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In this paper, we establish through an openness condition the metric subregularity of a multimapping with normal ω (·) -regularity of either the graph or values. Various preservation results for prox-regular and subsmooth sets are also provided. [ABSTRACT FROM AUTHOR]

Sets of bilinear constraints are important in various machine learning models. Mathematically, they are hyperbolas in a product space. In this paper, we give a complete formula for projections onto sets of bilinear constraints or hyperbolas in a general Hilbert space. [ABSTRACT FROM AUTHOR]

Abstract We study a mechanical system with a finite number of degrees of freedom, subjected to perfect time-dependent frictionless unilateral (possibly nonconvex) constraints with inelastic collisions on active constraints. The dynamic is described in the form of a second-order measure differential inclusion. Under some regularity assumptions on the data, we establish several properties of the set of admissible positions, which is not necessarily convex but assumed to be uniformly prox-regular. Our approach does not require any second-order information or boundedness of the Hessians of the constraints involved in the problem and are specific to moving sets represented by inequalities constraints. On that basis, we are able to discretize our problem by the time-stepping algorithm and construct a sequence of approximate solutions. It is shown that this sequence possesses a subsequence converging to a solution of the initial problem. This methodology is not only used to prove an existence result but could be also used to solve numerically the vibroimpact problem with time-dependent nonconvex constraints.

This paper mainly studies the quadratic growth and the strong metric subregularity of the subdifferential of a function that can be represented as the sum of a function twice differentiable in the extended sense and a subdifferentially continuous, prox-regular, twice epi-differentiable function. For such a function, which is not necessarily prox-regular, it is shown that the quadratic growth, the strong metric subregularity of the subdifferential at a local minimizer, and the positive definiteness of the subgradient graphical derivative at a stationary point are equivalent. In addition, other characterizations of the quadratic growth and the strong metric subregularity of the subdifferential are also given. Besides, properties of functions twice differentiable in the extended sense are examined. [ABSTRACT FROM AUTHOR]

In this paper, we provide in a general Hilbert space new characterizations of uniform prox-regularity involving outside but sufficiently close points of considered sets. We show that the complement of a prox-regular set is nothing but the union of closed balls with common radius. We derive from this that the prox-regularity of a given closed set is equivalent to the semiconvexity property of its distance function. Various estimates involving the metric projection mapping to a prox-regular set are also established. [ABSTRACT FROM AUTHOR]

In the setting adopted by Edmond and Thibault [Relaxation of an optimal control problem involving a perturbed sweeping process, Math. Program. 104 (2005), 347-373], we study a class of perturbed sweeping processes. Under suitable assumptions, we obtain two solution existence theorems for the perturbed sweeping processes with the constraint sets being prox-regular sublevel sets. The results are applied to the behavior of some concrete mechanical sweeping processes, which appear for the first time in this paper. [ABSTRACT FROM AUTHOR]

This paper introduces a new class of equilibrium problems named general regularized nonconvex mixed equilibrium problem. By using the auxiliary principle technique, some predictor–corrector methods for solving such class of general regularized nonconvex mixed equilibrium problems are suggested and analyzed. The study of convergence analysis of the proposed iterative algorithms requires either jointly pseudomonotonicity or partially mixed relaxed and strong monotonicity property of the operator and bifunction involved in the considered problem. As a consequence of our main results, we provide the correct versions of the algorithms and results presented in the literature. [ABSTRACT FROM AUTHOR]

