An iterative Method for solving H-differentiable

An iterative Method for solving H-differentiable inclusions

¨l Gaydu Michae

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Abstract. This paper deals with the solving of variational inclusions T (x) 3 0 that possess a certain generalized differentiability property which has been recently developed by Pang in [13]. Thus, using both a recent extension of the proximal point method and a linearization of set-valued mapping T , we propose a new iterative process which encompasses the method of Arag´on and al in [1]. Afterwards, we show the local convergence of suggested method and we investigate its variational perturbations. Key words: Variational inclusions, proximal point method, H-differentiability, positively homogeneous mappings, metric regularity, variational perturbations, set-convergence. AMS 2010 Subject Classification: 49J53, 49J40.

1

Introduction

It is well known that variational inclusions of the form T (x) 3 0

(1)

modelize a wide variety of mathematical problems such as complementarity problems, systems of nonlinear equations, variational inequalities. It can characterize some optimality or equilibrium conditions and thus have many practical applications. We find them for example within the economic theory framework (to deal with Nash or Walras equilibrium) but also in engineering (with the transport problems for instance). For more informations on the vast field of modelized phenomena by these inclusions, reader may consult the monograph of Facchinei and Pang [9], the article of Ferris and Pang [10]. In this paper, we present a new iterative process in order to approximate the solutions of variational inclusion (1), where T : X → → Y is a set-valued mapping (acting from a Banach space X to the subsets of another Banach space Y ) which enjoys an certain “Hdifferentiability” property. The starting point of our study is based on a recent extension of the proximal point method which has been proposed and studied by Arag´on and al 1

LAMIA, Dept. of Mathematics, Universit´e des Antilles et de la Guyane, Pointe-`a-Pitre, Guadeloupe (France), [email protected]. The author is supported by Contract EA4540.

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in [1, 2, 3]. Actually in [1], the authors study the convergence of a general version of → Y has certain the proximal point algorithm for solving the problem (1) when T : X → metrically regular properties. For this purpose, they choose a sequence of Lipschitz continuous function gn : X → Y (such that gn (0) = 0) and show that if the Lipschitz constants λn are upper bounded by one over twice the regularity modulus of T around the reference solution x¯ then we can find a neighborhood U of x¯ such that for any initial point x0 ∈ U , there exists a sequence xn satisfying 0 ∈ gn (xn+1 − xn ) + T (xn+1 )

for n = 0, 1, 2, . . .

(2)

which linearly converges to x¯. Furthermore, the convergence is superlinear when λn goes to 0. Inspired by [11], the originality of our approach resides in the linearization of multivalued term T (xn+1 ) in (2) via a recent concept of generalized differentiability for setvalued mappings which has been developed and studied by Pang in [6, 13, 14]. Recall that, a set-valued mapping F : X → → Y is said to be strictly H-differentiable at x¯ (according to Pang [13]) if for any δ > 0, there exists a neighborhood U of x¯ such that F (x) ⊂ F (x0 ) + H(x − x0 ) + δkx − x0 kIB

for all x, x0 ∈ U,

(3)

where H is the derivative of F (not necessarily unique) which is a positively homogeneous set-valued mapping and IB is the closed unit ball of Y . Thus, when xn+1 and xn are close enough to x¯, previous relation yields F (xn+1 ) ⊂ F (xn ) + H(xn+1 − xn ) + δkxn+1 − xn kIB.

(4)

In this work, we assume that problem (1) has a solution x¯. Then when T enjoys the strict H-differentiability property at x¯, coupling both (2) and (4), we obtain for any fixed δ small enough the following process: 0 ∈ gn (xn+1 − xn ) + T (xn ) + H(xn+1 − xn ) + δkxn+1 − xn kIB

for n = 0, 1, 2, . . . , (5)

which relaxes method (2) and can be applied without metric regularity assumption on T. In the following Section, we present some background material. Section 3 is about the statement and the proof of the local convergence of our method while in Section 4 we investigate the variational perturbations of this one.

