Nonlinear Analysis. On the co-derivative of normal cone mappings to inequality systems

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1 Nonlinear Analysis ) Contents lists available at ScienceDirect Nonlinear Analysis journal hoepage: On the co-derivative of noral cone appings to inequality systes R. Henrion a,, J. Outrata b, T. Surowiec c a Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, Berlin, Gerany b Institute of Inforation Theory and Autoation, Praha 8, Czech Republic c Huboldt University Berlin, Unter den Linden 6, Berlin, Gerany article info abstract Article history: Received 22 July 2008 Accepted 6 Noveber 2008 MSC: 90C30 49J53 Keywords: Mordukhovich coderivative Noral cone apping Calness Inequality constraints The paper deals with co-derivative forulae for noral cone appings to sooth inequality systes. Both the regular Linear Independence Constraint Qualification satisfied) and nonregular Mangasarian Froovitz Constraint Qualification satisfied) cases are considered. A ajor part of the results relies on general transforation forulae previously obtained by Mordukhovich and Outrata. This allows one to derive exact forulae for general sooth, regular and polyhedral, possibly nonregular systes. In the nonregular, nonpolyhedral case a generalized transforation forula by Mordukhovich and Outrata applies, however, a ajor difficulty consists in checking a calness condition of a certain ultivalued apping. The paper provides a translation of this condition in ters of uch easier to verify constraint qualifications. The final section is devoted to the situation where the calness condition is violated. A series of exaples illustrates the use and coparison of the presented forulae Elsevier Ltd. All rights reserved. 1. Introduction The Mordukhovich co-derivative has becoe an iportant tool for the characterization of stability and optiality in variational analysis. We refer to the basic onograph [1] for definitions, properties, calculus rules and applications of this object. When dealing with generalized equations or variational inequalities, the ultivalued appings for which the co-derivative is to be calculated are typically given by noral cones N Ω to certain closed sets Ω. For copleentarity probles, e.g., Ω = R n +, an explicit, ready-to-use forula for the co-derivative D n + is available. However, in any applications, Ω is often ore coplicated than just R n +. For exaple, Ω ay be a general polyhedron or a set described by a finite nuber of sooth inequalities. In such cases see [2,1]), given that certain constraint qualifications hold true, the existence of convenient calculus rules for the co-derivative allow siilar forulae to be obtained as well. If Ω is defined via a sooth inequality syste satisfying the Linear Independence Constraint Qualification LICQ), then the co-derivative D N Ω can be led back to the well-known forula for D n + with the addition of a second order ter and a linear transforation. In the nonregular case, i.e., LICQ is violated, a slightly ore coplicated transforation forula involving a union over nonuniquely defined ultipliers) can be applied provided that the Mangasarian Froovitz Constraint Qualification MFCQ) holds as well as the additional requireent that a certain associated ultifunction is cal see [3]). In general, this transforation forula holds true as an inclusion only, thus leaving a gap between the precise expression for the coderivative and the one cofortably calculated fro the forula. Closing this gap aounts to calculating the co-derivative This work was supported by the DFG Research Center Matheon Matheatics for key technologies in Berlin and by the Grant Agency of the Czech Acadey of Sciences Grant No. IAA ). Corresponding author. E-ail addresses: henrion@wias-berlin.de R. Henrion), outrata@utia.cas.cz J. Outrata), surowiec@ath.hu-berlin.de T. Surowiec) X/$ see front atter 2008 Elsevier Ltd. All rights reserved. doi: /j.na

