GENERALIZED DIFFERENTIATION WITH POSITIVELY HOMOGENEOUS MAPS: APPLICATIONS IN SET-VALUED ANALYSIS AND METRIC REGULARITY

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1 GENERALIZED DIFFERENTIATION WITH POSITIVELY HOMOGENEOUS MAPS: APPLICATIONS IN SET-VALUED ANALYSIS AND METRIC REGULARITY C.H. JEFFREY PANG Abstract. We propose a new concept of generalized dierentiation of setvalued maps that captures rst order information. This concept encompasses the standard notions of Fréchet dierentiability, strict dierentiability, calmness and Lipschitz continuity in single-valued maps, and the Aubin property and Lipschitz continuity in set-valued maps. We present calculus rules, sharpen the relationship between the Aubin property and coderivatives, and study how metric regularity and open covering can be rened to have a directional property similar to our concept of generalized dierentiation. Finally, we discuss the relationship between the robust form of generalization dierentiation and its one sided counterpart. Contents 1. Introduction 1 2. Preliminaries and notation 3 3. Generalized dierentiability of single-valued maps 4 4. Generalized dierentiability of set-valued maps 8 5. Calculus of T -dierentiability The Mordukhovich criterion Metric regularity and open covering Strict T -dierentiability from outer T -dierentiability 22 References Introduction We say that S is a set-valued map (or a multi-valued function, or multifunction) if for all x X, S(x) is a subset of Y, and we denote set-valued maps by S : X Y. Many problems of feasibility, control, optimality and equilibrium are set-valued in nature, and are best treated with methods in set-valued analysis. The texts [4, 5, 21] contain much of the theory of set-valued analysis. Set-valued analysis serves as a foundation for the theory of dierential inclusions [3, 2], control theory [13] and variational analysis [35, 27, 12], which in turn have many applications in applied Date: February 16, Mathematics Subject Classication. 26E25, 46G05, 46T20, 47H04, 49J50, 49J52, 49J53, 54C05, 54C50, 54C60, 58C06, 58C07, 58C20, 58C25, 90C31 IIIOR/MS classication words: Mathematics (Functions, Sets). Key words and phrases. multi-function dierentiability, metric regularity, coderivatives, calmness, Lipschitz continuity. 1

2 2 C.H. JEFFREY PANG mathematics. We refer to these texts for the abundant bibliography on the history of set-valued analysis. The main contribution of this paper is to introduce a new concept of generalized dierentiation (Denitions 4.1 and 4.6) using positively homogeneous maps. Any reasonable denition of a derivative for set-valued maps has to describe changes in the set in terms of the input variables. Using the Pompieu-Hausdor distance (a metric on the space of compact sets), one obtains the classical denition of Lipschitz continuity of set-valued maps. The concept introduced in this paper provides a more precise tool that incoporates rst order information in a set-valued map, encompassing the standard notions of Fréchet dierentiability, strict dierentiability, calmness and Lipschitz continuity in single-valued maps, and the Aubin property and Lipschitz continuity in set-valued maps. We illustrate how this new concept relates to, and extends, existing methods in variational and set-valued analysis. To motivate our discussion, we revisit the relation between the Clarke subdierential and Clarke Jacobian and the nonsmooth behavior of functions that can be traced back to [17]. Other than the rst four sections which provide the necessary background for the rest of the paper, the last four sections can be read in any order. In Section 3, we recall generalized dierentiation for single-valued functions, which was rst proposed by Ioe [16, 17]. We relate the generalized derivative to common notions in classical and variational analysis, paying particular attention to the Clarke subdierential and the Clarke Jacobian. In Section 4, we dene generalized dierentiation for set-valued maps, and illustrate the lack of relation between our generalized derivatives and the notions of set-valued derivatives based on the tangent cones, namely semidierentiability [29] and proto-dierentiability [34]. We present calculus rules in Section 5. The Aubin property (See Denition 4.7), which is commonly attributed to [1], is a method of analyzing local Lipschitz continuity of set-valued maps. In Section 6, we revisit the classical relationship between the Aubin property and the coderivatives of a set-valued map. This relationship is referred to as the Mordukhovich criterion in [35]. Since the coderivatives of a set-valued map can be calculated in many applications and enjoy an eective calculus, this relationship is an important tool in the study of the Lipschitz properties of set-valued maps. We will show that the coderivatives actually gives more information on the local Lipschitz continuity property in our language of generalized derivatives. It is well known that the Aubin property is related to metric regularity and open covering [6, 26, 30]. Open covering is sometimes known as linear openness. Metric regularity is important in the analysis of solutions to ȳ S( x), while open covering studies local covering properties of a set-valued map. Both metric regularity and open covering can be viewed as a study of set-valued maps whose inverse have the Aubin property. For more on metric regularity, we refer the reader to [35, 27, 19, 20]. In Section 7, we take a new look at metric regularity and open covering in view of our denitions of the generalized derivatives. We study metric regularity and open covering in a much broader framework, illustrating that a directional behavior similar to that in our denition of generalized derivatives is present in metric regularity and open covering. In Section 8, we discuss how the (basic and strict) generalized derivatives dened in Sections 3 and 4 relate to each other. As particular cases, we obtain an equivalent

3 GENERALIZED DIFFERENTIATION WITH POSITIVELY HOMOGENEOUS MAPS 3 criterion for strict dierentiability of set-valued maps, and a relationship between calmness and Lipschitz continuity in both single-valued and set-valued maps. As far as we are aware, the relation between calmness and Lipschitz continuity in set-valued maps was rst discussed in [24, 32]. 2. Preliminaries and notation Throughout this paper, we shall assume that X and Y are Banach spaces. In most cases, we follow the notation of [35]. Given two sets A, B X, the notation A + B stands for the Minkowski sum of two sets, dened by A + B := {a + b a A, b B}. The notation A B is interpreted as A + ( B). We use, : X X R, where ζ, x := ζ(x), to denote the usual dual relation. In Hilbert spaces (and hence in R n ),, reduces to the usual inner product. The notation x x means that D x x, with sequences lying in D X. The closed ball with center x and radius r is denoted by B(x, r), while B denotes the ball with center 0 and radius 1. We say that S is a set-valued map from X to Y if S(x) Y for all x X, and a set-valued map is denoted by S : X Y. The set-valued map S is closed-valued if S(x) is closed for all x X, and it is convex-valued if S(x) is convex for all x X. A closed set-valued map is a map whose graph is closed. We say that C X is a cone if 0 C and λx C for all λ > 0 and x C. The graph of a set-valued map gph(s) X Y is the set {(x, y) y S(x)}. The set-valued map S 1 : Y X is dened by S 1 (y) := {x y S(x)}, and gph(s 1 ) = {(y, x) (x, y) gph(s)}. Denition 2.1. A set-valued map T : X Y is positively homogeneous if T (0) is a cone, and T (kw) = kt (w) for all k > 0 and w X. It is clear that T is positively homogeneous if and only if gph(t ) is a cone. If T 1, T 2 : X Y are two set-valued maps such that T 1 (w) T 2 (w) for all w X, then we write this property as T 1 T 2. We denote the set-valued map T ( ) : X Y to be T ( )(w) := T ( w). We recall the denition of inner limits of a set valued map. Denition 2.2. When S : X Y is a set-valued map, we say that lim inf x x S(x) := {y Y lim d(y, S(x)) = 0} is the inner limit of S(x) when x x. x x We recall the denitions of outer and inner semicontinuity. Denition 2.3. For a closed-valued mapping S : X Y and a point x X: (1) S is upper semicontinuous at x if for any open set U such that S( x) U, there exists some η > 0 such that if x x < η, then S(x) U. (2) S is outer semicontinuous at x if for any open set U such that S( x) U and U ρb = X for some ρ > 0, there exists some η > 0 such that if x x < η, then S(x) U. (3) S is inner semicontinuous at x if S( x) lim inf x x S(x). (4) S is continuous at x if it is both outer and inner semicontinuous there.