2

Background material

Throughout, X and Y are Banach spaces, F : X → → Y is a set-valued mapping from X into the subsets of Y and gph F = {(x, y) ∈ X × Y | y ∈ F (x)} is the graph of F . The 2

distance from a point x to a set C is denoted by d(x, C) = inf y∈C kx−yk. The closed unit ball (of Y or X according to the context) is designated by IB. We denote IB r (a) the closed ball of radius r centered at a while IB r represents the closed ball of radius r centered at 0. The inverse of F , denoted by F −1 : Y → → X is defined as x ∈ F −1 (y) ⇔ y ∈ F (x). The excess from a set A to a set B is defined by e(A, B) = supx∈A d(x, B). We say that a set C ⊂ X is locally closed at z ∈ C if there exists a constant ρ > 0 such that the set C ∩ IB ρ (z) is closed. The concept of generalized differentiation we are dealing with strongly relies on positively homogeneous set-valued mappings the definition of which can be found below. → Y be a set-valued mapping. It is called positively homogeDefinition 2.1. Let H : X → neous if H(0) 3 0 and H(λx) = λH(x) for all x ∈ X and λ > 0. One can immediately note that a mapping is positively homogeneous if and only if its graph is a cone and that the inverse of a positively homogeneous mapping is another positively homogeneous mapping. Graphical derivatives of set-valued mappings, introduced by Aubin [4], are positively homogeneous set-valued mappings and so are sublinear mappings (i.e., set-valued mappings such that their graph is a convex cone). To be able to work efficaciously with positively homogeneous mappings we need the following tool known as the outer norm. Definition 2.2. Let H : X → → Y be a positively homogeneous mapping. The outer norm of H is |H|+ = sup sup kyk, (6) kxk≤1 y∈H(x)

with the convention that sup kyk = −∞. y∈∅

Note that an equivalent (and useful) formulation of (6) is given by |H|+ = inf{κ > 0 | H(IB) ⊂ κIB}.

(7)

From (7), it follows that if |H|+ < ∞ then for all x ∈ X, H(x) ⊂ |H|+ kxkIB, and consequently for any z ∈ H(x), kzk ≤ |H|+ kxk. Now, let us present a regularity concept for set-valued mapping which has been useful in this work. Definition 2.3. A mapping F : X → → Y is said to be metrically regular at x¯ for y¯ if F (¯ x) 3 y¯ and there exist some positive constants κ, a and b such that d(x, F −1 (y)) ≤ κd(y, F (x))

for all x ∈ IB a (¯ x), y ∈ IB b (¯ y ). 3

(8)

The infimum of κ for which (8) holds is the regularity modulus denoted regF (¯ x|¯ y ); for a mapping F : X → Y , the absence of metric regularity at x ¯ for y ¯ is characterized → by regF (¯ x|¯ y ) = ∞. The metric regularity of a mapping F at x¯ for y¯ is known to be equivalent to the Aubin continuity (also called Lipschitz-Like property) of the inverse F −1 at y¯ for x¯ (see, e.g., [12, 15]). Finally, we state the following set-valued generalization of the Banach fixed point established by Dontchev and Hager in [7]. It will play an important role in the proof of our convergence theorem. Theorem 2.4. (Banach Fixed Point Theorem). Let (X, ρ) be a complete metric space, and consider a set-valued mapping Φ : X → → X, a point x¯ ∈ X, and nonnegative scalars α and θ be such that 0 ≤ θ < 1, the sets Φ(x) ∩ IB α (¯ x) are closed for all x ∈ IB α (¯ x) and the following conditions hold: (i) d(¯ x, Φ(¯ x)) < α(1 − θ); T (ii) e(Φ(u) IB α (¯ x), Φ(v)) ≤ θρ(u, v) for all u, v ∈ IB α (¯ x). Then Φ has a fixed point in IB α (¯ x). That is, there exists x ∈ IB α (¯ x) such that x ∈ Φ(x).

3

Local convergence of the method

Let us begin this section by stating and proving a useful technical lemma in establishing our main result. Lemma 3.1. Consider T : X → → Y a set-valued map that is strictly H-differentiable at x¯ for some positively homogeneous map H : X → → Y . If x¯ is a solution of problem (1) then for all δ > 0 there exists a positive constant a such that for all x ∈ IB a (¯ x) there are some elements z ∈ T (x) and u ∈ IB such that −z − δ k x¯ − x k u ∈ H(¯ x − x). Proof. The strict H-differentiability of T at x¯ ensure that for any δ > 0 there exists a positive constant a such that T (x) ⊂ T (x0 ) + H(x − x0 ) + δkx − x¯kIB

∀x, x0 ∈ IB a (¯ x).