2 1214 R. Henrion et al. / Nonlinear Analysis ) fro scratch. Iportant exaples where precise forulae for the co-derivative could be obtained in the nonregular case are general polyhedra where LICQ ay be violated see [4]) and the second order cone, which also adits no description satisfying LICQ see [5]). The ai of this paper is to provide explicit, readily-applicable co-derivative forulae for noral cone appings to possibly nonregular inequality systes. The first part reviews soe precise expressions for the co-derivative in the regular and nonregular polyhedral setting. Siilar to the reduction approach by Bonnans and Shapiro [6], p. 240), it is illustrated how nonregularity of the given set Ω can be shifted in certain special situations to the sipler iage set, which ight happen to be polyhedral, thus allowing one to apply the previously entioned co-derivative forula for certain nonregular, nonpolyhedral sets Ω as well. A second part of the paper deals with the transforation forula for the nonregular systes entioned above. The forula is first used to derive an alternative explicit expression for the co-derivative in case of Ω being polyhedral possibly nonregular). Soe exaples are then furnished with the intent of contrasting the forula s ease-of-use versus its possible lack of precision when copared to that which one obtains by applying the precise forula. In the polyhedral case, the entioned calness condition required for the application of the transforation forula happens to be autoatically satisfied. However, for nonlinear inequality systes this is no longer true and one is therefore required to verify calness. The original calness condition is forulated for a ultifunction of coplicated structure involving prial and dual variables. A ajor part of the paper is therefore devoted to a reforulation of this condition as a constraint qualification, i.e., in ters of prial variables only. More precisely, by associating the respective equality syste with the original inequality syste describing Ω, one then checks calness of this equality syste along with that for all its subsystes. A reasonable constraint qualification ensuring this property is derived for the special case that the nuber of binding inequalities exceeds the spacial diension. The final section of the paper addresses the ost coplicated situation: where MFCQ holds true but the entioned calness condition is violated. Partial, albeit less precise, upper estiates for the coderivative are then discussed for this situation. 2. Soe concepts and tools of variational analysis We start with the definitions of the ain objects in our investigation. For a closed set Λ R n and a point x Λ, the Fréchet noral cone to Λ at x Λ is defined by ˆN Λ x) := {x R n x, x x o x x ) x Λ}. The Mordukhovich noral cone to Λ at x Λ results fro the Fréchet noral cone in the following way: N Λ x) := Lisup x x,x Λ ˆN Λ x). The Lisup in the definition above is the upper liit of sets in the sense of Kuratowski-Painlevé, cf. [7]. For a ultifunction Φ : R n R p, consider a point of its graph: x, y) gph Φ. The Mordukhovich noral cone induces the following co-derivative D Φ x, y) : R p R n of Φ at x, y): D Φ x, y) y ) ={x R n x, y ) N gph Φ x, y)} y R p. Siilarly, the ore eleentary Fréchet noral cone induces the Fréchet co-derivative ˆD Φ x, y) y ) ={x R n x, y ) ˆNgph Φ x, y)} y R p. The relation between both concepts is given by D Φ x, ȳ) ȳ ) = Lisup x,y) x,ȳ) y Φx) y ȳ ˆD Φ x, y) y ). Moreover, the co-derivative enjoys the following robustness property: D Φ x, ȳ) ȳ ) = Lisup D Φ x, y) y ). x,y) x,ȳ) y Φx) y ȳ A ultifunction Z : Y X between etric spaces is said to be cal at a point ȳ, x) belonging to its graph, if there exist L,ε > 0, such that dx, Zȳ)) Ldy, ȳ) x Zy) B x,ε) y B ȳ,ε). Here d and B refer to the distances and balls with corresponding radii in the respective etric space. For the special ultifunction Z : R n R R p, defined by Z α, β) := {x R p G 1 x) = α, G 2 x) β},

3 R. Henrion et al. / Nonlinear Analysis ) where G 1 : R p R n and G 1 : R p R are continuous appings, it is easy to see that calness of Z at 0, 0, x) for soe x satisfying G 1 x) = 0 and G 2 x) = 0 is equivalent with the existence of L,ε > 0, such that dx, Z0, 0)) L G 1i x) + ) [G 2i x)] + i i x B x,ε). 1) Here, [y] + := ax{y, 0}. 3. On the co-derivative of noral cone appings 3.1. Regular constraint systes The following theore recalls a basic transforation forula for co-derivatives that was established first as an inclusion by Mordukhovich and Outrata in [2] Theore 3.4) and later as an equality in [1] Theore 1.127): Theore 3.1. Let C = F 1 P), where F : R n R is twice continuously differentiable and P R is soe closed subset. Consider points x C and v N C x). If the Jacobian F x) is surjective, then ) D N C x, v)v ) = λ i 2 F i x) v + T F x) D N P F x), λ ) F x) v ). 