Isolated calmness and sharp minima via H\"older graphical derivatives

The paper utilizes H\"older graphical derivatives for characterizing H\"older strong subregularity, isolated calmness and sharp minimum. As applications, we characterize H\"older isolated calmness in linear semi-infinite optimization and H\"older sharp minimizers of some penalty functions for constrained optimization.


Introduction
This paper continues our previous work [18] and utilizes Hölder graphical derivatives (sometimes referred to as Studniarski derivatives) for characterizing certain regularity properties of set-valued mappings and real-valued functions.
In the next Section 2, we discuss q-order (q > 0) positively homogeneous mappings and q-order graphical (contingent) derivatives. The definitions and statements mostly follow the corresponding linear ones in [8]. Two norm-like quantities are used for quantifying Hölder graphical derivatives. One of them is a generalization of the wellknown outer norm of a positively homogeneous mapping, while the other seems new and allows to simplify some statements (and proofs) even in the linear case.
In Section 3, Hölder graphical derivatives are used for characterizing Hölder strong subregularity, isolated calmness and sharp minimum. In particular, we give characterizations of Hölder sharp minimizers in terms of Hölder graphical derivatives of the subdifferential mapping. The characterizations from Section 3 are used in Sections 4 and 5 to characterize Hölder isolated calmness in linear semi-infinite optimization and sharp minimizers of ℓp penalty functions, respectively.
Our basic notation is standard, see, e.g., [8,22]. Throughout the paper, X and Y are normed spaces. We use the same notation · for norms in all spaces. If not explicitly stated otherwise, products of normed spaces are assumed equipped with the maximum norms, e.g., (x, y) := max{ x , y }, (x, y) ∈ X × Y . If X is a normed space, its topological dual is denoted by X * , while ·, · denotes the bilinear form defining the pairing between the two spaces. Symbols R, R + and N denote the sets of all real numbers, all nonnegative real numbers and all positive integers, respectively. For the empty subset of R + , we use the conventions sup ∅ = 0 and inf ∅ = +∞. Given an α ∈ R, we denote α + := max{0, α}.
For an extended-real-valued function f : X → R ∪ {+∞}, its domain and epigraph are defined, respectively, by dom f := {x ∈ X | f (x) < +∞} and epi f := {(x, α) ∈ X × R | f (x) ≤ α}. A set-valued mapping F : X ⇒ Y between two sets X and Y is a mapping, which assigns to every x ∈ X a subset (possibly empty) F (x) of Y . We use the notations gph F := {(x, y) ∈ X × Y | y ∈ F (x)} and dom F := {x ∈ X | F (x) = ∅} for the graph and the domain of F , respectively, and F −1 : Y ⇒ X for the inverse of F . This inverse (which always exists with possibly empty values at some y) is defined by F −1 (y) := {x ∈ X | y ∈ F (x)}, y ∈ Y . Obviously dom F −1 = F (X).
Recall that a mapping F : X ⇒ Y is outer semicontinuous (cf., e.g., [8]) at x ∈ X if lim sup i.e., if gph F ∋ (x k , y k ) → (x, y), then y ∈ F (x). This is always the case when gph F is closed.
Throughout the paper, we assume the order of all Hölder properties to be determined by a fixed number q > 0.
If q = 1, we simply say the H is positively homogeneous. The graph of a positively homogeneous mapping is a cone. This is obviously not the case when q = 1.
The next simple fact is a direct consequence of the definition.
For a q-order positively homogeneous mapping H : X ⇒ Y , we define two normlike quantities: (2.1) If dim Y < ∞ and H is outer semicontinuous at 0, the two conditions are equivalent.
If dim X < ∞ and H −1 is outer semicontinuous at 0, the two conditions are equivalent.
Proof Assertions (i)-(vi) and the first parts of assertions (vii) and (viii) are direct consequences of (2.1) and Definition 2.1. For instance, in the case of assertion (ii) using definitions (2.1) we have: To prove the second part of (vii), we need to show that, under the assumptions, . Then u k → 0 as k → ∞ and v k = 1 (k ∈ N). Without loss of generality, v k → v as k → ∞ and v = 1. Furthermore, by Definition 2.1, (u k , v k ) ∈ gph H (k ∈ N) and, thanks to the outer semicontinuity of H, v ∈ H(0).
The proof of the second part of (viii) is similar. Let dim X < ∞, H is outer semicontinuous at 0, and H ⊖ q = 0. By (2.1), there exists a sequence (x k , y k ) ∈ gph H (k ∈ N) such that y k / x k q → 0 as k → ∞. Without loss of generality, x k = 0 for all k ∈ N, and u k := x k / x k → u with u = 1, while v k := y k / x k q → 0. By Definition 2.1, (u k , v k ) ∈ gph H (k ∈ N) and, thanks to the outer semicontinuity of ⊓ ⊔ Assertions (i), (v), (vii) and (viii) in Proposition 2.3 generalize and expand the corresponding parts of [8, Propositions 4A.6 and 5A.7, and Exercise 4A.9]. The above proof of the second part of (vii) largely follows that of the corresponding part of [8,Proposition 4A.6].
Next we briefly consider the case Y = X * .
Definition 2.4 A mapping H : X ⇒ X * is q-order positively definite if there exists a number λ > 0 such that The exact upper bound of all such λ > 0 is denoted by H * q .
In Definition 2.4, it obviously holds In general, the expression in (2.2) is nonnegative, and the case H * q = 0 means that H is not q-order positively definite.  (iii) If H is q-order positively homogeneous and p-order positively definite with some p > 0, then either dom H = {0} or p = q.

