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 Set-Valued and Variational AnalysisJournal Prestige (SJR): 0.641 Citation Impact (citeScore): 1Number of Followers: 0      Hybrid journal (It can contain Open Access articles) ISSN (Print) 1877-0533 - ISSN (Online) 1877-0541 Published by Springer-Verlag  [2469 journals]
• Nonconvex and Nonsmooth Approaches for Affine Chance-Constrained
Stochastic Programs

Abstract: Chance-constrained programs (CCPs) constitute a difficult class of stochastic programs due to its possible nondifferentiability and nonconvexity even with simple linear random functionals. Existing approaches for solving the CCPs mainly deal with convex random functionals within the probability function. In the present paper, we consider two generalizations of the class of chance constraints commonly studied in the literature; one generalization involves probabilities of disjunctive nonconvex functional events and the other generalization involves mixed-signed affine combinations of the resulting probabilities; together, we coin the term affine chance constraint (ACC) system for these generalized chance constraints. Our proposed treatment of such an ACC system involves the fusion of several individually known ideas: (a) parameterized upper and lower approximations of the indicator function in the expectation formulation of probability; (b) external (i.e., fixed) versus internal (i.e., sequential) sampling-based approximation of the expectation operator; (c) constraint penalization as relaxations of feasibility; and (d) convexification of nonconvexity and nondifferentiability via surrogation. The integration of these techniques for solving the affine chance-constrained stochastic program (ACC-SP) is the main contribution of this paper. Indeed, combined together, these ideas lead to several algorithmic strategies with various degrees of practicality and computational efforts for the nonconvex ACC-SP. In an external sampling scheme, a given sample batch (presumably large) is applied to a penalty formulation of a fixed-accuracy approximation of the chance constraints of the problem via their expectation formulation. This results in a sample average approximation scheme, whose almost-sure convergence under a directional derivative condition to a Clarke stationary solution of the expectation constrained-SP as the sample sizes tend to infinity is established. In contrast, sequential sampling, along with surrogation leads to a sequential convex programming based algorithm whose asymptotic convergence for fixed- and diminishing-accuracy approximations of the indicator function can be established under prescribed increments of the sample sizes.
PubDate: 2022-05-07

• On the Generalized Jacobian of the Inverse of a Lipschitzian Mapping

Abstract: The objective of this short note is to provide an estimate of the generalized Jacobian of the inverse of a Lipschitzian mapping when Clarke’s inverse function theorem applies. Contrary to the classical $$\mathcal {C}^{1}$$ case, inverting matrices of the generalized Jacobian is not enough. Simple counterexamples show that our results are sharp.
PubDate: 2022-05-06

• Convergence of Constant Step Stochastic Gradient Descent for Non-Smooth
Non-Convex Functions

Abstract: Abstract This paper studies the asymptotic behavior of the constant step Stochastic Gradient Descent for the minimization of an unknown function, defined as the expectation of a non convex, non smooth, locally Lipschitz random function. As the gradient may not exist, it is replaced by a certain operator: a reasonable choice is to use an element of the Clarke subdifferential of the random function; another choice is the output of the celebrated backpropagation algorithm, which is popular amongst practioners, and whose properties have recently been studied by Bolte and Pauwels. Since the expectation of the chosen operator is not in general an element of the Clarke subdifferential of the mean function, it has been assumed in the literature that an oracle of the Clarke subdifferential of the mean function is available. As a first result, it is shown in this paper that such an oracle is not needed for almost all initialization points of the algorithm. Next, in the small step size regime, it is shown that the interpolated trajectory of the algorithm converges in probability (in the compact convergence sense) towards the set of solutions of a particular differential inclusion: the subgradient flow. Finally, viewing the iterates as a Markov chain whose transition kernel is indexed by the step size, it is shown that the invariant distribution of the kernel converge weakly to the set of invariant distribution of this differential inclusion as the step size tends to zero. These results show that when the step size is small, with large probability, the iterates eventually lie in a neighborhood of the critical points of the mean function.
PubDate: 2022-04-08

• What is a Lipschitzian Manifold'

Abstract: Abstract We propose a definition of Lipschizian manifold that is more precise than the notion of Lipschitzian parameterization. It is modelled on the notion of differentiable manifold. We also give a notion of Lipschitzian submanifold and compare it with a notion devised by R.T. Rockafellar (Ann. I.H.P. Sect. C 2(3), 167–184, 1985). Among the examples we mention, the case of the graph of a maximally monotone operator and of the subjet of a convex function are the most notable.
PubDate: 2022-04-05

• Calmness of the Solution Mapping of Navier-Stokes Problems Modeled by
Hemivariational Inequalities

Abstract: The main purpose of this paper is to find conditions for Hölder calmness of the solution mapping, viewed as a function of the boundary data, of a hemivariational inequality governed by the Navier-Stokes operator. To this end, a more abstract model is studied first: a class of parametric equilibrium problems defined by trifunctions. The presence of trifunctions allows the extension of the monotonicity notions in the theory of equilibrium problems.
PubDate: 2022-04-05

