Primal characterizations

Metric regularity of set-valued mappings is one of the corner stones of variational analysis.

The property is regarded as a natural extension to set-valued mappings of the regularity estimates provided by the classical Banach-Schauder open mapping theorem (for linear op-erators) and the Lyusternik-Graves theorem (for nonlinear opop-erators) [47,48,65,109,129].

The Robinson-Ursescu theorem gives an important example of this property, in particu-lar, a closed convex set-valued mappingF is metrically regular at a point x¯∈ domF for

¯

y∈F(¯x) if and only ify¯is an interior point ofrangeF.

The following concept ofmetric regularity with functional modulus on a setcharacterizes the stability of mappings at points in their image and has played a central role, implicitly and explicitly, in our analysis of convergence of Picard iterations [4,59,103]. In particular, the key insight into condition (b)of Theorem 3.1.1 is the connection to metric regularity of set-valued mappings (cf., [50,129]). This approach to the study of algorithms has been advanced by several authors [2,3,70,74,122]. We modify the concept ofmetric regularity with functional modulus on a set suggested in [66, Definition 2.1 (b)] and [67, Definition 1 (b)] so that the property is relativized to appropriate sets for iterative methods.

Definition 2.2.1 (metric regularity on a set). [103, Definition 2.5] Let F : X ⇒ Y , U ⊂ X, V ⊂ Y. The mapping F is called metrically regular with gauge µ on U ×V relative toΛ⊂X if

dist x, F⁻¹(y)∩Λ

≤µ(dist (y, F(x))) (2.4) holds for all x ∈U∩Λ andy ∈V with 0< µ(dist (y, F(x))). When the set V consists of a single point,V ={y}, then¯ F is said to be metrically subregular fory¯on U with gauge µrelative to Λ⊂X.

When µ is a linear function (that is, µ(t) = κt,∀t∈ [0,∞)), one says “with constant κ” instead of “with gauge µ(t) =κt”. When Λ =X, the quantifier “relative to” is dropped.

When µ is linear, the infimum of κ for which (2.4) holds is called the modulus of metric regularity onU ×V.

The conventional concept of metric regularity [10, 50, 129] (and metric regularity of order ω, respectively [86]) at a point x¯ ∈ X for y¯ ∈ F(¯x) corresponds to the setting in Definition2.2.1 whereΛ =X,U andV areneighborhoods ofx¯and y, respectively, and the¯ gauge functionµ(t) =κt(µ(t) =κt^ω for metric regularity of orderω <1) for allt∈[0,∞), withκ >0. The infimum of κ over all neighborhoods U and V such that (2.4) is satisfied is the regularity modulus ofF atx¯for y¯and denoted by reg(F; ¯x|¯y).

The flexibility of choosing the setsU andV in Definition2.2.1allows the same definition and terminology to cover well-known relaxations of metric regularity such as metric sub-regularity (U is a neighborhood ofx¯ andV ={y}¯ [50]. In this case, the infimum ofκ over all neighborhoodsU ofx¯ such that (2.4) is satisfied is the modulus of metric subregularity ofF at x¯ for y¯and denoted by subreg(F; ¯x|¯y).) andmetric hemi/semiregularity (U ={¯x}

and V is a neighborhood of y¯ [109, Definition 1.47]). For our purposes, we will use the flexibility of choosing U and V in Definition2.2.1 to exclude the reference point x¯ and to isolate the image point y. This is reminiscent of the Kurdyka-Łojasiewicz (KL) property¯ [25] for functions which requires that the subdifferential possesses a sharpness property near (but not at) critical points of the function. However, since the restriction of V to a point features prominently in our development, we retain the terminologymetric subregularityto

ease the technicality of the presentation. The reader is cautioned, however, that our usage of metric subregularity does not precisely correspond to the usual definition (see [50]) since we do not require the domainU to be a neighborhood.

The metric regularity of a set-valued mappingF can be used for measuring the “condi-tioning” of the generalized equation: for a giveny∈Y,

findx∈X such thaty∈F(x). (2.5)

Inequality (2.4) then provides an estimate of how far a pointxcan be from the solution set of (2.5) corresponding to the right-hand side y; this distance is bounded from above by a multiple κ of the “residual” dist(y, F(x)). In other words, the presence of metric regularity ofF at x¯fory¯∈F(¯x)means that (2.5) is, from a certain perspective, well-posed around there. This conditioning is stable under small perturbations on F [48, 49], where quantitative estimates of how large a perturbation can be before metric regularity breaks down are also established.

Metric regularity admits several equivalent descriptions to (2.4). Recall that [65, p.510]

F is calledmetrically graph-regular at x¯ fory¯∈F(¯x) if there exist positive numbersκ and δ such that

dist x, F⁻¹(y)

≤d_κ((x, y),gphF), ∀x∈B_δ(¯x), y∈B_δ(¯y), (2.6) where

d_κ((x, y),gphF) := inf

(u,w)∈gphF(dist(x, u) +κdist(y, w)).