2) i=1 Here, the F i are the coponents of F and λ is the unique solution of the equation T F x) λ = v, i.e., λ = F x) T F x) ) 1 F x) v. The value of transforation forula 2) relies on the fact that, starting with the co-derivative for noral cone appings to siple objects such as an orthant), one ay pass to nonlinearly transfored constraint systes such as differentiable inequalities). So, for instance, if C ={x R n F i x) 0 i = 1,...,)}, where the F i are twice continuously differentiable and x C satisfies the Linear Independence Constraint Qualification, then, putting F := F 1,...,F ) T P := R, one ay calculate D N C fro D via Theore 3.1. To do so, one ay access the following representation see, e.g., [8,3, 4, Cor. 3.5)]) for any x,v) gr : { D x, v)v if i : vi v ) = i 0 {x x i = 0 i I 1, x i 0 i I 2 } else where I 1 := {i x i < 0} {i v i = 0,v i < 0}, I 2 := {i x i = 0,v i = 0,v i > 0}. Forula 2) is of use even in the linear case: 3) Corollary 3.1. Let C := {x R n Ax b}, where b R and A is soe atrix of order, n) having rank. Then, for x C and v N C x), it holds that { D N C x, v)v if i : λ ) = i ai,v 0 {A T x x i = 0 i Ĩ 1, x i 0 i Ĩ 2 } else where Ĩ 1 := {i a i, x < b i } {i λ i = 0, a i,v < 0}, Ĩ 2 := {i a i, x = b i, λ i = 0, a i,v > 0}, λ = ) AA T 1 A v and the ai refer to the rows of A. Proof. Putting Fx) := Ax b, 2) yields that D N C x, v)v ) = A T D A x b, λ ) Av ). Now, the result follows fro 3). The full-rank assuption in the corollary can in fact be localized, thus allowing the forula to be applied to any regular polyhedra defined by possibly any inequalities. Of course in this case, the atrix A ust then replaced by the subatrix corresponding to active inequalities.

4 1216 R. Henrion et al. / Nonlinear Analysis ) Nonregular constraint systes polyhedral iage sets Corollary 3.1 does not, however, apply to nonregular polyhedra. For exaple, one cannot derive a co-derivative forula for the polyhedral set x 3 ax{ x 1, x 2 }. Nevertheless, by using the well-known representation of polyhedral noral cone appings fro Dontchev and Rockafellar [8], proof of Theore 2), one ay derive an explicit co-derivative forula for arbitrary polyhedra C := {x R n Ax b}, where b R and A is soe atrix of order, n). To this ai, denote by a i the rows of A and consider arbitrary x C and v N C x). Then, v = A T λ for soe λ R +. Define, the index sets I := {i a i, x =b i } and J := {j λ j > 0}, then it is easily noted that J I. Finally, with each I I associate its characteristic index set χi ) consisting of those indices j I such that for all h R n the following iplication holds true: a i, h 0 i I \ I ), a i, h =0 i I ) a j, h =0. Clearly, I χi ) I, χi ) χi ) for I I and I = χi ) if the subatrix {a i } i I has full rank. Given this setting, the following relations hold true [4], Prop. 3.2 and Cor. 3.4): Theore 3.2. With the notation introduced above, one has that { D N C x, v) v ) = x x, v ) } P I1,I 2 Q I1,I 2, 4) J I 1 I 2 I where P I1,I 2 = con {a i i χ I 2 ) \ I 1 }+span {a i i I 1 } Q I1,I 2 ={h R n a i, h = 0 i I 1 ), a i, h 0 i χ I 2 ) \ I 1 )} and con and span refer to the convex conic and linear hulls, respectively. A ore convenient expression avoiding the union above can be used in the following upper estiation: D N C x, v) v ) con {a i i χ I a v ) I b v ) ) \ I a v )}+span {a i i I a v )} 5) if a i,v = 0 for all i J and a i,v 0 for all i χj) \ J, whereas D N C x, v) v ) = otherwise. Here, I a v ) := {i I a i,v =0}, I b v ) := {i I a i,v > 0}. Corollary 3.2. D N C x, v) 0) = span {a i i I}. Proof. Since 0 Q I1,I 2 for any index sets I 1, I 2, it follows fro 4) that D N C x, v) 0) = P I1,I 2 = P I,I. J I 1 I 2 I Here, the last equality relies on the fact that P I1,I 2 P I3,I 4, whenever I 1 I 3 and I 2 I 4. To illustrate these characterizations, consider the following two exaples: Exaple 3.1. Let C := Ax 0, where A := Put x := 0 and v := a 1 + a 2 = 0, 0, 2). Then, I ={1, 2, 3, 4}, J = {1, 2} and χj) = I. Referring to 5), the condition a i,v = 0 for all i J and a i,v 0 for all i χj) \ J reduces to v = 0. Moreover, I a 0) = I and I b 0) =. Consequently, D N C x, v) v ) = for v 0. On the other hand, by Corollary 3.2, D N C x, v) 0) = span {a i i I} =I A T = R 3. Exaple 3.2. In the previous exaple, put x := 0 and v := a 1 + a 3 = 1, 1, 2). Then, I ={1, 2, 3, 4}, J = {1, 3} and χj) = J. Now, the condition a i,v = 0 for all i J and a i,v 0 for all i χj) \ J reduces to v = 1 v = 2 v 3. Consequently, D N C x, v) v ) = if this last identity is violated. If it holds true, then D N C x, v) 0) = R 3 by Corollary 3.2 and { D con {a2, a N C x, v) t, t, t) 4 }+span {a 1, a 3 } if t > 0 span {a 1, a 3 } if t < 0. This follows easily fro 5), fro the definition of A and fro the already stated identity χ{1, 3}) = {1, 3}.