We systematically study the local single-valuedness of the Bregman proximal mapping and local smoothness of the Bregman–Moreau envelope of a nonconvex function under relative prox-regularity—an extension of prox-regularity—which was originally introduced by Poliquin and Rockafellar. As Bregman distances are asymmetric in general, in accordance with Bauschke et al., it is natural to consider two variants of the Bregman proximal mapping, which, depending on the order of the arguments, are called left and right Bregman proximal mapping. We consider the left Bregman proximal mapping first. Then, via translation result, we obtain analogue (and partially sharp) results for the right Bregman proximal mapping. The class of relatively prox-regular functions significantly extends the recently considered class of relatively hypoconvex functions. In particular, relative prox-regularity allows for functions with a possibly nonconvex domain. Moreover, as a main source of examples and analogously to the classical setting, we introduce relatively amenable functions, i.e. convexly composite functions, for which the inner nonlinear mapping is component-wise smooth adaptable, a recently introduced extension of Lipschitz differentiability. By way of example, we apply our theory to locally interpret joint alternating Bregman minimization with proximal regularization as a Bregman proximal gradient algorithm, applied to a smooth adaptable function. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a regularized nonconvex mixed equilibrium problem and suggest iterative algorithms for solving such a problem by using the auxiliary principle technique. The convergence analysis of the proposed iterative algorithms is discussed either under pseudomono-tonicity or partially mixed relaxed and strong monotonicity of type (I) property of the bifunctions involved in the formulation. We also point out some fatal errors in [M.A. Noor et al.: On non-convex bifunction variational inequalities. Optim. Lett. 6, 1477-1488 (2012)] and [M.A. Noor et al.: Some iterative methods for solving nonconvex bifunction equilibrium variational inequalities. J. Appl. Math. Volume 2012, Article ID 280451]. Finally, we present the correct version of the results presented in these references. [ABSTRACT FROM AUTHOR]

The paper introduces and studies the notions of Lipschitzian and Hölderian full stability of solutions to three-parametric variational systems described in the generalized equation formalism involving nonsmooth base mappings and partial subgradients of prox-regular functions acting in Hilbert spaces. Employing advanced tools and techniques of second-order variational analysis allows us to establish complete characterizations of, as well as directly verifiable sufficient conditions for, such full stability properties under mild assumptions. Furthermore, we derive exact formulas and effective quantitative estimates for the corresponding moduli. The obtained results are specified for important classes of variational inequalities and variational conditions in both finite and infinite dimensions. [ABSTRACT FROM AUTHOR]

We propose ZeroFPR, a nonmonotone linesearch algorithm for minimizing the sum of two nonconvex functions, one of which is smooth and the other possibly nonsmooth. ZeroFPR is the first algorithm that, despite being fit for fully nonconvex problems and requiring only the black-box oracle of forward-backward splitting (FBS)|namely evaluations of the gradient of the smooth term and of the proximity operator of the nonsmooth one|achieves superlinear convergence rates under mild assumptions at the limit point when the linesearch directions satisfy a Dennis{Moré condition, and we show that this is the case for Broyden's quasi-Newton directions. Our approach is based on the forward-backward envelope (FBE), an exact and strictly continuous penalty function for the original cost. Extending previous results we show that, despite being nonsmooth for fully nonconvex problems, the FBE still enjoys favorable first- and second-order properties which are key for the convergence results of ZeroFPR. Our theoretical results are backed up by promising numerical simulations. On large-scale problems, by computing linesearch directions using limited-memory quasi-Newton updates our algorithm greatly outperforms FBS and its accelerated variant (AFBS). [ABSTRACT FROM AUTHOR]

In this paper, we propose two iterative algorithms for finding the minimum-norm solution of a split minimization problem. We prove strong convergence of the sequences generated by the proposed algorithms. The iterative schemes are proposed in such a way that the selection of the step-sizes does not need any prior information about the operator norm. We further give some examples to numerically verify the efficiency and implementation of our new methods and compare the two algorithms presented. Our results act as supplements to several recent important results in this area. [ABSTRACT FROM AUTHOR]

A local convergence result for an abstract descent method is proved. The sequence of iterates is attracted by a local (or global) minimum, stays in its neighborhood, and converges within this neighborhood. This result allows algorithms to exploit local properties of the objective function. In particular, the abstract theory in this paper applies to the inertial forward-backward splitting method: iPiano—a generalization of the Heavy-ball method. Moreover, it reveals an equivalence between iPiano and inertial averaged/alternating proximal minimization and projection methods. Key for this equivalence is the attraction to a local minimum within a neighborhood and the fact that, for a prox-regular function, the gradient of the Moreau envelope is locally Lipschitz continuous and expressible in terms of the proximal mapping. In a numerical feasibility problem, the inertial alternating projection method significantly outperforms its non-inertial variants. [ABSTRACT FROM AUTHOR]

We synthesize and unify notions of regularity, both of individual sets and of collections of sets, as they appear in the convergence theory of projection methods for consistent feasibility problems. Several new characterizations of regularities are presented which shed light on the relations between seemingly different ideas and point to possible necessary conditions for local linear convergence of fundamental algorithms. [ABSTRACT FROM AUTHOR]

