About intrinsic transversality of pairs of sets

The article continues the study of the ‘regular’ arrangement of a collection of sets near a point in their intersection. Such regular intersection or, in other words, transversality properties are crucial for the validity of qualification conditions in optimization as well as subdifferential, normal cone and coderivative calculus, and convergence analysis of computational algorithms. One of the main motivations for the development of the transversality theory of collections of sets comes from the convergence analysis of alternating projections for solving feasibility problems. This article targets infinite dimensional extensions of the intrinsic transversality property introduced recently by Drusvyatskiy, Ioffe and Lewis as a sufficient condition for local linear convergence of alternating projections. Several characterizations of this property are established involving new limiting objects defined for pairs of sets. Special attention is given to the convex case

FUTURE CHALLENGES FOR VARIATIONAL ANALYSIS

Abstract. I will also discuss open problems and current challenges for the subject Boris Mordukhovich has played a key role in the development of modern Variational Analysis (VA) and its Applications. Modern non-smooth analysis is now roughly thirty-five years old. In this paper I shall attempt to analyse (briefly): where the subject stands today, where it should be going, and what it will take to get there? Summary: the first order theory is rather impressive, as are some applications. The second order theory is by comparison somewhat underdeveloped and wanting. “It is not knowledge, but the act of learning, not possession but the act of getting there, which grants the greatest enjoyment. When I have clarified and exhausted a subject, then I turn away from it, in order to go into darkness again; the never-satisfied man is so strange if he has completed a structure, then it is not in order to dwell in it peacefully,but in order to begin another. I imagine the world conqueror must feel thus, who, after one kingdom is scarcely conquered, stretches out his arms for others.”—Carl Friedrich Gauss (1777-1855) 1 1. Preliminaries I intend to discuss First-Order Theory, and then Higher-Order Theory mainly second-order and only mention passingly higher-order theory which really devolves to second-order theory. I’ll finish by touching on Applications of Variational Analysis both inside and outside Mathematics, mentioning both successes and limitations or failures. Each topic leads to open questions even in the convex case (CA). Some ar

A convergent relaxation of the Douglas–Rachford algorithm

This paper proposes an algorithm for solving structured optimization problems, which covers both the backward–backward and the Douglas–Rachford algorithms as special cases, and analyzes its convergence. The set of fixed points of the corresponding operator is characterized in several cases. Convergence criteria of the algorithm in terms of general fixed point iterations are established. When applied to nonconvex feasibility including potentially inconsistent problems, we prove local linear convergence results under mild assumptions on regularity of individual sets and of the collection of sets. In this special case, we refine known linear convergence criteria for the Douglas–Rachford (DR) algorithm. As a consequence, for feasibility problem with one of the sets being affine, we establish criteria for linear and sublinear convergence of convex combinations of the alternating projection and the DR methods. These results seem to be new. We also demonstrate the seemingly improved numerical performance of this algorithm compared to the RAAR algorithm for both consistent and inconsistent sparse feasibility problems.Team Raf Van de Pla

Some New Characterizations of Intrinsic Transversality in Hilbert Spaces

© 2020, The Author(s). Motivated by a number of questions concerning transversality-type properties of pairs of sets recently raised by Ioffe and Kruger, this paper reports several new characterizations of the intrinsic transversality property in Hilbert spaces. New results in terms of normal vectors clarify the picture of intrinsic transversality, its variants and sufficient conditions for subtransversality, and unify several of them. For the first time, intrinsic transversality is characterized by an equivalent condition which does not involve normal vectors. This characterization offers another perspective on intrinsic transversality. As a consequence, the obtained results allow us to answer a number of important questions about transversality-type properties