Metric subregularity for proximal generalized equations in Hilbert spaces

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 results of Robinson-Ursescu theorem type for prox-regular multifunctions.

Prox-regularity approach to generalized equations and image projection

“…The key of our approach is a suitable extension of the famous Robinson−Ursescu Theorem [35,41], which asserts for two Banach spaces X and Y , a multimapping M : X ⇒ Y with a closed convex graph is metrically regular at x for y ∈ M (X) if and only if y ∈ int M (X). Over the years, many researchers have weakened the convexity assumption in Robinson−Ursescu theorem (see, e.g., [17,18,44,47]). Our result is concerned with the class of normally ω(•)-regular sets, a new class which encompasses prox-regular sets in the Hilbert setting and (σ, δ)-subsmooth sets [44] in a general Banach space.…”

Nonlinear Metric Subregularity

“…The word "outer" is used to emphasize that only points outside the set S(f ) are taken into account. The word "uniform" emphasizes the nonlocal (non-limiting) character of |∇f | ⋄ ρ (x, y) involved in definition (15). Observe that the definition of the modified strict outer slope (14) contains under max a nonlocal (when (x, y) is fixed) component f (x, y)/d(x,x).…”

“…Constants (11)-(15) are nonnegative and can be infinite. In (13)- (15), the usual convention that the infimum of the empty set equals +∞ is in force. In these definitions, we have not only x →x and f (x, y) ↓ 0, but also the metric on X × Y used in the definitions of the corresponding ρ-slopes changing with the contribution of the y component diminishing as ρ ↓ 0.…”

Perturbation of error bounds

“…Many authors have recently studied error bounds in connection with the metric regularity and subregularity (cf. [21]) as well as Aubin property and calmness of setvalued mappings: [3,15,16,26,33,34,36,[39][40][41]54,56,58,59,68,69]. The connections between the error bounds and weak sharp minima were studied in [13].…”