⊓ ⊔
Given a set-valued mapping H : X ⇒ Y and a function f : X → Y , their sum H + f is a set-valued mapping from X to Y defined by The next statement characterizes perturbed positively homogeneous mappings. It generalizes [8, Theorem 5A.8] (and is accompanied by a much shorter proof). Theorem 2.6 Let both H : X ⇒ Y and f : X → Y be q-order positively homogeneous.
Then H + f is q-order positively homogeneous. Moreover,

⊓ ⊔
Given a set-valued mapping F : X ⇒ Y , its q-order graphical derivative at (x,ȳ) ∈ gph F is a set-valued mapping DqF (x,ȳ) : X ⇒ Y defined for all x ∈ X by DqF (x,ȳ) is sometimes referred to as q-order upper Studniarski derivative [26, Definition 3.1] of F at (x,ȳ). When q = 1, it reduces to the standard graphical (contingent) derivative; cf. [2,3,8,16,22]. Clearly, DqF (x,ȳ) is a q-order positively homogeneous mapping with closed graph, and Given a function f : X → Y and a pointx ∈ dom f , we write Dqf (x) instead of Dqf (x, f (x)). If Dqf (x) is single-valued, i.e. the limit exists for all x ∈ X, we say that f is q-order Hadamard directionally differentiable atx.
The next proposition provides a sum rule for q-order graphical derivatives. It is a direct consequence of the definitions of q-order graphical derivative and q-order Hadamard directional differentiability.
Given a function f : X → R ∪ {+∞}, its q-order Hadamard directional subderivative [24,25] atx ∈ dom f is defined for all x ∈ X by (cf. [ (2.6) If f is Lipschitz continuous nearx and 0 < q ≤ 1, the above definition takes a simpler form: Observe that the function f ′ q (x; ·) : X → R ∪ {±∞} is lower semicontinuous and q-order positively homogeneous in the sense that for all x ∈ X and λ > 0. We are going to use for characterizing this function the following norm-like quantity: The next statement is a direct consequence of the definitions. It uses the epigraph- Note that the graph of the latter mapping is the epigraph epi f of f . We use the same notation for the epigraph and the epigraphical mapping.
is lower semicontinuous. This proves the opposite implication.

⊓ ⊔
The next corollary is a consequence of Propositions 2.7 and 2.8.