4 4 C.H. JEFFREY PANG We caution that the terminology used to denote upper semicontinuity and outer semicontinuity is not consistent in the literature. Outer semicontinuity is better suited to handle set-valued maps with unbounded value sets S(x). For example, the set-valued map S : R R 2 dened by S(θ) := {(t cos θ, t sin θ) t 0} is not upper semicontinuous anywhere but is outer semicontinuous, and in fact continuous, everywhere. When S( x) is bounded, upper and outer semicontinuity are equivalent. We will not use upper semicontinuity in this paper. 3. Generalized differentiability of single-valued maps The emphasis of this section is the generalized dierentiability of single-valued maps f : X Y. Much of the theory is already in [17], but we concentrate on the key results that we will extend for the set-valued case in later sections. We now begin with our rst denition of generalized dierentiability. Denition 3.1. Let T : X Y be a set-valued map. We say that f : X Y is T -dierentiable at x if for any δ > 0, there exists a neighborhood V of x such that f(x) f( x) + T (x x) + δ x x B for all x V. We say that f : X Y is strictly T -dierentiable at x if T : X Y is positively homogeneous, and for any δ > 0, there exists a neighborhood V of x such that f(x) f(x ) + T (x x ) + δ x x B for all x, x V. The map T : X Y is referred to as a prederivative when f is T -dierentiable, and as the strict prederivative when f is strictly T -dierentiable in [17, Denition 9.1]. The denition of T -dierentiability includes the familiar concepts of dierentiability and Lipschitz continuity as special cases. Example 3.2. (Examples of T -dierentiability) Let f : X Y be a single-valued map. (1) When T : X Y is a (single-valued) linear map, T -dierentiability is precisely Fréchet dierentiability with derivative T, and strict T -differentiability is precisely strict dierentiability with derivative T. (2) When T : X Y is dened by T (w) = κ w B, T -dierentiability is precisely calmness with modulus κ, and strict T -dierentiability is precisely Lipschitz continuity with modulus κ. Strict T -dierentiability is more robust than T -dierentiability. The function f : R R dened by f(x) = x 2 sin( 1 x 2 ) is Fréchet dierentiable at 0 but not Lipschitz there. With the right choice of T, T -dierentiability can also handle inequalities. Example 3.3. (Inequalities with T -dierentiability) Let f : X R be a singlevalued map. (1) f : X R is calm from below at x with modulus κ if and only if f is T -dierentiable there, where T : X R is dened by T (w) = [ κ w, ). (2) The vector v X is a Fréchet subgradient of f at x if and only if f is T -dierentiable there, where T : X R is dened by T (w) = [ v, w, ).

5 GENERALIZED DIFFERENTIATION WITH POSITIVELY HOMOGENEOUS MAPS 5 (3) The convex set C X is a subset of the Fréchet subdierential of f at x if and only if f is T -dierentiable there, where T : X R is dened by T (w) = [sup v C v, w, ). While the map T : X Y need not be positively homogeneous, most results in this paper are obtained for this particular case. It is clear from the denitions that if f is T 1 -dierentiable at x, and T 2 satises T 2 T 1, then f is T 2 -dierentiable at x as well. At this point, we mention connections to other notions of other generalized derivatives for single-valued functions close to the denition of (strict) T -differentiability. Semidierentiability, as is recorded in [35, Denition 7.20], can be traced back to [28], and is equivalent to the case where T : X Y for the case when T is continuous and single-valued (see [35, Section 9D]). Semidierentiability for single-valued maps is not to be confused with semidierentiability for set-valued maps dened in Denition We now look at a slightly nontrivial example involving the Clarke subdierential [7]. Denition 3.4. [9, Section 2.1] Let X be a Banach space. Suppose f : X R is locally Lipschitz. The Clarke generalized directional derivative of f at x in the direction v X is dened by f ( x; v) = lim sup t 0,x x f(x + tv) f(x), t where x X and t is a positive scalar. The Clarke subdierential of f at x, denoted by C f( x), is the convex subset of the dual space X given by {ζ X f ( x; v) ζ, v for all v X}. The Clarke subdierential enjoys a mean value theorem. For C X, dene the set C, w by { c, w c C}. The following result is due to Lebourg [23], which has since been generalized in other ways. Theorem 3.5. [23] (Nonsmooth mean value theorem) Suppose x 1, x 2 X and f is Lipschitz on an open set containing the line segment [x 1, x 2 ]. Then there exists a point u (x 1, x 2 ) such that f(x 2 ) f(x 1 ) C f(u), x 2 x 1. We show how the Clarke subdierential relates to T -differentiability. For extensions and a more general treatment of the following result, we refer the reader to [17, Sections 9, 10] Theorem 3.6. (Clarke subdierential and T -dierentiability) Let f : X R be a Lipschitz function, and C be a convex subset of X. If f is strictly T -dierentiable at x, where T : X R is dened by T (w) = C, w, then C f( x) C. The converse holds if the map x C f(x) is outer semicontinuous at x, which is the case when X = R n.