• Stability of Minimization Problems and the Error Bound Condition

Abstract: Abstract It is well known that Error Bound conditions provide some (usually linear or sublinear) rate of convergence for gradient descent methods in unconstrained and, in certain cases, in constrained optimization. We prove that the same conditions in constrained optimization guarantee stability of minimization problems: if we slightly change the function and the set then the solution set can not change much. Both the function and the set are not necessarily convex. We obtain an upper bound for the Hausdorff halfdistance between solutions via the function from the Error bound condition. In a real Hilbert space or in $$\mathbb {R}^{n}$$ these results generalize known results about stability of convex functionals.
PubDate: 2022-04-01

• Limits of Eventual Families of Sets with Application to Algorithms for the
Common Fixed Point Problem

Abstract: We present an abstract framework for asymptotic analysis of convergence based on the notions of eventual families of sets that we define. A family of subsets of a given set is called here an “eventual family” if it is upper hereditary with respect to inclusion. We define accumulation points of eventual families in a Hausdorff topological space and define the “image family” of an eventual family. Focusing on eventual families in the set of the integers enables us to talk about sequences of points. We expand our work to the notion of a “multiset” which is a modification of the concept of a set that allows for multiple instances of its elements and enable the development of “multifamilies” which are either “increasing” or “decreasing”. The abstract structure created here is motivated by, and feeds back to, our look at the convergence analysis of an iterative process for asymptotically finding a common fixed point of a family of operators.
PubDate: 2022-03-23

• Characterizations of Some Transversality-Type Properties

Abstract: Abstract Primal characterizations (necessary and sufficient conditions) and slope characterizations of subtransversality, intrinsic transversality and transversality are obtained. The metric nature of intrinsic transversality is established. The relation of intrinsic transversality and tangential transversality is clarified. The equivalence of our characterization of intrinsic transversality and the primal characterization of intrinsic transversality of Thao et al. in the setting of Hilbert spaces is proved, while in general Banach spaces our characterization is less restrictive.
PubDate: 2022-03-23

• Set-Valued Evenly Convex Functions: Characterizations and C-Conjugacy

Abstract: Abstract In this work we deal with set-valued functions with values in the power set of a separated locally convex space where a nontrivial pointed convex cone induces a partial order relation. A set-valued function is evenly convex if its epigraph is an evenly convex set, i.e., it is the intersection of an arbitrary family of open half-spaces. In this paper we characterize evenly convex set-valued functions as the pointwise supremum of its set-valued e-affine minorants. Moreover, a suitable conjugation pattern will be developed for these functions, as well as the counterpart of the biconjugation Fenchel-Moreau theorem.
PubDate: 2022-03-12

• Erratum to: New Constraint Qualifications and Optimality Conditions for
Second Order Cone Programs

Abstract: Abstract 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.
PubDate: 2022-03-01
DOI: 10.1007/s11228-021-00573-5

• R-Regularity of Set-Valued Mappings Under the Relaxed Constant Positive
Linear Dependence Constraint Qualification with Applications to Parametric
and Bilevel Optimization

Abstract: Abstract The presence of Lipschitzian properties for solution mappings associated with nonlinear parametric optimization problems is desirable in the context of, e.g., stability analysis or bilevel optimization. An example of such a Lipschitzian property for set-valued mappings, whose graph is the solution set of a system of nonlinear inequalities and equations, is R-regularity. Based on the so-called relaxed constant positive linear dependence constraint qualification, we provide a criterion ensuring the presence of the R-regularity property. In this regard, our analysis generalizes earlier results of that type which exploited the stronger Mangasarian–Fromovitz or constant rank constraint qualification. Afterwards, we apply our findings in order to derive new sufficient conditions which guarantee the presence of R-regularity for solution mappings in parametric optimization. Finally, our results are used to derive an existence criterion for solutions in pessimistic bilevel optimization and a sufficient condition for the presence of the so-called partial calmness property in optimistic bilevel optimization.
PubDate: 2022-03-01
DOI: 10.1007/s11228-021-00578-0

• Error Bounds for Approximate Solutions of Abstract Inequality Systems and
Infinite Systems of Inequalities on Banach Spaces

Abstract: 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 the Lipschitz condition) around a point x0. Under some mild conditions, we establish results on the existence of the solutions, and the error bound properties for approximate solutions of abstract inequality systems. Applications of these results to finite/infinite systems of inequalities/equalities on Banach spaces are presented and the error bound properties of approximate solutions of finite/infinite systems of inequalities/equalities are also established. Our results extend the corresponding results in [3, 18, 19].
PubDate: 2022-03-01
DOI: 10.1007/s11228-020-00551-3

• Perturbation Techniques for Convergence Analysis of Proximal Gradient
Method and Other First-Order Algorithms via Variational Analysis