(9)

Take an element x ∈ IB a (¯ x). Using both the fact that x¯ is a solution of (1) together with relation (9), we obtain 0 ∈ T (¯ x) ⊂ T (x) + H(¯ x − x) + δk¯ x − xkIB. Therefore, there exist some elements z ∈ T (x), u ∈ IB such that −z − δk¯ x − xku ∈ H(¯ x − x).

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(10)

The following theorem establishes the local convergence of the iterative method we consider in this paper. Theorem 3.2. Consider T : X → → Y a set-valued mapping, x¯ a solution of (1) and → H : X → Y a positively homogenous mapping. Assume the following conditions are held: (i) T is strictly H-differentiable at x¯, (ii) H is outer bonded i.e., |H|+ < ∞, (iii) H is metrically regular at 0 for 0 and its graph is locally closed around (0, 0). Choose a sequence of single-valued functions gn : X → Y with gn (0) = 0 which are Lipschitz continuous in a same neighborhood of 0 for all n ∈ N, with Lipschitz constants λn satisfying sup λn < 1/(2 reg(H; 0|0)). Then, there exists a positive constant η such n∈N

x), there is a sequence xn satisfying (5) which linearly that for each initial point x0 ∈ IB η2 (¯ converges to x¯. Proof. Set λ := supn∈N λn . The assertion (iii) and the choice of gn ensure the existence of positive constants κ, α, β, δ, b and µ such that d(x, H −1 (y)) ≤ κd(y, H(x)) for all (x, y) ∈ IB α × IB β ,

h

2κ(λ + δ) < 1, i h gph H ∩ IB µ × IB µ is closed set; i

(11) (12) (13)

and for all x, x0 ∈ IB b kgn (x) − gn (x0 )k ≤ λn kx − x0 k for n = 0, 1, 2, . . .

(14)

The assumption (i) for its part, implies the existence of a constant a > 0 such that T (x) ⊂ T (x0 ) + H(x − x0 ) + δkx − x0 kIB for all x, x0 ∈ IB a (¯ x).

(15)

Now we pose n η := min a, b, α,

o µ β , µ, λ + δ + |H|+ λ + δ + |H|+

(16)

Pick x0 ∈ IB η/2 (¯ x) ⊂ IB a (¯ x). If x0 = x¯ then T (x0 ) 3 0 and there is nothing to do (since we have found a solution of (1), our algorithm stops). Otherwise (x0 6= x¯) thanks to Lemma 3.1 (and its proof) there exist some elements z0 ∈ T (x0 ), u0 ∈ IB satisfying −z0 − δk¯ x − x0 ku0 ∈ H(¯ x − x0 ).

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(17)

κ(λ + δ) and consider the positive scalar α0 := γkx0 − x¯k together 1 − κ(λ + δ) with the following set-valued mapping Φ0 : X → → X defined by   Φ0 (x) = H −1 − g0 (x − x0 ) − z0 − δkx − x0 ku0 + x0 .

Then, pose γ :=

Before showing that the set-valued mapping Φ0 satisfies the assumptions of Theorem 2.4, let us begin by proving that −g0 (x − x0 ) − z0 − δkx − x0 ku0 ∈ IB β Indeed, for all x ∈ IB α0 (¯ x) we have



− g0 (x − x0 ) − z0 − δkx − x0 ku0 =

for all x ∈ IB α0 (¯ x).

(18)



g0 (0) − g0 (x − x0 ) − z0 − δkx − x0 ku0



≤ − z0 − δk¯ x − x0 ku0 + δk¯ x − x0 ku0 − δkx − x0 ku0 + kg0 (0) − g0 (x − x0 )k

 



≤ − z0 − δk¯ x − x0 ku0 + δ k¯ x − x0 k − kx − x0 k u0 + kg0 (0) − g0 (x − x0 )k



≤ − z0 − δk¯ x − x0 ku0 + δk¯ x − xk + kg0 (0) − g0 (x − x0 )k (19) In addition, by relation (12) we have 0 < γ < 1 and consequently α0 < η/2. Therefore x − x0 ∈ IB η ⊂ IB b . Then, using (19), the assumption (ii) together with relations (14) and (17) we obtain



x − x0 k + δk¯ x − xk + λ0 kx − x0 k

− g0 (x − x0 ) − z0 − δkx − x0 ku0 ≤ |H|+ k¯ ≤ (λ + δ + |H|+ )η ≤ β, and (18) is well checked. Now, we will show that the first assumption in the Theorem 2.4 is held. Thanks to