6 6 C.H. JEFFREY PANG Proof. If f is strictly T -dierentiable at x, then for any direction h X, the Clarke directional derivative is max v, h = f ( x; h) v f( x) f(x + th) f(x) = lim sup x x,t 0 t max v, h. v C Suppose on the contrary that v C f( x)\c. Then there exists a direction h and some α R such that v, h > α but max v C v, h α. This is a contradiction, which shows that C f( x) C. We now prove the converse. It is well-known that when X = R n, the Clarke subdierential mapping of a Lipschitz function is outer semicontinuous. See [35] for example. Suppose that C f( x) C. By outer semicontinuity, given any δ > 0, there is some ɛ > 0 such that C f(x) C + δb for all x B( x, ɛ). Theorem 3.5 states that for any points x 1, x 2 B( x, ɛ), there is an x (x 1, x 2 ) such that f(x 1 ) f(x 2 ) C f(x ), x 1 x 2. This immediately implies that f(x 1 ) f(x 2 )+ C, x 1 x 2 +δ x 1 x 2 B, and hence strict T -dierentiability. It is well known that in nite dimensions, the Clarke subdierential is the limit of gradients taken over where the function is dierentiable. We now recall Rademacher's theorem. Theorem 3.7. (Rademacher's Theorem) Let O R n be open, and let f : O R m be Lipschitz. Let D be the subset of O consisting of the points where F is dierentiable. Then O\D is a set of measure zero in R n, In particular, D is dense in O, i.e., cl D O. Closely related to the Clarke subdierential is the Clarke Jacobian that was rst introduced in [8]. Denition 3.8. Let f : O R m be Lipschitz, with O R n open, and let D O consist of the points where f is dierentiable. The Clarke Jacobian at x is dened by f( x) := conv{a R m n : x i x with x i D, f(x i ) A}. It is clear from Rademacher's Theorem that the Clarke Jacobian is a nonempty, compact set of matrices. If f : R n R is Lipschitz, then it is well-known that the Clarke Jacobian reduces to the Clarke subdierential. The following result is equivalent to [17, Proposition 10.9], and is a generalization of a result that is well known for m = 1. Theorem 3.9. (Clarke Jacobian and T -dierentiability) Let f : O R m be Lipschitz on an open set O R n, and let D be the subset of O where f is dierentiable. Let D D be such that D\D is of measure zero. At each x O, f is T -dierentiable at x, where T : R n R m is dened by T (w) := {Aw A f( x)}.

7 GENERALIZED DIFFERENTIATION WITH POSITIVELY HOMOGENEOUS MAPS 7 Proof. In the case O = R n, a useful equivalent condition for a set C R n to be of measure zero is this: with respect to any vector w 0, C is of measure zero if and only if the set {τ x + τw C} R is of measure zero (in the one-dimensional sense) for all x outside a set of measure zero. When x O and τ is small enough that the line segment from x to x + τw lies in O, the function ϕ(t) = f(x + tw) is Lipschitz continuous for t [0, τ]. Lipschitz τ continuity guarantees that ϕ(τ) = ϕ(0)+ 0 ϕ (t)dt, and in this integral a negligible set of t values can be disregarded. Thus, the integral is unaected if we concentrate on t values such that x + tw D, in which case ϕ (t) = f(x + tw)(w). For any ɛ > 0, we can reduce O so that whenever A = f(x) at some x O\D, then A f( x) + ɛb. We have f(x + τw) = ϕ(τ) = ϕ(0) + = f(x) + f(x) + τ 0 τ 0 τ 0 ϕ (t)dt f(x + tw)(w)dt T (w) + ɛ w Bdt f(x) + T (τw) + ɛ τw B. The case where f(x+tw) does not exist for all t can be treated easily by perturbing x. This establishes the T -dierentiability of f. In Theorem 3.9, it is clear that there can be no closed convex valued positively homogeneous map T T such that f is T -dierentiable at x. For convenience, we dene strict T -dierentiability on a set U as follows. Denition We say that f : X Y is strictly T -dierentiable on U X if for all x, x U, we have f(x) f(x ) + T (x x ). Given any map T : X Y and constant δ > 0, we shall denote (T + δ) : X Y to be the map (T + δ)(w) := T (w) + δ w B. It is clear from the denitions that f is strictly T -dierentiable at x if and only if for any δ > 0, we can nd an open neighborhood U δ of x such that f is strictly (T + δ)-dierentiable on U δ. For single-valued maps, T -dierentiability behaves well under intersections. Proposition (Intersections) Let f : X Y. (1) If f : X Y is both strictly T 1 -dierentiable and strictly T 2 -dierentiable on an open set U X, then f is strictly (T 1 T 2 )-dierentiable on U. (2) If f : X Y is T 1 -dierentiable and T 2 -dierentiable at x X, then f is (T 1 T 2 )-dierentiable at x. An analogous statement holds for strict T -dierentiability. A function strictly T -dierentiable at a point has added structure. Proposition (T = T ( ) in strict T -dierentiability) Let f : X Y be a single-valued function that is strictly T -dierentiable on an open set neighborhood

8 8 C.H. JEFFREY PANG U X. Then f is strictly (T T ( ))-dierentiable on U. Hence we can assume that T satises T = T ( ). Similarly, f is strictly T -dierentiable at x implies f is (T T ( ))-dierentiable there. We can again assume T = T ( ). Proof. For any x, x U, we have f(x) f(x ) T (x x ). Reversing the role of x and x gives f(x) f(x ) T (x x). This gives f(x) f(x ) T (x x ) T ( (x x )), which gives the required conclusion. The conclusion T = T ( ) is also elementary. The statement on T -dierentiability at a point has a similar proof. We refer the reader to Section 5 for calculus of T -dierentiable functions, and to Corollary 8.4 for the relation between T -dierentiability and strict T -dierentiability. For more information on T -dierentiability, we refer the reader to [17], especially Sections 9 and 10 there. 4. Generalized differentiability of set-valued maps In this section, we move on to dene the generalized dierentiability of setvalued maps and state some basic properties. Here is the rst denition of the dierentiability of a set-valued map. Denition 4.1. (a) Let T : X Y be a set-valued map. We say that S : X Y is outer T -dierentiable at x if for any δ > 0, there exists a neighborhood V of x such that S(x) S( x) + T (x x) + δ x x B for all x V. It is inner T -dierentiable at x if the formula above is replaced by S( x) S(x) T (x x) + δ x x B for all x V. It is T -dierentiable at x if it is both outer T -dierentiable and inner T -dierentiable. (b) Let T : X Y be a positively homogeneous set-valued map. We say that S : X Y is strictly T -dierentiable at x if for any δ > 0, there exists a neighborhood V of x such that S(x) S(x ) + T (x x ) + δ x x B for all x, x V. It is elementary that if S : X Y is a single-valued map S : X Y, then the denitions of outer T -dierentiability, inner T -dierentiability and T - dierentiability in Denition 4.1(a) coincide. The case T (w) := κ w B, where κ 0 is nite, is well studied in variational analysis. In the denitions of calmness and Lipschitz continuity below, we recall the notation for κ commonly used in variational analysis. Calmness was rst referred to as upper Lipschitzian by Robinson [31], who established this property for polyhedral mappings. Denition 4.2. (a) We say that S : X Y is calm at x if there exists a neighborhood V of x and κ 0 such that S(x) S( x) + κ x x B for all x V. The inmum of all constants κ is the calmness modulus, denoted by calm S( x). Calmness at x is equivalent to outer T -dierentiability there, where T : X Y is dened by T (w) := calm S( x) w B.

