Two Classes of Merit Functions for Infinite-Dimensional Second Order Complimentary Problems

In this article, we extend two classes of merit functions for the second-order complementarity problem (SOCP) to infinite-dimensional SOCP. These two classes of merit functions include several popular merit functions, which are used in nonlinear complementarity problem, (NCP)/(SDCP) semidefinite complementarity problem, and SOCP, as special cases. We give conditions under which the infinite-dimensional SOCP has a unique solution and show that all these merit functions provide an error bound for infinite-dimensional SOCP and have bounded level sets. These results are very useful for designing solution methods for infinite-dimensional SOCP.     

Constrained stochastic controllability of infinite-dimensional linear systems

The main purpose of this article is to investigate the problem of (ε, δ)-stochastic controllability for linear systems of evolution type in infinite-dimensional spaces, wherein the controls are subjected to norm-bounded constrained sets. Some basic prerequisites of infinite-dimensional measures, in particular, Gaussian distributed type, are discussed. Corresponding to this measure, various properties of (ε, δ)-stochastic attainable sets in Hilbert spaces are studied. Necessary and sufficient conditions for (ε, δ)-stochastic controllability with respect to Hilbert space valued linear systems are obtained. Relationships with the deterministic counterpoint are noted. Pursuit game problems are also considered. Examples on systems governed by stochastic linear partial differential equations and stochastic differential delay equations are given for illustration.     

Constraint qualifications and optimality conditions for nonconvex semi-infinite and infinite programs

The paper concerns the study of new classes of nonlinear and nonconvex optimization problems of the so-called infinite programming that are generally defined on infinite-dimensional spaces of decision variables and contain infinitely many of equality and inequality constraints with arbitrary (may not be compact) index sets. These problems reduce to semi-infinite programs in the case of finite-dimensional spaces of decision variables. We extend the classical Mangasarian–Fromovitz and Farkas–Minkowski constraint qualifications to such infinite and semi-infinite programs. The new qualification conditions are used for exact calculations of the appropriate normal cones to sets of feasible solutions for these programs by employing advanced tools of variational analysis and generalized differentiation. In the further development we derive first-order necessary optimality conditions for infinite and semi-infinite programs, which are new in both finite-dimensional and infinite-dimensional settings.     

Preservation of persistence and stability under intersections and operations,part 1: Persistence

New results about convergence of sets and functions in possibly infinite-dimensional spaces are derived in a simple and unified way from two results dealing with the continuity with respect to a parameter of the intersection of two convex sets depending on this parameter.     

Construction of non-Gaussian surface measures by Malliavin’s method

We generalize the Airault-Malliavin theorem on the existence of surface measures on infinite-dimensional spaces with Gaussian measures on surfaces. We prove that the sets of capacities generated by Sobolev classes on infinite-dimensional spaces are dense. Translated fromMatematicheskie Zametki, Vol. 65, No. 3, pp. 377–388, March, 1999.     

A relaxed self-adaptive CQ algorithm for the multiple-sets split feasibility problem

The multiple-sets split feasibility problem (MSFP) is to find a point belongs to the intersection of a family of closed convex sets in one space, such that its image under a linear transformation belongs to the intersection of another family of closed convex sets in the image space. Many iterative methods can be employed to solve the MSFP. Jinling Zhao et al. proposed a modification for the CQ algorithm and a relaxation scheme for this modification to solve the MSFP. The strong convergence of these algorithms are guaranteed in finite-dimensional Hilbert spaces. Recently López et al. proposed a relaxed CQ algorithm for solving split feasibility problem, this algorithm can be implemented easily since it computes projections onto half-spaces and has no need to know a priori the norm of the bounded linear operator. However, this algorithm has only weak convergence in the setting of infinite-dimensional Hilbert spaces. In this paper, we introduce a new relaxed self-adaptive CQ algorithm for solving the MSFP where closed convex sets are level sets of some convex functions such that the strong convergence is guaranteed in the framework of infinite-dimensional Hilbert spaces. Our result extends and improves the corresponding results.     