5 R. Henrion et al. / Nonlinear Analysis ) Fig. 1. Illustration of the boundary of the constraint set in Exaple 3.3. We cobine the previous results for general linear and regular nonlinear constraint systes in order to calculate the co-derivative in a special nonregular, nonlinear setting. We assue that C := {x R n AFx) b}, 6) where F : R n R s is twice continuously differentiable, b R and A is soe atrix of order, s). Suppose also that F x) is surjective. Note that in order to calculate D N C, we cannot invoke Theore 3.1 because surjectivity of AF) x) ay be violated. Nevertheless, we ay rewrite the constraint set as C := F 1 P), P := {y R s Ay b} 7) and then apply Theore 3.1, recalling that we are able to calculate D N P via Theore 3.2. We illustrate this fact in the following exaple: Exaple 3.3. Let C := {x 1, x 2, x 3 ) x 3 x 1 + x x4, 2 x3 + 1 x 2 x 3 ) 2 }. Evidently, C can be equivalently represented by the nonlinear inequality syste x 1 x 3 1 x4 + 2 x 3 0 x 1 + x x4 + 2 x 3 0 x 3 1 x 2 + x x 3 0 x x 2 x x 3 0. Fig. 1 illustrates the boundary of this constraint set. At x = 0 C, all inequalities are active, so their gradients cannot be linearly independent the nonregularity can also be recognized fro Fig. 1, where the graph exhibits four creases eeting at x). This prevents an application of Theore 3.1. However, we ay write C in the for 7), where b = 0, A is as in Exaple 3.1 and Fx) = x 1 + x x4, 2 x3 + 1 x 2 x 3, 2 x 3) T. Evidently, F0) = I 3 is surjective. As a noral vector v N C 0) choose for exaple v = 1, 1, 2). Because of 2 F i 0) = 0 for i = 1, 2, 3, Theore 3.1 provides the forula D N C 0, v)v ) = D N P 0, v) v ). Hence, we ay use for D N C 0, v) exactly the sae estiates as obtained in Exaple Nonregular constraint systes the use of calness In [3] Th. 3.1), it was shown how the assuption of calness for a certain ultifunction allows one to weaken the surjectivity condition in a result like Theore 3.1 to a condition that, in the setting of Theore 3.1, aounts to the Mangasarian Froovitz Constraint Qualification MFCQ). Specifying those ideas to our setting, one gets the following generalization of Theore 3.1: Theore 3.3. Consider the set C = {x R n F i x) 0 i = 1,...,)}, where F : R n R is twice continuously differentiable. Fix soe x C and v N C x) such that, without loss of generality, F x) = 0 and suppose that the following two constraint qualifications are fulfilled: 1. The rows of { F x)} are positively linearly independent i.e., MFCQ is satisfied at x)