9 GENERALIZED DIFFERENTIATION WITH POSITIVELY HOMOGENEOUS MAPS 9 (b) We say that S : X Y is Lipschitz at x if there exists a neighborhood V of x and κ > 0 such that S(x) S(x ) + κ x x B for all x, x V. The inmum of all constants κ is the Lipschitz modulus, denoted by lip S( x). Lipschitz continuity at x is equivalent to strict T -dierentiability there, where T : X Y is dened by T (w) := lip S( x) w B. Consider the case where there is some κ 0 such that T : X Y satis- es T (w) κ w B for all w X. It follows straight from the denitions that if S : X Y is outer T -dierentiable and maps to compact sets, then it is outer semicontinuous. The same relations hold for inner T -dierentiability and inner semicontinuity, and for T -dierentiability and continuity. We remark that strict T -dierentiability implies strict continuity in the sense of [35, Denition 9.28], but not vice versa. Their dierence is analogous to the dierence between upper semicontinuity and outer semicontinuity. We motivate this denition of set-valued dierentiability with Proposition 4.4, whose proof is straightforward. We now recall the Pompieu-Hausdor distance. Denition 4.3. For C, D X closed and nonempty, the Pompieu-Hausdor distance between C and D is the quantity d(c, D) := inf{η C D + ηb, D C + ηb}. The Pompieu-Hausdor distance is a metric on compact subsets of X. In fact, the motivation of calmness and Lipschitz continuity in Denition 4.2 comes from the Pompieu-Hausdor distance. To motivate the denition of T -dierentiability, we note the following result. Proposition 4.4. (Single-valued T -dierentiability) Suppose S : X Y is closedvalued. (1) Let T : X Y be a single-valued map. Then S is T -dierentiable at x if and only if for any δ > 0, there is a neighborhood V of x such that d(s( x) + T (x x), S(x)) < δ x x for all x V. (2) Let T : X Y be a single-valued map such that T = T ( ). Then S is strictly T -dierentiable at x if and only if for any δ > 0, there is a neighborhood V of x such that d(s(x ) + T (x x ), S(x)) < δ x x for all x, x V. We now make a remark on the Pompieu-Hausdor distance that is in the spirit of the main idea in this paper. Remark 4.5. (More precise measurement of sets) We can rewrite the Pompieu- Hausdor distance as d(c, D) = inf{η C D + E 1, D C + E 2, E 1 ηb, E 2 ηb}. In certain situations, it might be useful to study sets E 1 and E 2 for which C D + E 1 and D C + E 2 instead of just taking them to be ηb. As is well-known in set-valued analysis, setting restrictions on the range gives a sharper analysis at the points of interest. We make the following denitions with this in mind.

10 10 C.H. JEFFREY PANG Denition 4.6. Let S : X Y be a set-valued map such that ȳ S( x). (1) Let T : X Y be a set-valued map. We say that S is pseudo outer T - dierentiable at x for ȳ if for any δ > 0, there exists neighborhoods V of x and W of ȳ such that S(x) W S( x) + T (x x) + δ x x B for all x V. It is pseudo inner T -dierentiable at x for ȳ if S( x) W S(x) T (x x) + δ x x B for all x V. It is pseudo T -dierentiable at x for ū if it is both pseudo outer T -differentiable and pseudo inner T -dierentiable there. (2) Let T : X Y be a positively homogeneous set-valued map. We say that S is pseudo strictly T -dierentiable at x for ȳ if for any δ > 0, there exists neighborhoods V of x and W of ȳ such that S(x) W S(x ) + T (x x ) + δ x x B for all x, x V. Again, the case T (w) := κ w B is of particular interest. The Aubin property was rst introduced as pseudo-lipschitzness in [1]. Denition 4.7. Let S : X Y be a set-valued map such that ȳ S( x). (1) We say that S is calm at x for ȳ if there exists neighborhoods V of x and W of ȳ, and κ 0 such that S(x) W S( x) + κ x x B for all x V. The inmum of all such constants κ is the calmness modulus, denoted by calm S( x ȳ). Calmness is precisely pseudo outer T -dierentiability, where T : X Y is dened by T (w) := calm S( x ȳ) w B. (2) We say that S : X Y has the Aubin Property at x for ȳ if there exists neighborhoods V of x and W of ȳ, and κ 0 such that S(x) W S(x ) + κ x x B for all x, x V. The inmum of all such constants κ is the graphical modulus, denoted by lip S( x ȳ). The Aubin property is also known as the pseudo-lipschitz property and as the Lipschitz-like property. The Aubin property is precisely strict pseudo T -dierentiability, where T : X Y is dened by T (w) := lip S( x ȳ) w B. We present an example to show that Propositions 3.11 and 3.12 cannot be extended for set-valued maps in a straightforward manner. Example 4.8. (Failure of intersections in set-valued T -dierentiability) Let S : R 2 R 2 be dened by S(x, y) = {x} R. Let T 1, T 2 : R 2 R 2 be dened by T 1 (x, y) = (x, y) and T 2 (x, y) = (x, 0). It is clear that S is (pseudo) strictly T 1 -dierentiable and T 2 -dierentiable at all points, but it is not (pseudo) (T 1 T 2 )- dierentiable anywhere. Example 4.9. (Failure of set-valued T ( )-dierentiability) Let S : R R be dened by S(x) := (, x], and let T : R R be dened by { {0} if w 0 T (w) = [ w, w] if w 0.

11 GENERALIZED DIFFERENTIATION WITH POSITIVELY HOMOGENEOUS MAPS 11 Here, S is strictly T -dierentiable at x for all x R, and it is pseudo strictly T -dierentiable at x for x for all x R. However, S is neither outer (or inner) T ( )-dierentiable anywhere, nor pseudo outer (or inner) T ( )-dierentiable at x for x for any x R. We remark that while inner T -dierentiability is dened so that Proposition 4.4 holds, this denition of inner T -dierentiability does not satisfy the property that strict T -dierentiability implies inner T -dierentiability in general. It actually implies inner T ( )-dierentiability. The same holds for pseudo strict T - dierentiability and pseudo inner T -dierentiability, or more correctly, pseudo inner T ( )-dierentiability. An example where this occurs is the function S : R R as dened in Example 4.9. Fortunately, inner T -dierentiability does not play a huge role in this paper. Much of the current methods for set-valued dierentiation are motivated by looking at the tangent cones of the graph of the set-valued map. See the discussion in [4, Chapter 5] on the dierent forms of set-valued dierentiation obtained by taking dierent kinds of tangent cones of the graph. The notions of semidierentiability [29] and proto-dierentiability [34] are based on this idea. We shall only recall the denition of semidierentiability. See also the techniques in [4, Chapter 5]. We now point out the lack of relation between pseudo T -dierentiability and these methods by observing the nite dimensional case. We remark on the denition of a limit. Let S : D R m be a set-valued map, where D R n. If x cl(d)\d, then we say that lim x x S(x) is the value at x of a continuous extension of S onto D { x}. Such an extension is unique. For a more complete treatment on limits in nite dimensional set-valued maps, see [35]. Denition [29] For a set-valued map S : R n R m, let x dom(f ), ȳ S( x) and w R n \{0}. If the limit S( x + τw) ȳ lim τ 0,w w τ exists, then we say that it is the semiderivative at x for ȳ and w. If the semiderivative exists for every vector w R n \{0}, then F is semidierentiable at x for ȳ with derivative DS( x ȳ) : R n R m dened by DS( x ȳ)( w) being equal to the limit dened above. The denition of semidierentiability for set-valued maps is not to be confused with the semidierentiability dened for single-valued maps earlier. Clearly, the limit in the denition above is a cone, and the derivative DS( x ȳ) is a positively homogeneous map. We present an example where S : R R 2 is not pseudo DS( x ȳ)-dierentiable at x for ȳ. Example (Semidierentiability and T -dierentiability 1) Consider the setvalued map S : R R 2 dened by { {x} [0, ) if x 0 S(x) = {(α, α) 0 α x} ( {x} [ x, ) ) if x 0. This map is semidierentiable at 0 for 0, with semiderivative DS(0 0) : R R 2 dened by DS(0 0)(w) = {0} [0, ) for all w R. However, S is not DS(0 0)- dierentiable at 0 for 0, because at any point in (0, α) {0} (0, ), we can nd a neighborhood U α of (0, α) such that S(x) U α is [{x} (, )] U α.