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relations (11), (14), (16), (17) and (18) it is clear that d(¯ x, Φ0 (¯ x)) = d(¯ x − x0 , H −1 (−g0 (¯ x − x0 ) − δk¯ x − x0 ku0 − z0 )) ≤ κd(−g0 (¯ x − x0 ) − z0 − δk¯ x − x0 ku0 , H(¯ x − x0 ))




≤ κ g0 (0) − g0 (¯ x − x0 ) − z0 − δk¯ x − x0 ku0 − − z0 − δk¯ x − x0 ku0 ≤ κλ0 k¯ x − x0 k < κ(λ + δ)k¯ x − x0 k = α0 (1 − κ(λ + δ)). (20) Let us prove that the second assumption in Theorem 2.4 is also satisfied. Take arbitrary u, v ∈ IB α0 (¯ x) ⊂ IB η/2 (¯ x). It is not difficult to see that both u − x0 and v − x0 belong to the ball IB η ⊂ IB b , and using again (11), (14), (16), (17) and (18) we have e(Φ0 (u) ∩ IB α0 (¯ x), Φ0 (v)) =

sup

d(ξ, Φ0 (v))

ξ∈Φ0 (u)∩IB α0 (¯ x)

sup d(ξ − x0 , H −1 (−g0 (v − x0 ) − δkv − x0 ku0 − z0 )) ξ∈Φ0 (u)∩IB η/2 (¯ x) n ≤ sup κd(−g0 (v − x0 ) − δkv − x0 ku0 − z0 , H(ξ − x0 )) | o ξ − x0 ∈ H −1 (−g0 (u − x0 ) − δku − x0 k − z0 ) ∩ IB α

 

≤ κ g0 (u − x0 ) − g0 (v − x0 ) + δ ku − x0 k − kv − x0 k u0 ≤ κkg0 (u − x0 ) − g0 (v − x0 )k + κδ ku − x0 k − kv − x0 k · ku0 k ≤

≤ κλ0 ku − vk + κδku − vk ≤ κ(λ + δ)ku − vk. (21) Remains to verify that for all x ∈ IB α0 (¯ x), Φ0 (x) ∩ IB α0 (¯ x) are closed sets.

(22)

For this purpose, consider a sequence (zn0)n∈N ∈ Φ0 (x) ∩ IB α0 (¯ x) which converges to an  0 0 −1 element z ∈ X. Thereby, zn − x0 ∈ H − g0 (x − x0 ) − z0 − δkx − x0 ku0 , and thanks to (13), (16), (18), it follows that z 0 ∈ Φ0 (x). In addition, the ball IB α0 (¯ x) is closed therefore the element z 0 lies in this one and consequently Φ0 (x) ∩ IB α0 (¯ x) is well a closed set.

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Then, Theorem 2.4 provides us with the existence of a fixed point x1 ∈ IB α0 (¯ x) for mapping Φ0 , i.e., the existence of an element x1 ∈ IB α0 (¯ x) such that 0 ∈ g0 (x1 − x0 ) + z0 + H(x1 − x0 ) + δkx1 − x0 ku0 . Hence a fortiori x1 satisfies both kx1 − x¯k ≤ γkx0 − x¯k and 0 ∈ g0 (x1 − x0 ) + T (x0 ) + H(x1 − x0 ) + δkx1 − x0 kIB. Now, take such a x1 which belongs to IB η/2 (¯ x) and repeat the proof by replacing x0 by x1 , then the same arguments as in (17), (18), (20), (21) and (22) yield the existence of an element x2 ∈ IB η/2 (¯ x) satisfying both kx2 − x¯k ≤ γkx1 − x¯k and 0 ∈ g1 (x2 − x1 ) + T (x1 ) + H(x2 − x1 ) + δkx2 − x1 kIB. Thus, the induction process is clear, by posing   −1 Φn (x) = H − gn (x − xn ) − zn − δkx − xn kun + xn , αn = γkxn − x¯k for n = 0, 1, 2, . . . (with xn 6= x¯) and having nremarked that αn < αn−1o< . . . < α1 < α0 < η/2, without difficulty, we build a sequence x0 , x1 , x2 , x3 , . . . , xn , . . . statistfying (5), which linearly converges to x¯. Finally, It is worth pointing out that even in simple cases, the H-differentiability for a set-valued mapping F : X → → Y at x¯ does not imply the metric regularity property at x¯ for y¯, as shown in the following remark. Remark 3.3. Consider the set-valued mapping    [−x − 1; +∞[   F (x) = [0; +∞[     [x − 1; +∞[