Abstract: Abstract We develop new 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 set-valued map, together with a local version of the proper separation of stationary value condition, is a sufficient condition to ensure the linear convergence of the algorithm. The equivalence of the calmness condition to the one for the canonically perturbed stationary point set-valued map is proved, and this equivalence allows us to derive some sufficient conditions for calmness by using some recent developments in variational analysis. These sufficient conditions are different from existing results (especially, those error-bound-based ones) in that they can be easily verified for many concrete application models. Our analysis is focused on the fundamental proximal gradient (PG) method, and it enables us to show that any accumulation of the sequence generated by the PG method must be a stationary point in terms of the proximal subdifferential, instead of the limiting subdifferential. This result finds the surprising fact that the solution quality found by the PG method is in general superior. Our analysis also leads to some improvement for the linear convergence results of the PG method in the convex case. The new perturbation technique can be conveniently used to derive linear rate convergence of a number of other first-order methods including the well-known alternating direction method of multipliers and primal-dual hybrid gradient method, under mild assumptions.
PubDate: 2022-03-01
DOI: 10.1007/s11228-020-00570-0

• Primal Superlinear Convergence of Sqp Methods in Piecewise

Abstract: Abstract 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 the noncriticality condition with the second-order sufficient condition, this primal superlinear convergence is equivalent with an appropriate version of the Dennis-Moré condition. We also recover Bonnans’ result in (Appl. Math. Optim. 29, 161–186, 1994) for the primal-dual superlinear of the basic SQP method for this class of composite problems under the second-order sufficient condition and the uniqueness of Lagrange multipliers. To achieve these goals, we first obtain an extension of the reduction lemma for convex Piecewise linear-quadratic functions and then provide a comprehensive analysis of the noncriticality of Lagrange multipliers for composite problems. We also establish certain primal estimates for KKT systems of composite problems, which play a significant role in our local convergence analysis of the quasi-Newton SQP method.
PubDate: 2022-03-01
DOI: 10.1007/s11228-021-00580-6

• The Stationary Point Set Map in General Parametric Optimization Problems

Abstract: Abstract 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 of the results in two preceding papers of the authors [J. Optim. Theory Appl. 180 (2019), 91–116 (Part 1); 117–139 (Part 2)], which were obtained for a problem with just one inequality constraint, now has an adequate extension for problems having finitely many equality and inequality constraints. Our main tool is an estimate of B. S. Mordukhovich and R. T. Rockafellar [SIAM J. Optim. 22 (2012), 953–986; Theorem 3.3] for a second-order partial subdifferential of a composite function. The obtained results are illustrated by three examples.
PubDate: 2022-03-01
DOI: 10.1007/s11228-020-00557-x

• Sufficient Conditions for Metric Subregularity of Constraint Systems with
Applications to Disjunctive and Ortho-Disjunctive Programs

Abstract: Abstract 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 on applications to disjunctive programs, where (directional) pseudo-normality is characterized via an extremal condition. This, in turn, yields efficient tools to verify pseudo-normality and the metric subregularity constraint qualification, which include, but are not limited to, Robinson’s result on polyhedral multifunctions and Gfrerer’s second-order sufficient condition for metric subregularity. Finally, we refine our study by defining the new class of ortho-disjunctive programs which comprises prominent optimization problems such as mathematical programs with complementarity, vanishing or switching constraints.
PubDate: 2022-03-01
DOI: 10.1007/s11228-020-00569-7

• The Obstacle Problem at Zero for the Fractional p-Laplacian

Abstract: Abstract 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 at small balls and at big balls.
PubDate: 2022-03-01
DOI: 10.1007/s11228-020-00562-0

• Calmness of a Perturbed Cournot Oligopoly Game with Nonsmooth Cost
Functions

Abstract: Abstract 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 on a mathematical induction on the number of players. It is concluded that the methodology used for the proofs can be replicated, in order to study the metric subregularity and calmness of multifunctions in a more general way.
PubDate: 2022-03-01
DOI: 10.1007/s11228-021-00577-1

• Second-Order Lagrange Multiplier Rules in Multiobjective Optimal Control
of Semilinear Parabolic Equations

Abstract: Abstract 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 necessary optimality conditions, via second-order Lagrange multiplier rules of Fritz-John type, for local weak Pareto solutions of the problems.
PubDate: 2022-03-01
DOI: 10.1007/s11228-020-00555-z

• Calmness and Calculus: Two Basic Patterns

Abstract: Abstract We establish two types of estimates for generalized derivatives of set-valued mappings which carry the essence of two basic patterns observed throughout the pile of calculus rules. These estimates also illustrate the role of the essential assumptions that accompany these two patters, namely calmness on the one hand and (fuzzy) inner calmness* on the other. Afterwards, we study the relationship between and sufficient conditions for the various notions of (inner) calmness. The aforementioned estimates are applied in order to recover several prominent calculus rules for tangents and normals as well as generalized derivatives of marginal functions and compositions as well as Cartesian products of set-valued mappings under mild conditions. We believe that our enhanced approach puts the overall generalized calculus into some other light. Some applications of our findings are presented which exemplary address necessary optimality conditions for minimax optimization problems as well as the calculus related to the recently introduced semismoothness* property.
PubDate: 2022-03-01
DOI: 10.1007/s11228-021-00589-x

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