12 12 C.H. JEFFREY PANG One may expect that if S is T -dierentiable at x for ȳ, then DS( x ȳ) T. The following example shows that this is not the case. Example (Semidierentiability and T -dierentiability 2) Consider S : R R dened by S(x) = R. Clearly S is pseudo T -dierentiable at x for any y R, where T : R R is dened by T (w) = 0 for all w R. But the semiderivative DS( x ȳ) : R R is equal to S, which gives DS( x ȳ) T for all x, ȳ R. In the nite dimensional case, semidierentiability implies proto-dierentiability [35, Exercise 8.43]. The examples showing the lack of relation between semidierentiability and pseudo T -dierentiability apply for proto-dierentiability as well. 5. Calculus of T -differentiability In this section, we prove a chain rule and a sum rule similar to the coderivative calculus in [35, Section 10H]. Let us rst introduce the outer norm of a positively homogeneous set-valued map. Denition 5.1. If T : X Y is a positively homogeneous map, then the outer norm of T is T + := sup sup x 1 y T (x) y = sup{ y y T (x), x 1}. Clearly T + < implies T (0) = {0}. When T (0) = {0}, T + = calm T (0 0) = calm T (0). Here is a chain rule for T -dierentiable functions. Theorem 5.2. (Chain rule) Let F : X Y, G : Y Z, and z G F ( x). Suppose the following conditions hold (1) F is pseudo outer T x y -dierentiable at x for y for all y in G 1 ( z) F ( x). (2) G is pseudo strictly T y z -dierentiable at y for z for all y in G 1 ( z) F ( x). (3) The set G 1 ( z) F ( x) Y is compact. (4) The map (x, z) G 1 (z) F (x) is outer semicontinuous at ( x, z). (5) α := sup y G 1 ( z) F ( x) T x y + is nite. (6) For all y G 1 ( z) F ( x), T y z (0) = {0}, and β is nite, where β := sup lip T y z (0). y G 1 ( z) F ( x) Then G F is pseudo outer T -dierentiable at x for z, where T : X Z is dened by (5.1) T := T y z T x y. y G 1 ( z) F ( x) The function G F is pseudo strictly T -dierentiable for T : X Y dened in (5.1) if in statement (1), F were pseudo strictly T x y -dierentiable at x for y instead. Proof. We shall prove only the result for F being pseudo outer T -dierentiable. The proof for pseudo strict T -dierentiability is almost exactly the same. Choose any δ > 0. Since (2) holds, for each y G 1 ( z) F ( x), we can nd some open convex neighborhoods V y of y and W of z such that G(y ) W G(y ) + T y z (y y ) + δ y y B for all y, y V y.

13 GENERALIZED DIFFERENTIATION WITH POSITIVELY HOMOGENEOUS MAPS 13 Next, since (1) holds, for each y G 1 ( z) F ( x), we can nd open convex neighborhoods U of x and V y V y of y such that F (x) V y [(F ( x) V y) + T x y (x x) + δ x x B] V y for all x U. Since G 1 ( z) F ( x) is compact by (3), there are nitely many y i G 1 ( z) F ( x) such that G 1 ( z) F ( x) i V y i. By taking nitely many intersections if necessary, the neighborhoods U and W can be assumed to be independent of y i. Let V := i V y i. Since the map (x, z) G 1 (z) F (x) is outer semicontinuous at ( x, z) by (4), we can reduce U and W if necessary so that G 1 (W ) F (U) V. This implies that for any x U, We have (5.2) G(F (x)) W = G(F (x) V ) W ( = G ( ) ) F (x) V yi W. G(F (x) V yi ) W G ( [(F ( x) V y ) i ) + T x yi (x x) + δ x x B] V yi W G ( F ( x) V y ) ( i + Tyi z T x yi (x x) + δ x x B ) +δ T x yi (x x) + δ x x B B i G F ( x) + T yi z T x yi (x x) + δβ x x B + δ(α + δ) x x B = G F ( x) + T yi z T x yi (x x) + δ(α + β + δ) x x B. We may assume that δ is small enough so that α + β + δ is bounded from above by some nite constant θ. Choosing y i over all i gives G ( ) F (x) V yi W i ( ) G F ( x) + T yi z T x yi (x x) + δθ x x B G F ( x) + i y G 1 ( z) F ( x) This completes the proof of the theorem. T y z T x y (x x) + δθ x x B. If G : Y Z were pseudo outer T -dierentiable, then we can still obtain a result for the case where F : X Y is single-valued. Proposition 5.3. (Chain rule) Let f : X Y and G : Y Z, ȳ = f( x), and z G(ȳ). Suppose the following conditions hold (1) f is T x ȳ -dierentiable at x for ȳ. (2) G is pseudo outer Tȳ z -dierentiable at ȳ for z. (3) T x ȳ + is nite. (4) Tȳ z (0) = {0}, and lip Tȳ z (0) is nite. Then G f is pseudo outer T -dierentiable at x for z, where T : X Z is dened by T = Tȳ z T x ȳ.