F :R→ → R defined below by if x ≤ −1, if − 1 < x < 1, if x ≥ 1.

It is easy to see that the set-valued mapping F is not metrically regular at 0 for 0 since for any δ ∈]0, 1[ and x ∈] − δ, δ[, the quantity d(x, F −1 (y)) is infinite as soon as y ∈] − δ, 0[ 8

while d(y, F (x)) is always finite. However, for above δ, we can find a real number a = 1 so that F (x) = F (x0 ) for all x, x0 ∈ IB a . Hence a fortiori F (x) ⊂ F (x0 ) + H(x − x0 ) + δkx − x0 kIB. Consequently, condition (3) is satisfied (for all H : X → → Y positively homogenous) and F is well strictly H-differentiable at 0. Thus, in a certain sense, the consideration of such proposed method in (5) can be an interesting alternative for approximating the solutions of (1) when T does not have the metric regularity property around its solutions.

4

Variational perturbations

This short section is devoted to the study variational perturbations of our method. More specifically, we replace the mappings T : X → →Y,H :X → → Y and the positive constant δ in (5) with sequences of set-valued mappings Tn , Hn and a sequence of positive bounded numbers δn (which can be determined as in (12)). Thus, we are led to associate to the inclusion (5) the following method submitted to variational perturbations: 0 ∈ gn (xn+1 − xn ) + Tn (xn ) + Hn (xn+1 − xn ) + δn kxn+1 − xn kIB.

(23)

First of all, let us recall that the lower and upper limits of a sequence An of subsets of a normed space, can be defined as follows: lim inf An n

= {x ∈ X | lim sup d(x, An ) = 0} n→∞

= {x ∈ X | ∃xn ∈ An with xn → x}; and lim sup An = {x ∈ X | lim inf d(x, An ) = 0} n→∞

n

= {x ∈ X | ∃n1 < n2 < . . . in N, ∃xnk ∈ Ank with xnk → x}. Now, we introduce the definition of set-convergence which is useful in the remainder. Definition 4.1. A sequence of set Dn converges to a set D according to Painlev´eKuratowski (or more simply, we also say that Dn set-convergences to D) if (a) A ⊂ lim inf An ; n

(b)

lim sup An ⊂ A. n

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Remark 4.2. When (a) is true, the sequence Dn lower-set converges to the set D while the upper-set convergence of Dn to D is equivalent to assertion (b). Proposition 4.3. Let T : X → → Y be a set-valued mapping, let Tn : X → → Y be a sequence → of set-valued mappings, and let Hn : X → Y be a sequence of positively homogeneous setvalued maps. Consider a sequence gn : X → Y of functions such that gn (0) = 0 and which are continuous in a same neighborhood of 0, for all n ∈ N. Let xn be a sequence satisfying (23). If the sequence gphTn upper set-converges to gphT and if there exists a constant M > 0 such that |Hn |+ ≤ M for all n ∈ N, then any cluster point of xn is a solution to (1). Proof. Consider a cluster point x˜ of the sequence xn satisfying (23), then there exists a subsequence xnk of xn which converges to x˜ and satisfying 0 ∈ gnk (xnk +1 − xnk ) + Tnk (xnk ) + Hnk (xnk +1 − xnk ) + δnk kxnk +1 − xnk kIB. Pose δ˜ := sup δn . Since Hn is uniformly outer bounded with the same positive constant n