6 1218 R. Henrion et al. / Nonlinear Analysis ) The ultifunction Mϑ) := {x,λ) Fx), λ) + ϑ Gr } is cal at 0, x, λ ) for all λ 0 with T F x) λ = v. Then, { ) D N C x, v)v ) λ i 2 F i x) v + T F x) D 0, λ ) F x) v )}. λ 0, T F x) λ= v i=1 As a first application of Theore 3.3, we recover an alternative estiate of 4) and 5) in ters of dual ultipliers) rather than prial characteristic index sets) objects. Corollary 3.3. Let C := {x R n Ax b}, where b R and A is soe atrix of order, n). Fix soe x C and v N C x) with A x = b. If there exists soe ξ R n such that Aξ <0 coponent-wise), then, with the notation of Theore 3.2, D N C x, v)v ) con {a i a i,v > 0}+span {a i a i,v =0}, whenever there exists soe λ 0 such that A T λ = v and λ i a i,v = 0 for all i = 1,...,. Otherwise, D N C x, v)v ) =. Proof. In the setting of Theore 3.3, put Fx) := Ax b. Observe, that the existence of ξ R n such that Aξ < 0 iplies via Gordan s Theore the first constraint qualification of the Theore. Moreover, the ultifunction considered in the second constraint qualification happens to be polyhedral, so it is cal by Robinson s well-known upper Lipschitz result for polyhedral ultifunctions. Hence, one ay conclude that D N C x, v)v ) A T D 0, λ ) Av ). 8) λ 0,A T λ= v Fro 3) we derive that the union on the right-hand side takes place only if λ i a i,v = 0 for all i = 1,...,, in which case A T D 0, λ ) Av ) ={A T x x i = 0 i : λ i = 0, a i,v < 0; x i 0 i : λ i = 0, a i,v > 0} = con {a i a i,v > 0}+span {a i a i,v =0}. This yields the assertion of the corollary. The last corollary provides a ore handy forula for calculating the coderivative of noral cone appings to polyhedra when copared to Theore 3.2, where characteristic index sets need to be calculated, on the other hand, it ay be less precise than the latter in certain circustances. This shall be illustrated by revisiting Exaples 3.1 and 3.2: Exaple 3.4 Exaple 3.2 Revisited). With the data fro Exaple 3.2, the only λ 0 with A T λ = v is λ = 1, 0, 1, 0). Hence, by Corollary 3.3, D N C x, v) v ) a 1,v = a 3,v =0 v = 1 v = 2 v. 3 Moreover, span {a 1, a 2, a 3, a 4 }=R 3 if t = 0 D N C x, v)t, t, t) con {a 2, a 4 }+span {a 1, a 3 } if t > 0 span {a 1, a 3 } if t < 0. Thus, we copletely recover the results of Exaple 3.2 obtained via Theore 3.2. Exaple 3.5 Exaple 3.1 Revisited). With the data fro Exaple 3.1, there are three possibilities for λ 0 with A T λ = v: λ = 0, 0, 1, 1), λ = 1, 1, 0, 0) and λ = r, r, s, s) for r, s > 0 and r + s = 1. Now, Corollary 3.3 iplies span {a 1, a 2, a 3, a 4 }=R 3 if v = 0 con {a 1 }+span {a 3, a 4 } if v = 2 v = 0, 3 v < 1 0 D N C x, v)v con {a ) 2 }+span {a 3, a 4 } if v = 2 v = 0, 3 v > 1 0 con {a 3 }+span {a 1, a 2 } if v = 1 v = 0, 3 v < 2 0 con {a 4 }+span {a 1, a 2 } if v = 1 v = 3 0,v < 2 0 else. In contrast to this result, the application of Theore 3.2 in Exaple 3.1 has shown that D N C x, v)v ) = whenever v 0. In other words, the forula of Corollary 3.3 creates soe additional artificial expressions in the coderivative forula.