14 14 C.H. JEFFREY PANG Proof. Choose some δ > 0. By condition (1), we can nd a neighborhood U of x such that f(x) f( x) + T (x x) + δ x x B for all x U. Let V be such that f( x) + T (x x) + δ x x B V for all x U. By (2), we can shrink U and V if necessary so that there is a neighborhood W of z such that G(y) W G(ȳ) + Tȳ z (y ȳ) + δ y ȳ B for all y V. A calculation similar to (5.2) concludes the proof. From the chain rule, we can infer a sum rule. Corollary 5.4. (Sum rule) Let S i : X Y for i = 1,..., p, and ȳ p i=1 S i( x). Dene F : X Y p by F (x) = (S 1 (x),..., S p (x)), and dene g : Y p Y to be the linear map mapping to the sum of the p elements in Y p. Suppose the following conditions hold. (1) S i is pseudo outer Tx y ī i -dierentiable at x for y i whenever y i S i ( x) and y y p = ȳ. (2) The set {(y 1,..., y p ) y i S i ( x), y y p = ȳ} Y p is compact. (3) The map (x, y) g 1 (y) F (x) is outer semicontinuous at ( x, ȳ). (4) α i := sup yi Π i(g 1 ( z) F ( x)) T ī x yi + is nite for each i = 1,..., p. Here, Π i : Y p Y is the projection onto the ith coordinate. Then g F : X Y, which is the sum of the maps S i, is T -dierentiable at x for ȳ, where T : X Y is dened by ( ) (5.3) T := Tx y ī i. y y p = ȳ y i S i ( x) The function g F is pseudo strictly T -dierentiable for T : X Y dened in (5.3) if in statement (1), S i were pseudo strictly T ī x y i -dierentiable at x for y i instead. Proof. For y i S i ( x) for all i, dene T (y1,...,y p) : X Y p by T (y1,...,y p)(w) := ( T 1 x y 1 (w),..., T p x y p (w) ). Condition (1) implies that the map F is pseudo outer T (y1,...,y p)-dierentiable at x for (y 1,..., y p ). We now proceed to apply the chain rule in Theorem 5.2. Since g is a linear function, the conditions for g needed for the chain rule are satised. The rest of the conditions in this result are just the appropriate conditions in the chain rule rephrased. The case of pseudo strict T -dierentiability is similar. Note that we have focused on pseudo (outer/ strict) T -dierentiability so far in this section. The relation between pseudo (strict/ outer/ inner) T -dierentiability and (strict/ outer/ inner) T -dierentiability is illustrated by the following theorem. We say that S : X Y is locally compact around x dom(f ) if there is a neighborhood O of x and a compact set C Y such that S(O) C. Theorem 5.5. (T -dierentiability from pseudo T -dierentiability) Let S : D Y be a closed-valued outer semicontinuous map on a closed domain D X. Suppose S is locally compact around x D. Then S is outer T -dierentiable at x if and only if S is pseudo outer T -dierentiable at x for all ȳ S( x). i

15 GENERALIZED DIFFERENTIATION WITH POSITIVELY HOMOGENEOUS MAPS 15 An analogous statement holds for (strict/ inner) T -dierentiability and pseudo (strict/ inner) T -dierentiability. Remark 5.6. In fact, the hypothesis of outer semicontinuity in Theorem 5.5 can be weakened: We can assume that S is closed at x: For every y / S( x), there are neighborhoods U of x and V of y such that S(x) V = for all x U. If S is outer semicontinuous, then gph(s) is closed by [4, Proposition 1.4.8], which implies that S is closed at x. The proof of Theorem 5.5 can be easily adapted from the proof of [27, Theorem 1.42], which traces its roots in the nite dimensional case to [33]. Other calculus rules that are important are the Cartesian product of set-valued maps, and rules for unions. These two operations are simple to formulate and prove. For intersections of set-valued maps, we feel it is more eective to look at the normal cones of the intersections of the graph of the appropriate functions and apply the Mordukhovich criterion. See Section 6. We close this section by referring the reader to [33] for more applications of set-valued chain rules. 6. The Mordukhovich criterion As we have seen in Section 4, the Aubin property gives a sharper analysis for the Lipschitz continuity of set-valued maps. An eective tool for calculating the graphical modulus (for the Aubin property) is the coderivative (Dention 6.2), which is a generalization of the adjoint linear operator of linear functions. Coderivatives enjoy an eective calculus, and can be easily calculated for set-valued maps whose graphs are dened by smooth maps. The relationship between the Aubin property and the coderivatives is referred to as the Mordukhovich criterion in [35]. For a history of the Mordukhovich criterion, see the bibliography in [35], which in turn cited [10, 18, 22, 25, 37], and also Commentaries , and of [27]. The aim of this section is to show that coderivatives in fact give directional behavior that is captured in the language of pseudo strict T -dierentiability. We now recall the classical denition of the normal cones and coderivatives in nite dimensions. Denition 6.1. Let C R n and x C. A vector v is normal to C at x in the regular sense, or a regular normal, written v ˆN C ( x), if v, x x o( x x ) for x C. It is normal to C at x in the general sense, or simply a normal vector, written v N C ( x), if there are sequences x i C x and v i v with v i ˆN C (x i ). Denition 6.2. Consider a mapping S : R n R m, a point x dom(s) and ȳ S( x). The coderivative at x for ȳ is the mapping D S( x ȳ) : R m R n dened by v D S( x ȳ)(z) (v, z) N gph(s) ( x, ȳ). In the case where S is a smooth map, the coderivative D S( x S( x)) is the adjoint of the derivative mapping there. Next, we recall the denition of the regular subdierential and general subdifferential, which are important in the proof in the main result of this section. Denition 6.3. Consider a function f : R n R and a point x R n. For a vector v R n,

16 16 C.H. JEFFREY PANG (a) v is a regular subgradient of f at x, written v ˆ f( x), if f(x) f( x) + v, x x + o( x x ); (b) v is a (general) subgradient of f at x, written v f( x), if there are sequences x i x and v i ˆ f(x i ) with v i v and f(x i ) f( x). (c) The sets ˆ f( x) and f( x) are the regular subdierential (or Fréchet subgradient) and the (general) subdierential (or limiting subdierential, or Mordukhovich subdierential) at x respectively. We now present our main result of this section. Theorem 6.4. (Mordukhovich criterion revisited) Consider S : R n R m, x dom(s) and ȳ S( x). Suppose gph(s) is locally closed at ( x, ȳ). If D S( x ȳ)(0) = {0}, or equivalently, D S( x ȳ) + <, then S is pseudo strictly T - dierentiable at x for ȳ, where T : R n R m is dened by where κ : R n R + is dened by T (w) = κ(w)b m {0}, κ(w) := max { v, w v D S( x ȳ)(b m )} = max { v, w (v, z) N gph(s) ( x, ȳ) for some z 1 }. The assumptions of Theorem 6.4 are the same as that of the Mordukhovich Criterion as stated in [35, Theorem 9.40], but the conclusion is improved. The Mordukhovich criterion establishes the equivalence between the Aubin property (with graphical modulus κ := max w 1 κ(w)) and the outer norm of the coderivative. Theorem 6.4 shows that the worst case Lipschitzian behavior need not occur in all directions. Proof. (of Theorem 6.4) We highlight only the parts we need to add to the proof of suciency as given in [35, Theorem 9.40], though with some changes to the variable names. Let κ = D S( x ȳ) +. By the denition of κ and the coderivatives, for any θ > 0, there exists δ 0 > 0 and ɛ 0 > 0 for which N gph(s) (ˆx, ŷ) (R n B m ) ( κ + θ)b n B m for all ˆx B( x, δ 0 ), ŷ B(ȳ, ɛ 0 ). (The above statement is just [35, 9(22)] rephrased.) The mapping to normal cones is outer semicontinuous, so (6.1) N gph(s) (ˆx, ŷ) [( κ + θ)b n B m ] N gph(s) ( x, ȳ) + θb n+m for all ˆx B( x, δ 0 ), ŷ B(ȳ, ɛ 0 ). Dene the map dỹ(x) : R n R by dỹ(x) := d(ỹ, S(x)). In their proof, the condition in [35, 9(23)] can be written more precisely as: if v ˆ dỹ(ˆx) with ˆx x δ 0 (6.2) and dỹ (ˆx) + ỹ ȳ ɛ 0, then there exists z B with (v, z) N gph(s) (ˆx, ŷ), where ŷ is any point of S(ˆx) nearest to ỹ, that is ŷ ỹ = dỹ(ˆx). Moving on, they proved that the map (x, y) d(y, S(x)) is Lipschitz continuous near ( x, ȳ), or more precisely, there exists some λ > 0 and µ > 0 such that (6.3) d(y, S(x)) λ( x x + y ȳ ) when x x µ and y ȳ µ.