M for all n (using the convexity of the unit ball of Y ), the prior inclusion become ˜ n +1 − xn kIB + Tn (xn ). −gnk (xnk +1 − xnk ) ∈ (M + δ)kx k k k k Therefore, there is an element vnk ∈ IB such that ˜ n +1 − xn kvn ∈ Tn (xn ), −gnk (xnk +1 − xnk ) − (M + δ)kx k k k k k that is, 

 ˜ n +1 − xn kvn ∈ gphTn . xnk , −gnk (xnk +1 − xnk ) − (M + δ)kx k k k k   ˜ Moreover, it is clear that xnk , −gnk (xnk +1 − xnk ) − (M + δ)kxnk +1 − xnk kvnk converges to (˜ x, 0). Then, (˜ x, 0) ∈ lim sup gphTn ⊂ gphT n

and the proof is complete. Remark 4.4. The result above can allow us to solve problem (1) without particular assumptions on T . Indeed, whenever we can approach T in (1) with a sequence of set-valued mappings Tn which are Hn -differentiable, in a natural way, the consideration of such a method in (23) could be an interesting tool for solving (1) as soon as Hn is uniformly outer bounded and gphTn upper-set converges to gphT . Obiviously, provided that we know (or we have a means to find) a cluster point for (23). Acknowledgments. The author has highly appreciated the comments and suggestions of the referee and would like to thank him/her for his/her remarks which helped him to improve the quality of this work. 10

References ´ n Artacho, A. L. Dontchev and M. H. Geoffroy, Convergence [1] F. J. Arago of the proximal point method for metrically regular mappings. ESAIM Proc. 17, (2007) pp. 1–7. ´ n Artacho and M. H. Geoffroy, Uniformity and inexact version [2] F. J. Arago of a proximal method for metrically regular mappings. J. Math. Anal. Appl. 335 (1), (2007) pp. 168–183. ´ n Artacho and M. Gaydu, A Lyusternik-Graves Theorem for the [3] F. J. Arago proximal point method, Comput. Optim. Appl. 52 (2012), no. 3, pp. 785–803. [4] J.-P. Aubin, Contingent derivatives of set-valued maps and existence of solutions to nonlinear inclusions and differential inclusions, Mathematical analysis and applications, Part A, pp. 159–229, Adv. in Math. Suppl. Stud., 7a, Academic Press, New York-London, 1981. [5] J.-P. Aubin, H. Frankowska. Set-valued analysis. Systems & Control: Foundations & Applications, 2. Birkhuser Boston, Inc., Boston, MA, 1990. xx+461 pp. [6] A. Daniilidis, C. H. Pang. Continuity of set-valued maps revisited in the light of tame geometry, J. Lond. Math. Soc. (2) 83 (2011), no. 3, pp. 637–658. [7] A. L. Dontchev, W. W. Hager, An inverse mapping theorem for set-valued maps. Proc. Amer. Math. Soc. 121 (1994) pp. 481–489. [8] A. L. Dontchev, R. T. Rockafellar. Implicit functions and solution mappings. A view from variational analysis. Springer Monographs in Mathematics. Springer, Dordrecht, 2009. xii+375 pp. [9] Facchinei, Francisco and Pang, Jong-Shi, Finite-dimensional variational inequalities and complementarity problems. Vol. I, Springer Series in Operations Research, Springer-Verlag, New York, 2003. [10] Ferris, M. C. and Pang, J. S., Engineering and economic applications of complementarity problems, SIAM Review. A Publication of the Society for Industrial and Applied Mathematics, vol.39 - 4 (1997) pp. 669–713. [11] M. Gaydu and M. H. Geoffroy, A Newton iteration for differentiable set-valued maps, J. Math. Anal. Appl. 399 (2013), no. 1, pp. 213–224. [12] B. Mordukhovich. Variational analysis and generalized differentiation. I. Basic theory. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 330. Springer-Verlag, Berlin, 2006. xxii+579 pp. 11

[13] C.H. J. Pang, Generalized differentiation with positively homogeneous maps: applications in set-valued analysis and metric regularity. Math. Oper. Res. 36 (2011), pp. 377–397. [14] C.H. J. Pang, Implicit multifunction theorems with positively homogeneous maps. Nonlinear Anal. 75 (2012), no. 3, pp. 1348–1361. [15] R. T. Rockafellar and R. J.-B. Wets, Variational Analysis, Springer-Verlag, Berlin, 1997.

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