In this paper we prove a theorem concerning the existence of solutions for a second order sweeping process with a Lipschitz single valued perturbation in Hilbert space. [ABSTRACT FROM AUTHOR]

This work focuses on the class of multivalued regularized equilibrium problems in the context of uniformly prox-regular sets introduced and studied in [Noor, M.A.: Multivalued regularized equilibrium problems. J. Global Optim. 35, 483-492 (2006)]. The algorithms and results presented in the paper cited above are investigated and analyzed and some fatal errors in them are detected. Meanwhile, the correct version of the corresponding algorithm and results is given. [ABSTRACT FROM AUTHOR]

In this paper, we turn our attention to formulating and studying a new class of variational inequalities in a nonconvex setting, called regularized nonconvex mixed variational inequalities. By using the auxiliary principle technique, some new predictor corrector methods for solving such class of regularized nonconvex mixed variational inequalities are suggested and analyzed. The study of convergence analysis of the proposed iterative algorithms requires either pseudomonotonicity or partially mixed relaxed and strong monotonicity of the operator involved in regularized nonconvex mixed variational inequalities. As a consequence of our main results, we provide the correct versions of the algorithms and results presented in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we deal with sweeping processes on (possibly infinite-dimensional) Riemannian Hilbert manifolds. We extend the useful notions (proximal normal cone, prox-regularity) already defined in the setting of a Hilbert space to the framework of such manifolds. Especially we introduce the concept of local prox-regularity of a closed subset in accordance with the geometrical features of the ambient manifold and we check that this regularity implies a property of hypomonotonicity for the proximal normal cone. Moreover we show that the metric projection onto a locally prox-regular set is single-valued in its neighborhood. Then under some assumptions, we prove the well-posedness of perturbed sweeping processes by locally prox-regular sets. [ABSTRACT FROM AUTHOR]

In this paper, our aim is to introduce a viscosity type algorithm for solving proximal split feasibility problems and prove the strong convergence of the sequences generated by our iterative schemes in Hilbert spaces. First, we prove strong convergence result for a problem of finding a point which minimizes a convex function f such that its image under a bounded linear operator A minimizes another convex function g. Secondly, we prove another strong convergence result for the case where one of the two involved functions is prox-regular. In all our results in this work, our iterative schemes are proposed by way of selecting the step sizes such that their implementation does not need any prior information about the operator norm because the calculation or at least an estimate of the operator norm $\|A\|$ is not an easy task. Finally, we give a numerical example to study the efficiency and implementation of our iterative schemes. Our results complement the recent results of Moudafi and Thakur (Optim. Lett. 8:2099-2110, 2014, doi:) and other recent important results in this direction. [ABSTRACT FROM AUTHOR]

Our aim in this paper is to introduce iterative algorithms and prove their strong convergence for solving proximal split feasibility problems and fixed point problems for $$k$$ -strictly pseudocontractive mappings in Hilbert spaces. The sequence generated by our first iterative scheme converges strongly to an approximate common solution of convex minimization feasibility problem and fixed point problem. Furthermore, our second algorithm generates a strongly convergent sequence to an approximate common solution of non-convex minimization feasibility problem and fixed point problem. In all our results in this work, our iterative schemes are proposed with a way of selecting the step-sizes such that their implementation does not need any prior information about the operator norm because the calculation or at least an estimate of the operator norm $$||A||$$ is very difficult, if not an impossible task. Our result complements many recent and important results in this direction. [ABSTRACT FROM AUTHOR]

Around a solution of an optimization problem, an 'identifiable' subset of the feasible region is one containing all nearby solutions after small perturbations to the problem. A quest for only the most essential ingredients of sensitivity analysis leads us to consider identifiable sets that are 'minimal'. This new notion lays a broad and intuitive variational-analytic foundation for optimality conditions, sensitivity, and active set methods. [ABSTRACT FROM AUTHOR]