Characterizations of normal, hyponormal and EP operators

In this paper normal and hyponormal operators with closed ranges, as well as EP operators, are characterized in arbitrary Hilbert spaces. All characterizations involve generalized inverses. Thus, recent results of S. Cheng and Y. Tian [S. Cheng, Y. Tian, Two sets of new characterizations for normal and EP matrices, Linear Algebra Appl. 375 (2003) 181-195] are extended to infinite-dimensional settings.     

Stability of uniform attractors of impulsive multi-valued semiflows

In this paper we investigate stability of uniformly attracting sets for semiflows generated by impulsive infinite-dimensional dynamical systems without uniqueness. Obtained abstract results are applied to weakly nonlinear parabolic system, whose trajectories have jumps at moments of intersection with certain surface in the phase space.     

On Quasi-Chebyshevity Subsets of Unital Banach Algebras

In this paper,first,we consider closed convex and bounded subsets of infinite-dimensional unital Banach algebras and show with regard to the general conditions that these sets are not quasi-Chebyshev and pseudo-Chebyshev.Examples of those algebras are given including the algebras of continuous functions on compact sets.We also see some results in C*-algebras and Hilbert C*-modules.Next,by considering some conditions,we study Chebyshev of subalgebras in C*-algebras.     

Uniform convexity and the splitting problem for selections

We continue to investigate cases when the Repovš–Semenov splitting problem for selections has an affirmative solution for continuous set-valued mappings. We consider the situation in infinite-dimensional uniformly convex Banach spaces. We use the notion of Polyak of uniform convexity and modulus of uniform convexity for arbitrary convex sets (not necessary balls). We study general geometric properties of uniformly convex sets. We also obtain an affirmative solution of the splitting problem for selections of certain set-valued mappings with uniformly convex images.     

Metric Regularity of Mappings and Generalized Normals to Set Images

The primary goal of this paper is to study some notions of normals to nonconvex sets in finite-dimensional and infinite-dimensional spaces and their images under single-valued and set-valued mappings. The main motivation for our study comes from variational analysis and optimization, where the problems under consideration play a crucial role in many important aspects of generalized differential calculus and applications. Our major results provide precise equality formulas (sometimes just efficient upper estimates) allowing us to compute generalized normals in various senses to direct and inverse images of nonconvex sets under single-valued and set-valued mappings between Banach spaces. The main tools of our analysis revolve around variational principles and the fundamental concept of metric regularity properly modified in this paper.     

The basic attractor and the remainder of homo- and heteroclinic orbits

In this paper, we discuss some issues in the dynamical systems theory of dissipative nonlinear partial differential equations (PDEs), on a bounded domain. A decomposition theorem says that attractors of PDEs can be decomposed into a basic attractor (a core) that attracts sets of positive measure, it attracts a prevalent set in phase space, and a remainder whose basin, up to sets that are attracted to the basic attractor, is shy, or of zero (infinite-dimensional) measure. If the basic attractor is low-dimensional and the remainder high-dimensional, then the dynamics can still be analyzed up to transients that are exponentially decaying toward the attractor in time. We focus on (ODE) examples of homo- and heteroclinic connections and show that generically these connections lie in the remainder but there exist exceptional cases where they lie in the basic attractor.     

Multiobjective optimization problems with equilibrium constraints

The paper is devoted to new applications of advanced tools of modern variational analysis and generalized differentiation to the study of broad classes of multiobjective optimization problems subject to equilibrium constraints in both finite-dimensional and infinite-dimensional settings. Performance criteria in multiobjective/vector optimization are defined by general preference relationships satisfying natural requirements, while equilibrium constraints are described by parameterized generalized equations/variational conditions in the sense of Robinson. Such problems are intrinsically nonsmooth and are handled in this paper via appropriate normal/coderivative/subdifferential constructions that exhibit full calculi. Most of the results obtained are new even in finite dimensions, while the case of infinite-dimensional spaces is significantly more involved requiring in addition certain “sequential normal compactness” properties of sets and mappings that are preserved under a broad spectrum of operations.     