The two descriptions (2.4) and (2.6) are equivalent with the sameκ (and possibly different δ), in particular, metric regularity ofF at x¯ for y¯is equivalent to metric graph-regularity ofF at x¯ for y¯[65, Proposition 4, p.510]. We also refer the reader to that paper for other equivalent descriptions of metric regularity. The main idea for these possibilities is that the definition of the conventional metric regularity would be qualitatively unchanged when reasonable restriction onx andy was added, for example, (x, y)∈/ gphF.

Dmitruk et al. [46] and Ioffe [63] showed the equivalence between the metric regularity and the linear openness property of a set-valued mapping F, which are determined by the first-order behaviour of the mapping and invariant under sufficiently small first-order perturbations [46,48,65]. The two properties are also equivalent to the Aubin property of the inverse mappingF⁻¹ thanks to Borwein and Zhuang [31] and Penot [123].

Definition 2.2.2. (i) A set-valued mapping F is linearly open atx¯ fory¯∈F(¯x) if there existκ≥0 andδ >0 such that

F(x+κρintB)⊃[F(x) +ρintB]∩Bδ(¯y), ∀x∈Bδ(¯x),∀ρ >0. (2.7) The infimum ofκover allδsuch that(2.7)is satisfied is the modulus of linear openness of F at x¯ for y¯and denoted by lop(F; ¯x|¯y).

(ii) F has the Aubin property at x¯ for y¯∈F(¯x) if there exist κ≥0 andδ >0 such that excess(F(x)∩B_δ(¯y), F(x⁰))≤κdist(x, x⁰), ∀x, x⁰∈B_δ(¯x). (2.8) The infimum of κ over all combinations of κ and δ such that (2.8) is satisfied is the Lipschitz modulus ofF at x¯ for y¯∈F(¯x) and denoted by lip(F; ¯x|¯y).

Proposition2.2.3. Metric regularity and linear openness of a set-valued mappingF at x¯ fory¯∈F(¯x)are equivalent. They are also equivalent to the Aubin property of the mapping F⁻¹ at y¯for x. Moreover, it holds¯

reg(F; ¯x|¯y) = lop(F; ¯x|¯y) = lip(F⁻¹; ¯y|¯x).

Metric subregularity also enjoys the relationships analogous to those stated in Propo-sition2.2.3 with the sub-versions of linear openness and Aubin properties. The interested reader is referred to [1,50,58,129].

Metric subregularity can also be characterized via the concept of local error bound of extended real-valued functions. A function f :X → R∪ {∞} having a local error bound at a pointx¯ with f(¯x) = 0 simply coincides with the set-valued mapping x7→[f(x),+∞) (∀x∈X) being metrically regular atx¯ for 0 [80, Proposition 9(ii)]. This transition allows one to deduce criteria for local error bounds of l.s.c. extended real-valued functions from those for metric subregularity.

In the finite dimensional setting E, the following proposition, taken from [50], charac-terizes metric subregularity in terms of the graphical derivative defined by (1.2).

Proposition2.2.4 (characterization of metric subregularity). Let T : E⇒E have locally closed graph at(¯x,y)¯ ∈gphT,F :=T−Id, andz¯:= ¯y−x. Then¯ F is metrically subregular at x¯ for z¯ with constant κ and some neighborhood U of x¯ satisfying U ∩F⁻¹(¯z) ={¯x} if and only if the graphical derivative satisfies

DF(¯x|¯z)⁻¹(0) ={0}. (2.9)

If, in addition, T is single-valued and continuously differentiable on U, then the two prop-erties hold if and only if ∇F has rank nat x¯ with

[[∇F(x)]^|]⁻¹

≤κ for all x on U. While the characterization (2.9) appears daunting, the property comes almost for free for polyhedral mappings.

Proposition2.2.5 (polyhedrality implies metric subregularity). [103, Proposition 2.6] Let Λ⊂Ebe an affine subspace andT : Λ⇒Λ. IfT is polyhedral and FixT∩Λ is an isolated point, {¯x}, then F := T −Id is metrically subregular at x¯ for 0 relative to Λ with some constantκ and some neighborhoodU of x¯ satisfying U ∩F⁻¹(0) ={¯x}.

The property characterized in Proposition 2.2.4 is known as the strong metric sub-regularity [50, Section 3I] while Proposition 2.2.5 characterizes its relative version. For completeness,F is strongly metrically subregular atx¯ for y¯∈F(¯x) (relative toΛ, respec-tively) if it is metrically subregular atx¯fory¯andx¯ is an isolated point ofF⁻¹(¯y) (relative to Λ, respectively). For certain applications in stability and numerical analysis, the strong metric subregularity is needed instead of metric subregularity due to its persistence under small perturbations onF.