17 GENERALIZED DIFFERENTIATION WITH POSITIVELY HOMOGENEOUS MAPS 17 Furnished with this, we can choose δ > 0 and ɛ > 0 small enough that 2δ min {µ, δ 0 }, ɛ min {µ, ɛ 0 /2} and λ (2δ + ɛ) ɛ 0 /2. If v ˆ dỹ(ˆx), ˆx x δ and ỹ ȳ ɛ, then by (6.3), we have dỹ(ˆx) λ( ˆx x + ỹ ȳ ) λ(2δ + ɛ) ɛ 0 2. Formula (6.2) then tells us that for any v ˆ dỹ(ˆx), there exists some z B such that (v, z) N gph(s) (ˆx, ŷ). By (6.1), there is some (ṽ, z) N gph(s) ( x, ȳ) such that (ṽ, z) (v, z) < θ. This gives ṽ D S( x ȳ)((1 + θ)b m 1 ), or 1+θ ṽ D S( x ȳ)(b m ). Further arithmetic gives v 1 1+θ ṽ v ṽ + θ κ 1+θ ṽ θ(1+ 1+θ ), so v D S( x ȳ)(b m )+ θ(1 + κ 1+θ )Bn. Let θ be θ(1 + κ 1+θ ). We thus have d ỹ(ˆx) D S( x ȳ)(b) + θb. The function dỹ( ) is Lipschitz, so the Clarke subdierential equals conv( dỹ( )) in B(ȳ, ɛ). Choose any two points x, x B( x, δ). By the nonsmooth mean value theorem (Theorem 3.5), we have dỹ(x) dỹ(x ) = v, x x for some v C dỹ(x τ ) = conv( dỹ(x τ )), where x τ = τx + (1 τ)x for some τ (0, 1). Now, v, x x max { ṽ, x x ṽ C dỹ(x τ )} max { ṽ, x x ṽ D S( x ū)(b) + θb } = κ(x x) + θ x x. This can be rephrased as dỹ(x ) dỹ(x) κ(x x) θ x x. As ỹ varies over all points in B(ȳ, ɛ), this readily gives S(x ) B(ȳ, ɛ) S(x) + T (x x) + θ x x B m, which is what we seek to prove. Rockafellar [33] established the relationship between the Aubin property of S at x for ȳ and the Lipschitz continuity properties of dỹ( ). (See [35, Exercise 9.37].) The key to the proof of Theorem 6.4 is that the nonsmooth mean value theorem on dỹ( ) gives us more information on the continuity properties of S. We close this section with a remark on Theorem 6.4. Remark 6.5. (More precise T -dierentiability) Suppose S : R n R m, and ȳ S( x). To obtain a better positively homogeneous map T : R n R m than the one stated in Theorem 6.4, one applies Theorem 6.4 on the map g S + f : R n R m, where f : R n R m and g : R m R m are linear maps and g is invertible. This approach is equivalent to looking at the normal cones of gph(g S + f), which can also be obtained by performing the linear map (x, y) (x, g(y) + f(x)) on gph(s) R n R m. By appealing to the calculus rules in Section 5, this gives another T : R n R m for which S is pseudo strictly T -dierentiable. If we choose nitely many {(f i, g i )} i, then we can dene T : R n R m by T (w) = T i (w), where i is determined uniquely by w. It is easy to see that S is T -dierentiable at x for ȳ. 7. Metric regularity and open covering For S : X Y, it is well known that the Aubin property of S 1 is related to the metric regularity and open covering properties of S. In this section, we study metric regularity and open covering in a more axiomatic manner with the help of

18 18 C.H. JEFFREY PANG T -dierentiability, proving new relations in these subjects. We caution that for much of this section, we need T : Y X instead of T : X Y. We begin with our denitions of generalized metric regularity and open covering. Denition 7.1. Let X and Y be Banach spaces and let S : X Y be a setvalued map where gph(s) is locally closed at ( x, ȳ), and T : Y X be positively homogeneous. (a) S is T -metrically regular at ( x, ȳ) gph(s) if for any δ > 0, there exist neighborhoods V of x and W of ȳ and r > 0 such that, for any x V and A rb, (or equivalently, for any set A rb containing exactly one element), (7.1) y (S(x) + A) W implies x S 1 (y) + (T + δ)(a). (b) S is (C, T )-metrically regular at ( x, ȳ) gph(s) if for any δ > 0, there exist neighborhoods V of x and W of ȳ and r > 0 such that, for any x V and t > 0, (7.1) holds for any set A rb of the form A = tc, where C Y is closed. Denition 7.2. Let X and Y be Banach spaces and let S : X Y be a setvalued map where gph(s) is locally closed at ( x, ȳ), and T : Y X be positively homogeneous. (a) S is a T -open covering at ( x, ȳ) gph(s) if for any δ > 0, there exist neighborhoods V of x and W of ȳ and r > 0 such that, for any x V and any set A rb (or equivalently, for any set A rb containing exactly one element), (7.2) (S(x) + A) W S ( x + (T + δ)(a) ). (b) S is a (C, T )-open covering at ( x, ȳ) gph(s) if for any δ > 0, there exist neighborhoods V of x and W of ȳ such that, for any x V and t > 0, (7.2) holds for any set A rb of the form A = tc, where C Y is closed. Open covering is sometimes known as linear openness. Setting C = B and T (w) := κ w B in both cases reduce the above denitions to the classical denitions of metric regularity and open covering with modulus κ, which will be clear by Proposition 7.8. Metric regularity with modulus κ is often written compactly as: For all κ > κ, there exists neighborhoods V of x and W of ȳ such that x V, y W implies d ( x, S 1 (y) ) κ d ( y, S(x) ). In practical problems, one might choose C to be a set dierent from the unit ball B, as this choice allows one to identify sensitive and less sensitive directions, and to use dierent norms. We have our following theorem generalizing the classical equivalence of pseudo strict T -dierentiability, T -metric regularity and T -open covering. Theorem 7.3. (T -metric regularity and T -openness) Let X and Y be Banach spaces and T : Y X be a positively homogeneous map. For S : X Y, the following are equivalent: (MR) S is T ( )-metric regular at ( x, ȳ) gph(s). (OC) S is a ( T ( ))-open covering at ( x, ȳ) gph(s). (IT) S 1 is pseudo strictly T -dierentiable at ȳ for x.