Slopes,Error Bounds and Weak Sharp Pareto Minima of a Vector-Valued Map

In this paper, we provide a detailed study of the upper and lower slopes of a vector-valued map recently introduced by Bednarczuk and Kruger. We show that these slopes enjoy most properties of the strong slope of a scalar-valued function and can be explicitly computed or estimated in the convex, strictly differentiable, linear cases. As applications, we obtain error bounds for lower level sets (in particular, a Hoffman-type error bound for a system of linear inequalities in the infinite-dimensional space setting, existence of weak sharp Pareto minima) and sufficient conditions for Pareto minima.     

Certain classes of weakly infinite-dimensional spaces and topological games

In [V.V. Fedorchuk, Questions on weakly infinite-dimensional spaces, in: E.M. Pearl (Ed.), Open Problems in Topology II, Elsevier, Amsterdam, 2007, pp. 637-645; V.V. Fedorchuk, Weakly infinite-dimensional spaces, Russian Math. Surveys 42 (2) (2007) 1-52] classes w-m-C of weakly infinite-dimensional spaces, 2?m?∞, were introduced. We prove that all of them coincide with the class wid of all weakly infinite-dimensional spaces in the Alexandroff sense. We show also that transfinite dimensions dimwm, introduced in [V.V. Fedorchuk, Questions on weakly infinite-dimensional spaces, in: E.M. Pearl (Ed.), Open Problems in Topology II, Elsevier, Amsterdam, 2007, pp. 637-645; V.V. Fedorchuk, Weakly infinite-dimensional spaces, Russian Math. Surveys 42 (2) (2007) 1-52], coincide with dimension dimw2=dim, where dim is the transfinite dimension invented by Borst [P. Borst, Classification of weakly infinite-dimensional spaces. I. A transfinite extension of the covering dimension, Fund. Math. 130 (1) (1988) 1-25]. Some topological games which are related to countable-dimensional spaces, to C-spaces, and some other subclasses of weakly infinite-dimensional spaces are discussed.     

Sufficient criteria of convexity

In this article a survey of criteria of convexity of sets and hypersurfaces inR ⁿ, in Riemannian and infinite-dimensional linear spaces, is made. Included are the following sections: support property; local convexity; local support; curvature of the boundary; section by planes; uniqueness of closest points; systems of sets. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 45, pp. 3–52, 1974.     

Calculus of sequential normal compactness in variational analysis

In this paper we study some properties of sets, set-valued mappings, and extended-real-valued functions unified under the name of “sequential normal compactness.” These properties automatically hold in finite-dimensional spaces, while they play a major role in infinite-dimensional variational analysis. In particular, they are essential for calculus rules involving generalized differential constructions, for stability and metric regularity results and their broad applications, for necessary optimality conditions in constrained optimization and optimal control, etc. This paper contains principal results ensuring the preservation of sequential normal compactness properties under various operations over sets, set-valued mappings, and functions.     

BV solutions of nonconvex sweeping process differential inclusion with perturbation

This paper is devoted to the study of differential inclusions, particularly discontinuous perturbed sweeping processes in the infinite-dimensional setting. On the one hand, the sets involved are assumed to be prox-regular and to have a variation given by a function which is of bounded variation and right continuous. On the other hand, the perturbation satisfies a linear growth condition with respect to a fixed compact subset. Finally, the case where the sets move in an absolutely continuous way is recovered as a consequence.     

Infinite-dimensional convex programming with applications to constrained approximation

In this paper, existence and characterization of solutions and duality aspects of infinite-dimensional convex programming problems are examined. Applications of the results to constrained approximation problems are considered. Various duality properties for constrained interpolation problems over convex sets are established under general regularity conditions. The regularity conditions are shown to hold for many constrained interpolation problems. Characterizations of local proximinal sets and the set of best approximations are also given in normed linear spaces.The author is grateful to the referee for helpful suggestions which have contributed to the final preparation of this paper. This research was partially supported by Grant A68930162 from the Australian Research Council.