In this paper our interest is in investigating properties and numerical solutions of Proximal Split feasibility Problems. First, we consider the problem of finding a point which minimizes a convex function $$f$$ such that its image under a bounded linear operator $$A$$ minimizes another convex function $$g$$ . Based on an idea introduced in Lopez (Inverse Probl 28:085004, ), we propose a split proximal algorithm with a way of selecting the step-sizes such that its implementation does not need any prior information about the operator norm. Because the calculation or at least an estimate of the operator norm $$\Vert A\Vert $$ is not an easy task. Secondly, we investigate the case where one of the two involved functions is prox-regular, the novelty of this approach is that the associated proximal mapping is not nonexpansive any longer. Such situation is encountered, for instance, in numerical solution to phase retrieval problem in crystallography, astronomy and inverse scattering Luke (SIAM Rev 44:169-224, ) and is therefore of great practical interest. [ABSTRACT FROM AUTHOR]

In this paper, we first study a nonsmooth steepest descent method for nonsmooth functions defined on a Hilbert space and establish the corresponding algorithm by proximal subgradients. Then, we use this algorithm to find stationary points for those functions satisfying prox-regularity and Lipschitz continuity. As an application, the established algorithm is used to search for the minimizer of a lower semicontinuous and convex function on a finite-dimensional space. A convergence theorem, as an extension and improvement of the existing converging result for twice continuously differentiable convex functions, is also presented therein. [ABSTRACT FROM AUTHOR]

This paper points out some fatal errors in the equivalent formulations used in Noor 2011 [Noor MA. Projection iterative methods for solving some systems of general nonconvex variational inequalities. Applied Analysis. 2011;90:777–786] and consequently in Noor 2009 [Noor MA. System of nonconvex variational inequalities. Journal of Advanced Research Optimization. 2009;1:1–10], Noor 2010 [Noor MA, Noor KI. New system of general nonconvex variational inequalities. Applied Mathematics E-Notes. 2010;10:76–85] and Wen 2010 [Wen DJ. Projection methods for a generalized system of nonconvex variational inequalities with different nonlinear operators. Nonlinear Analysis. 2010;73:2292–2297]. Since these equivalent formulations are the main tools to suggest iterative algorithms and to establish the convergence results, the algorithms and results in the aforementioned articles are not valid. It is shown by given some examples. To overcome with the problems in these papers, we consider a new system of extended regularized nonconvex variational inequalities, and establish the existence and uniqueness result for a solution of the aforesaid system. We suggest and analyse a new projection iterative algorithm to compute the unique solution of the system of extended regularized nonconvex variational inequalities which is also a fixed point of a nearly uniformly Lipschitzian mapping. Furthermore, the convergence analysis of the proposed iterative algorithm under some suitable conditions is studied. As a consequence, we point out that one can derive the correct version of the algorithms and results presented in the above mentioned papers. [ABSTRACT FROM AUTHOR]

The NC-proximal average is a parametrized function used to continuously transform one proper, lsc, prox-bounded function into another. Until now it has been defined for two functions. The purpose of this article is to redefine it so that any finite number of functions may be used. The layout generally follows that of Hare (SIAM J Optim 20(2):650-666, ), extending those results to the more general case and in some instances giving alternate proofs by using techniques developed after the publication of that paper. We conclude with an example examining the discontinuity of the minimizers of the NC-proximal average. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study a new system of general nonconvex variational inclusions involving four different nonlinear operators (SGNVI) and prove the equivalence between the SGNVI and a fixed point problem. By using this equivalent formulation, we establish the existence and uniqueness theorem for solution of the SGNVI. We use the foregoing equivalent alternative formulation and two nearly uniformly Lipschitzian mappings [InlineEquation not available: see fulltext.] and [InlineEquation not available: see fulltext.] to suggest and analyze some new two-step projection iterative algorithms for finding an element of the set of fixed points of the nearly uniformly Lipschitzian mapping [InlineEquation not available: see fulltext.], which is the unique solution of the SGNVI. Further, the convergence analysis of the suggested iterative algorithms under suitable conditions is studied. MSC: 47H05, 47J20, 49J40, 90C33. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of a new class of nonconvex variational inequalities, named general regularized nonconvex variational inequalities. By using the auxiliary principle technique, a new modified predictor-corrector iterative algorithm for solving general regularized nonconvex variational inequalities is suggested and analyzed. The convergence of the iterative algorithm is established under the partially relaxed monotonicity assumption. As a consequence, the algorithm and results presented in the paper overcome incorrect algorithms and results existing in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we investigate or analyze non-convex variational inequalities and general non-convex variational inequalities. Two new classes of non-convex variational inequalities, named regularized non-convex variational inequalities and general regularized non-convex variational inequalities, are introduced, and the equivalence between these two classes of non-convex variational inequalities and the fixed point problems are established. A projection iterative method to approximate the solutions of general regularized non-convex variational inequalities is suggested. Meanwhile, the existence and uniqueness of solution for general regularized non-convex variational inequalities is proved, and the convergence analysis of the proposed iterative algorithm under certain conditions is studied. [ABSTRACT FROM AUTHOR]