19 GENERALIZED DIFFERENTIATION WITH POSITIVELY HOMOGENEOUS MAPS 19 Proof. We rst note that x S 1 (y ) V can be rewritten as x V, y S(x), so (IT) is equivalent to: For any δ > 0, we can nd neighborhoods V of x and W of ȳ such that: (7.3) x } {{ V }, y S(x), y W, y W implies x S 1 (y) + (T + δ)(y y). }{{}}{{}}{{}}{{} (1) (2) (3) (4) (5) Take the set A to be A = {y y }. Rewriting (2) and (3) as y [S(x) + A] W, we get implies x V, A y W and y [S(x) + A] W x S 1 (y) + (T + δ)(a). We can assume W = B(ȳ, r ). If y B(ȳ, r r 2 ), then B(0, 2 ) y B(ȳ, r ). So choosing r = r 2 and writing (2) and (3) as y [S(x) + A] B(ȳ, r), we have T ( )-metric regularity in (MR), i.e., (7.4) implies x V, A B(0, r) and y [S(x) + A] B(ȳ, r) x S 1 (y) + (T + δ)(a). For the converse, setting A = {y y } in A B(0, r) gives y B(y, r). The formula y [S(x) + A] B(ȳ, r) can be written as y S(x), y B(ȳ, r). If y B(ȳ, r 2 ), then B(ȳ, r 2 ) B(y, r). This implies statement (7.3) with W = B(ȳ, r 2 ), which gives (IT) as needed. For the equivalence of (OC) and (IT), note that condition (5) is equivalent to y S(x (T +δ)(y y)). The steps in the proof follow exactly with this change. At this point, we prove the equivalence of (C, T )-metric regularity and T -metric regularity for some well chosen T : Y X. The properties of (C, T )-metric regularity (and hence (C, T )-open coverings) can therefore be obtained from the corresponding properties of T -metric regularity. Proposition 7.4. (Reduction of (C, T )-metric regularity to T -metric regularity) Let X and Y be Banach spaces, S : X Y be a set-valued map, and T : Y X be positively homogeneous. Suppose ɛb C RB for some ɛ, R > 0. Then S is (C, T )-metrically regular at ( x, ȳ) if and only if S is T -metrically regular there, where T : Y X is dened by T (w) = T (tc) for t = min{λ w λc}. Proof. We recall from the proof of Theorem 7.3, T -metric regularity is equivalent to: For all δ > 0, there is a neighborhood V of x and r > 0 such that (7.5) x V, y S(x), y B(ȳ, r), y y rb implies x S 1 (y)+(t +δ)(y y ). Next, we note that (C, T )-metric regularity is equivalent to: For any δ > 0, there is a neighborhood V of x and r > 0 such that x V, y (S(x) + tc) B(ȳ, r), tc rb implies x S 1 (y) + (T + δ)(tc), which is in turn equivalent to: For any δ > 0, there is a neighborhood V of x and r > 0 such that (7.6) implies x V, y S(x), y B(ȳ, r), y y tc, tc rb x S 1 (y) + (T + δ)(tc). Suppose S is T -metrically regular at ( x, ȳ). Let δ > 0, V and r be such that (7.5) holds, and suppose x V, y S(x) and y B(ȳ, r). Let t > 0 be such

20 20 C.H. JEFFREY PANG that y y tc. Then we have x S 1 (y) + (T + δ)(y y ), which implies x S 1 (y) + (T + δ)(tc). This in turn means that (7.6) holds, so S is (C, T )- metrically regular at ( x, ȳ). For the converse, note that the assumptions implicitly assume that C is bounded. We assume ɛb C RB and examine the condition (7.6). If y y rɛ R B, then the minimum t > 0 such that y y tc gives tc rb, so the condition tc rb is superuous in (7.6). This gives us the converse. We now look at the connection between pseudo T -outer dierentiability and metric subregularity. For more on metric subregularity, we refer the reader to [11]. We dene T -metric subregularity as follows. Denition 7.5. Let X and Y be Banach spaces, S : X Y be a set-valued map, and T : Y X be positively homogeneous. (1) S is T -metrically subregular at ( x, ȳ) gph(s) if for any δ > 0, there exists a neighborhood V of x and r > 0 such that, for any A rb, (or equivalently, for any set A rb containing exactly one element.) (7.7) x V, ȳ S(x) + A implies x S 1 (ȳ) + (T + δ)(a). (2) S is (C, T )-metrically subregular at ( x, ȳ) gph(s) provided that there exist a neighborhood V of x such that, for t > 0, (7.7) holds for A = tc rb. (or equivalently, for any set A rb containing exactly one element.) Again, the special case of C = B and T (w) := κ w B, T -metric subregularity and (C, T )-metric subregularity are equivalent to metric subregularity with modulus κ (which will be clear by Proposition 7.8), and can be written compactly as: For any κ > κ, there exists a neighborhood V of x such that x V implies d ( x, S 1 (ȳ) ) κ d ( ȳ, S(x) ). Here is a result on the equivalences between T -metric subregularity and pseudo outer T -dierentiability. Theorem 7.6. (Generalized metric subregularity) Let X and Y be Banach spaces, S : X Y be a closed set-valued map, and T : Y X be a positively homogeneous set-valued map. The following are equivalent: (MR ) S is T ( )-metric subregular at ( x, ȳ) gph(s). (IT ) S 1 is pseudo outer T -dierentiable at ȳ for x. Proof. Condition (IT ) can be written as: For any δ > 0, there exists a neighborhood V of x and r > 0 such that (7.8) x V, x S 1 (y ), y ȳ rb = x S 1 (ȳ) + (T + δ)(y ȳ). It is also clear that condition (MR ) can be written in this form. Here is a lemma on pseudo strict T -dierentiability amended from [35, Lemma 9.39], which is in turn attributed to [14]. Lemma 7.7. (Extended formulation of pseudo T -dierentiability) Consider a mapping S : X Y, a pair ( x, ȳ) gph(s) where gph(s) is locally closed, and a set D X containing x, and a positively homogeneous map T : X Y. Suppose that there is a δ > 0 such that (7.9) T (w) δ w B for all w X.