete sets, and moons are studied. The concepts of a B-sun and a local B-sun in an asymmetric space are introduced (a set is called a B-sun if its intersection