Hölder strong subregularity, isolated calmness and sharp minimum
In this section, Hölder graphical derivatives are used for characterizing Hölder strong subregularity, isolated calmness and sharp minimum.
The exact upper bound of all such τ > 0 is denoted by srgq F (x,ȳ). (ii) A mapping S : Y ⇒ X possesses q-order isolated calmness property at (ȳ,x) ∈ gph S with modulus τ > 0 if there exist neighbourhoods U ofx and V ofȳ such that The exact upper bound of all such τ > 0 is denoted by clmq S(ȳ,x).
The properties in the above definition are well known in the linear case q = 1 (see, e.g., [8]), but have also been studied in the general setting (also for not necessarily strong subregularity and not necessarily isolated calmness); cf. [7,9,19]. Because of the distance involved in the right-hand side of (3.1) (and also in its left-hand side in the case of the not strong version), the property in part (i) of Definition 3.1 is often referred to as q-order strong metric subregularity.
which justifies the word 'isolated' in the name of the property in Definition 3.1(ii). (iii) The moduli srgq F (x,ȳ) and clmq S(ȳ,x) are usually introduced to characterize the usual (not strong!) subregularity and (not isolated!) calmness. We do not consider these two weaker properties in the current paper. If a respective (strong or isolated) property in Definition 3.1 holds, then the corresponding modulus coincides with the conventional one. (iv) When V = Y , condition (3.1) is obviously implied by the following q-order strong graph subregularity property: (with the same τ and U ). It is not difficult to show that, when q ≥ 1, q-order (strong) subregularity in part (i) of Definition 3.1 implies q-order (strong) graph subregularity (with smaller τ and U ); cf. a characterization of subregularity in [ (v) There is some inconsistency in the literature concerning whether to place the constants τ and/or q, which determine the properties in Definition 3.1, in the left or right-hand sides of the inequalities (3.1) and (3.2) (and similar inequalities involved in related definitions); cf., e.g., [16]. This applies also to our own recent paper [18], where we placed q in the right-hand sides of the inequalities. Of course, the position of the constants does not effect the properties, but it has an effect on the values of the respective moduli. Our choice in the current paper is determined by our desire to produce the simplest relations between these moduli and the quantitative characteristics of Hölder graphical derivatives and more straightforward proofs.
The next proposition is an immediate consequence of Definition 3.1.

⊓ ⊔
The following statement provides a characterization of q-order strong subregularity of a mapping in terms of its q-order graphical derivative. (i) F is q-order strongly subregular at (x,ȳ); Proof Thanks to Proposition 3.4, we have the implication (i) ⇒ (ii) in general, and the equivalence (i) ⇔ (ii) when dim X < +∞ and dim Y < +∞. The implication (ii) ⇒ (iii) is an immediate consequence of Proposition 2.3(viii). The graph of DqF (x,ȳ) is closed by definition, hence, DqF (x,ȳ) −1 is outer semicontinuous at 0. Employing Proposition 2.3(viii) again, we conclude that (ii) ⇔ (iii) when dim X < +∞. ⊓ ⊔ Remark 3.6 A coderivative analogue (employing a special kind of limiting coderivative) of the equality in Corollary 3.5(iii) is used in [28,Theorem 5.2] to characterize nonlinear subregularity.
The next example illustrates application of Corollary 3.5 for checking Hölder strong subregularity as well as computation of the Hölder graphical derivative and relevant norm-like quantity. Of course, in this simple example, the same conclusions can be obtained directly from Definition 3.1(i).
Proof By Proposition 2.3(ii) and (2.5), we have The assertion follows from Propositions 3.3 and 3.4.

⊓ ⊔
Corollary 3.9 Let S : Y ⇒ X and (ȳ,x) ∈ gph S. Consider the following conditions: (i) S possesses q-order isolated calmness property at (ȳ,x); Proof Thanks to Corollary 3.8, we have the implication (i) ⇒ (ii) in general, and the equivalence (i) ⇔ (ii) when dim X < +∞ and dim Y < +∞. The implication (ii) ⇒ (iii) is an immediate consequence of Proposition 2.3(vii). The graph of D 1 q

⊓ ⊔
The following theorem shows that q-order strong subregularity enjoys stability under perturbations by functions with small q-order Hadamard directional derivatives.
Proof By Proposition 2.7 and Theorem 2.6, The assertion follows from Proposition 3.4 and Corollary 3.5.

⊓ ⊔
The next proposition is a consequence of Proposition 2.5(ii) and Proposition 3.4.
Given a function f : X → R ∪ {+∞}, it is easy to check (taking into account Remark 3.2(i)) that the q-order strong subregularity of its epigraphical mapping at (x, f (x)) reduces to the property in the next definition. (3.5) The exact upper bound of all such τ > 0 is denoted by shrpq f (x).
Ifx is not a q-order sharp minimizer of f , we have shrpq f (x) = 0. Remark 3.13 (i) The property in Definition 3.12 is also known as isolated local minimum with order q; cf. [24]. (ii) If f (x) = 0 andx is a q-order sharp minimizer of f , then shrpq f (x) coincides with the q-order error bound modulus erq f (x) of f atx.

