An Induction Theorem and Nonlinear Regularity Models

Abstract

A general nonlinear regularity model for a set-valued mapping F : X × R + ⇒ Y , where X and Y are metric spaces, is studied using special iteration procedures, going back to Banach, Schauder, Lyusternik and Graves. Namely, we revise the induction theorem from Khanh, J. Math. Anal. Appl., 118 (1986) and employ it to obtain basic estimates for exploring regularity/openness properties. We also show that it can serve as a substitution of the Ekeland variational principle when establishing other regularity criteria. Then, we apply the induction theorem and the mentioned estimates to establish criteria for both global and local versions of regularity/openness properties for our model and demonstrate how the definitions and criteria translate into the conventional setting of a set-valued mapping F : X ⇒ Y . An application to second-order necessary optimality conditions for a nonsmooth set-valued optimization problem with mixed constraints is provided.

Some New Characterizations of Intrinsic Transversality in Hilbert Spaces

“…The property has been known under quite a number of other names including regularity, strong regularity, property (U R) S , uniform regularity, strong metric inequality [25][26][27]36] and linear regular intersection [40]. Plenty of primal and dual space characterizations of transversality (especially in the Euclidean space setting) as well as its close connections to important concepts in optimization and variational analysis such as weak sharp minima, error bounds, conditions involving primal and dual slopes, metric regularity, (extended) extremal principles and other types of mutual arrangement properties of collections of sets have been established and extended to more general nonlinear settings in a series of papers by Kruger and his collaborators [23,[25][26][27][28][29][30]37,38]. Apart from classical applications of the property, for example, as a sufficient condition for strong duality to hold for convex optimization (Slater's condition) [8,9] or as a constraint qualification condition for establishing calculus rules for the limiting/Mordukhovich normal cones [44, page 265] and coderivatives (in connection with metric regularity, the counterpart of transversality in terms of set-valued mappings) [12,50], important applications have also been emerging in the field of numerical analysis.…”

A convergent relaxation of the Douglas–Rachford algorithm

“…Remark 2 It follows from Lemma 2 (i) & (iii) that the set U defined by (20) contains at least the ball B δ ′ (x), where…”

“…As a result, the question about convergence of a solving method can often be reduced to checking whether certain regularity properties of the problem data are satisfied. There have been a considerable number of papers studying these two ingredients of convergence analysis in order to establish sharper convergence criteria in various circumstances, especially those applicable to algorithms for solving nonconvex problems [5,12,13,19,26,27,[31][32][33]38,42,45]. This paper suggests an algorithm called T λ , which covers both the backwardbackward and the DR algorithms as special cases of choosing the parameter λ ∈ [0, 1], and analyzes its convergence.…”

“…The question of convergence for a given method can therefore be reduced to checking regularity properties of the problem data. There have been a considerable number of works studying the two ingredients of convergence analysis in order to provide sharper tools in various circumstances, especially in nonconvex cases, e.g., [5,13,14,20,27,28,[32][33][34]39,43,46]. This paper proposes an algorithm called T λ , which covers both the backward -backward/alternating projection and DR algorithms as special cases of choosing the parameter λ ∈ [0, 1], and reports its convergence results.…”

Regularity Properties in Variational Analysis and Applications in Optimisation

“…Motivated by Ioffe [8] and his subsequent publications [9][10][11], we use the classical iteration scheme going back to Banach, Schauder, Lyusternik and Graves to establish criteria for regularity properties of set-valued mappings and compare this approach with the one based on the Ekeland variational principle [12].…”