Metric Subregularity and Calmness for Nonconvex Generalized Equations in Banach Spaces

Abstract

Abstract. This paper concerns a generalized equation defined by a closed multifunction between Banach spaces, and we employ variational analysis techniques to provide sufficient and/or necessary conditions for a generalized equation to have the metric subregularity (i.e., local error bounds for the concerned multifunction) in general Banach spaces. Following the approach of Ioffe [Trans. Amer. Math. Soc., 251 (1979), pp. 61-69] who studied the numerical function case, our conditions are described in terms of coderivatives of the concerned multifunction at points outside the solution set. Motivated by the existing modulus representation and point-based criteria for the metric regularity, we establish the corresponding results for the metric subregularity. In the Asplund space case, sharper results are obtained.

Metric subregularity for proximal generalized equations in Hilbert spaces

“…His work has been followed by many others, and it is now well known that Ioffe's result is still true when f is a general proper semicontinuous function on X. In this line, in terms of the coderivatives, the authors [41] further extended the Ioffe's result to the case when F is a general closed multifunction and established the following result:…”

“…But, under the convexity assumption, the converse of each of these results does hold. Indeed, the authors [41] proved the following characterization: if F is a closed convex multifunction between Banach spaces X and Y then (GE) is metrically subregular at a ∈ F −1 (b) if and only if there exist η, δ ∈ (0, +∞) and ε ∈ (0, 1) such that (1.4) holds. It is a natural problem to ask whether the above characterization can been extended to a larger class of possibly nonconvex functions.…”

Proximal Point Algorithms on Hadamard Manifolds: Linear Convergence and Finite Termination

“…The metric subregularity (the terminology "error bond" is sometimes adopted instead of metric subregularity; see [19] for history of terminology) and its equivalent calmness counterpart for inverse mappings have been studied extensively in linear spaces. Various results in this direction and their applications can be found in [1,2,3,19,23,27,28,30,62,63] and references therein. Below, we extend the notion of metric subregularity (cf.…”

On metric subregularity for convex constraint systems by primal equivalent conditions

“…It is known that metric subregularity is closely related to calmness, error bounds, linear regularity and basic constraint qualification (BCQ) in optimization and has found a huge range of applications in areas of variational analysis and mathematical programming like optimality conditions, variational inequalities, subdifferential theory, sensitivity analysis of generalized equations and convergence analysis of algorithms for solving equations or inclusions. For these reasons, the concept of metric subregularity has been extensively studied by many authors (see [3,4,5,6,8,18,20,21,22,23,24,25]). In this paper, we mainly consider metric subregularity and targets at its primal criteria.…”