The next proposition is a consequence of Proposition 3.4 and Proposition 2.8(iii).
Proposition 3.14 Let f : If dim X < +∞, then (3.6) holds as equality.
The following lemma describing Hölder sharp minimizers in terms of the Hölder strong subregularity of the subdifferential mappings is a reformulation of [27, Theorem 4.1] in the convex setting. (i)x is a (q + 1)-order sharp minimizer of f with modulus ρ > 0; (ii) ∂f is q-order strongly subregular at (x, 0) with modulus τ > 0.
Next we give characterizations of Hölder sharp minimizers in terms of Hölder graphical derivatives of the subdifferential mapping. The theorem below is partially motivated by [1,Corollary 3.7], which provides a characterization of the strong subregularity in terms of the positive-definiteness of the graphical derivative. The modulus estimate in the following theorem is inspired by [20,Theorem 3.6], where a characterization of tilt stability of local minimizers for extended-real-valued functions is derived via the second-order subdifferential. (i)x is a (q + 1)-order sharp minimizer of f with modulus ρ > 0; (ii) Dq∂f (x, 0) is q-order positively definite with modulus λ > 0.
Of course, in this simple example, the same conclusions can be obtained directly from Definition 3.12. Moreover, shrp 2n f (0) = 1, i.e. the lower estimate in (3.7) is not sharp.
Comparing the statements of Proposition 3.11, Lemma 3.15 and Theorem 3.16, we arrive at the following corollary, which provides an important special case when the implication in Proposition 3.11 holds as equivalence.
Corollary 3.18 Let dim X < +∞, f : X → R ∪ {+∞} be lower semicontinuous and convex, andx ∈ dom f be a local minimizer of f . Then Dq∂f (x, 0) is q-order positively definite if and only if ∂f is q-order strongly subregular at (x, 0), and