Prox-regular subsets of Riemannian manifolds are introduced. A characterization of prox-regular sets based on the hypomonotonicity of the truncated limiting normal cone is obtained. Moreover, some properties of metric projection mapping and distance function corresponding to the proxregular sets are presented. [ABSTRACT FROM AUTHOR]

We prove that uniform second-order growth, tilt stability, and strong metric regularity of the subdifferential--three notions that have appeared in entirely different settings--are all essentially equivalent for any lower-semicontinuous, extended real-valued function. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we introduce and consider the concept of the prox-regularity of a multifunction. We mainly study the metric subregularity of a generalized equation defined by a proximal closed multifunction between two Hilbert spaces. Using proximal analysis techniques, we provide sufficient and/or necessary conditions for such a generalized equation to have the metric subregularity in Hilbert spaces. We also establish the results of Robinson–Ursescu theorem type for prox-regular multifunctions. [Copyright &y& Elsevier]

Abstract: Some new classes of extended general nonconvex set-valued variational inequalities and the extended general Wiener–Hopf inclusions are introduced. By the projection technique, equivalence between the extended general nonconvex set-valued variational inequalities and the fixed point problems as well as the extended general nonconvex Wiener–Hopf inclusions is proved. Then by using this equivalent formulation, we discuss the existence of solutions of the extended general nonconvex set-valued variational inequalities and construct some new perturbed finite step projection iterative algorithms with mixed errors for approximating the solutions of the extended general nonconvex set-valued variational inequalities. We also verify that the approximate solutions obtained by our algorithms converge to the solutions of the extended general nonconvex set-valued variational inequalities. The results presented in this paper extend and improve some known results from the literature. [ABSTRACT FROM AUTHOR]

Abstract: We consider a variable Krasnosel’skii–Mann algorithm for approximating critical points of a prox-regular function or equivalently for finding fixed-points of its proximal mapping . The novelty of our approach is that the latter is not non-expansive any longer. We prove that the sequence generated by such algorithm (via the formula , where is a sequence in ), is an approximate fixed-point of the proximal mapping and converges provided that the function under consideration satisfies a local metric regularity condition. [Copyright &y& Elsevier]

The idea of a finite collection of closed sets having “linearly regular intersection” at a point is crucial in variational analysis. This central theoretical condition also has striking algorithmic consequences: in the case of two sets, one of which satisfies a further regularity condition (convexity or smoothness, for example), we prove that von Neumann’s method of “alternating projections” converges locally to a point in the intersection, at a linear rate associated with a modulus of regularity. As a consequence, in the case of several arbitrary closed sets having linearly regular intersection at some point, the method of “averaged projections” converges locally at a linear rate to a point in the intersection. Inexact versions of both algorithms also converge linearly. [ABSTRACT FROM AUTHOR]

In this paper we discuss local uniqueness, continuity, and differentiability properties of solutions of parameterized variational inequalities (generalized equations). To this end we use two types of techniques. One approach consists in formulating variational inequalities in a form of optimization problem based on regularized gap functions, and applying a general theory of perturbation analysis of parameterized optimization problems. Another approach is based on a theory of contingent (outer graphical) derivatives and some results about differentiability properties of metric projections. [ABSTRACT FROM AUTHOR]

Relying on prox-regularity we suggest and analyze a class of iterative schemes for solving nonconvex variational inequalities, our result can be considered as an improvement and a significant extension of previously known results in this field. [Copyright &y& Elsevier]

Differential inclusions involving the normal cone to a moving set are investigated. A special attention is paid to the sweeping process associated with sets for which no regularity assumption is required. [Copyright &y& Elsevier]