q-order isolated calmness in linear semi-infinite optimization
In this section, we consider a canonically perturbed linear semi-infinite optimization problem: where x ∈ R n is the vector of variables, c ∈ R n , ·, · represents the usual inner product in R n , T is a compact Hausdorff space, and the function t → (a t , b t ) is continuous on T . In this setting, the pair (c, b) ∈ R n × C(T, R) is regarded as the perturbation parameter. The parameter space R n × C(T, R) is endowed with the uniform convergence topology through the maximum norm (c, b) := max{ c , b ∞ }, where · is the Euclidean norm in R n and b ∞ := max t∈T |b t |.
The feasible set and solution mappings corresponding to the above problem are defined, respectively, by From now on, we assume a point ((c,b),x) ∈ gph S to be given. We are going to consider also the partial solution mapping Sc : C(T, R) ⇒ R n given by Sc(b) = S(c, b) and the level set mapping and employ the following convex and continuous function: Observe that f (x) = 0, and The problem P (c, b) satisfies the Slater condition if there exists a pointx ∈ R n such that a t ,x < b t for all t ∈ T . The set of active indices at x ∈ F(b) is defined by The following lemma is an analogue of [18,Proposition 4.5].  (iv)x is a q-order sharp minimizer of f ; ( (ii) ⇒ (iii). Suppose that L does not possess q-order isolated calmness property at (( c,x ,b),x). To reach a contradiction with (ii), it suffices to show that, for the sequence {(b k , x k )} ⊂ gph F in Lemma 4.1, it holds x k ∈ Sc(b k ), k ∈ N, which readily follows from the KKT conditions (4.5) (by continuity, it is not restrictive to assume that P (c, b k ) satisfies the Slater condition).
(iv) ⇒ (i). By [18,Lemma 4.2] (with f ≡ 0), there exist a number M > 0 and neighbourhoods U ofx and V of (c,b) such that where 'co' stands for the convex hull. Letx be a q-order sharp minimizer of f . By Definition 3.12, condition (3.5) holds with some number τ > 0 and a smaller neighbourhood U if necessary. Without loss of generality, we assume that M > 1, and U is bounded: x −x < δ for some δ > 0 and all for some η t ≥ 0, t ∈ T b (x) , satisfying t∈T b (x) η t ≤ M and only finitely many being positive. Hence, in view of representation (4.8), and definitions (4.1) and (4.2), Recalling definition (4.3) and the fact that f (x) = 0, we have By Definition 3.1(ii), S possesses q-order isolated calmness property at ((c,b),x).
(iv) ⇔ (v) is immediate from Proposition 3.14. (ii) ⇒ (vi) and the opposite implication when T is finite, together with the equality (4.6) follow from Corollaries 3.8 and 3.9. It suffices to notice that, when T is finite, the parameter space C(T, R) is finite-dimensional. When q > 1, the equivalence (iv) ⇔ (vii) is a consequence of Theorem 3.16. In the case q ≥ 1, implication (iv) ⇒ (i) is a special case of [13,Theorem 2.2]. For the semi-infinite optimization model P (c, b), this implication was explicitly given, e.g., in [14,Proposition 4.2]. Indeed,x is a q-order sharp minimizer of f , then, using the notation of Definition 3.1, one has in particular i.e.,x is a strict local minimizer of P (c,b) in the sense of [14]. Since the Slater condition is equivalent to the extended Mangasarian-Fromovitz CQ (for this equivalence in relation to the linear SIP problem P (c, b) see, e.g., [10, Theorem 6.1] and [4, Theorem 2.1]), [14,Proposition 4.2] applies and gives (in particular) that S possesses the q-order isolated calmness property at ((c,b),x).
Next we recall the Extended Nürnberger Condition (ENC, in brief) [6, Definition 2.1]. Definition 4.4 ENC is satisfied at ((c,b),x) when P (c,b) satisfies the Slater condition, and there is no subset D ⊂ Tb(x) with |D| < n such that −c ∈ cone {a t , t ∈ D}.
The following lemma is [6, Theorem 2.1 and Lemma 3.1]). Lemma 4.5 Suppose that ENC is satisfied at ((c,b),x). Then (i) S is single valued and Lipschitz continuous in a neighbourhood of (c,b); Thanks to Lemma 4.5, we can show that the parameter c can be considered fixed in our analysis, provided that ENC holds at ((c,b),x).
x k =x for all k ∈ N. If ENC is satisfied at ((c,b),x), then, by Lemma 4.5, (b k , x k ) ∈ gph Sc for all k large enough. Hence, This completes the proof.
(ii) Suppose that f is q-order Hadamard directionally differentiable atx, m = 1 andx is a q-order sharp minimizer of (5.2) for some r > 0. Then (5.4) holds.
Proof (i) If K * (x) = ∅, then a(x) > 0 and b(x) ≤ 0. Therefore ρ 0 is well defined and nonnegative. Let r > ρ 0 . Let x = 0. As (ℓp) ′ q (x; ·) is positively homogeneous, without loss of generality, we assume that x = 1. Obviously, (g + i ) ′ q/p (x; x) ≥ 0 for all i ∈ I. It is easy to show that p for all i ∈ I(x).
Since f ′ q (x; ·) is proper, it follows that , we have f ′ q (x; x) > 0, and consequently, (ℓp) ′ q (x; x) > 0. If x ∈ K * (x), then b(x) ≤ 0. So, by definitions of ρ 0 and a(x), we have r i∈I(x) and thus it follows from (5.5) that So, by (5.3),x is a q-order sharp minimizer for (5.2). (ii) It follows from Corollary 2.9 that (5.5) holds as equality. The conclusion is verified by Proposition 3.14 and Proposition 2.8 (iv).
Furthermore, for all i ∈ I(x), u ∈ R n and t > 0, we obviously have Hence, if g i (i ∈ I(x)) is (q/p)-order Hadamard directionally differentiable atx, then so is g + i , and (g + i ) ′ q/p (x; x) = max{0, (g i ) ′ q/p (x; x)} for all x ∈ R n . Therefore, if all g i (i ∈ I(x)) are (q/p)-order Hadamard directionally differentiable atx, then (5.4) is equivalent to the following condition: f ′ q (x; x) > 0 for all x = 0 with (g i ) ′ q/p (x; x) ≤ 0, i ∈ I(x).
The following simple example shows the calculation of the exact upper bound of the 1-order sharp minimizer of the penalty problem (5.2).