7 R. Henrion et al. / Nonlinear Analysis ) We now turn to an application of Theore 3.3 in a nonlinear setting. The crucial calness condition required there has been investigated in [9] see Th. 2, Th. 6, Ex. 6). In general, the conditions for calness used there ay be difficult to verify. Therefore, we provide a different characterization here, where the calness property needs to be verified only for certain constraint systes defined as subsystes of the original inequality constraints in the space of x-variables. For the definition of calness used in the following, we refer to Section 2. Proposition 3.1. If for all I {1,...,} the ultifunctions H I α) ={x F i x) = α i i I), F i x) 0i I c )} are cal at 0, x), then the ultifunction M introduced in Theore 3.3 is cal at 0, x, λ ) for any λ specified there. Proof. Throughout this proof we use the 1-nor of vectors. Note first, that for I =, H I is trivially cal as a constant ultifunction. Hence, this special case can be excluded fro the assuption. Next, observe that, by F x) = 0, one has indeed 0, x) gr H I for all I {1,...,}. The calness assuption eans that for any I {1,...,}, there exist constants δ I,ε I, L I > 0 such that dx, H I 0)) L I α x B δi x) H I α) α : α i ε I,ε I ) i I). Putting δ := in δ I, ε := in ε I, L := ax L I, I {1,...,} I {1,...,} I {1,...,} one obtains that δ, ε, L > 0 and dx, H I 0)) L α x B δ x) H I α) α : α i ε, ε) i I), I {1,...,}. Due to F x) = 0, we ay further shrink δ>0 such that F i x) ε x B δ x) i {1,...,}. 10) Now, consider any λ 0. Then, λ 0) x, and so λ ) M0). We show that 9) dx,λ), M0)) L + 1) ϑ x,λ) Mϑ) B δ x) R ) ϑ = ϑ 1,ϑ 2 ) B ε 0) R. 11) This would prove the asserted calness of M at 0, x, λ ). To this ai, choose arbitrary ϑ = ϑ 1,ϑ 2 ) B ε 0) R and x,λ) Mϑ) B ε x) R ). Note first that x,λ) Mϑ) aounts to λ + ϑ 2 Fx) + ϑ 1 ). Accordingly, Fx) + ϑ 1 0, λ+ ϑ 2 0, λ i + ϑ 2i )F i x) + ϑ 1i ) = 0 i {1,...,}. 12) For the fixed x, define I x := {i {1,...,} F i x) + ϑ 1i = 0orF i x) 0}. Choose x H Ix 0) such that x x = dx, HIx 0)). Note that by definition of I x, F i x) <0 for all i I x ) c. Consequently, x B ε x) H Ix α) for α defined by α i := F i x) i I x ). Since also 10) ensures that α i ε, ε) for all i I x, we ay apply 9) to derive that dx, H Ix 0)) L α = L F i x). i I x Now, if i I x is such that F i x) + ϑ 1i = 0, then F i x) = ϑ 1i. Otherwise, by 12), F i x) + ϑ 1i < 0 and, by definition of I x, F i x) 0. This iplies F i x) ϑ 1i. In any case we ay conclude that x x = dx, HIx 0)) L ϑ 1. Next, define λ R by λ i := λ i + ϑ 2i if i I x and λ i := 0ifi I x ) c. Then, λ 0by12). Moreover, x H Ix 0) entails that F i x) = 0ifi I x and F i x) 0ifi I x ) c. In particular, λ i F i x) = 0 for all i {1,...,}. This eans that λ F x) ) and, hence, x, λ) M0). Finally, observe that, for i I x ) c, one has F i x) + ϑ 1i < 0 and, thus, by 12), λ i =ϑ 2i. This proves that λ λ = ϑ 2. Consequently, dx,λ), M0)) x,λ) x, λ) = x x + λ λ L ϑ 1 + ϑ 2 L + 1) ϑ which shows 11).

8 1220 R. Henrion et al. / Nonlinear Analysis ) We refer the reader to [10] for ethods to check calness of constraint systes like those given by the ultifunctions H I in the previous proposition. The next proposition shows how to get rid of inequalities for the verification of calness in the previous proposition. More precisely, calness ust only be checked for all equality subsystes. This proposition, which yields a slightly stronger result than needed, requires a technical lea, the proof of which is shifted to the Appendix Lea 3.1). Proposition 3.2. If for all I {1,...,} the ultifunctions H I α) := {x F i x) = α i i I)} are cal at 0, x), then the ultifunctions H I α) = { x F i x) = α i i I), F i x) α i i I c )} are also cal at 0, x) for all I {1,...,}. In particular, the ultifunctions H I α) introduced in Proposition 3.1 are cal at 0, x) for all I {1,...,}. Proof. We proceed by induction over the nuber of coponents of F. Consider first the case = 1. We either have I = or I = {1}. In the second case, one has HI = HI due to = 1, hence calness of HI follows fro that of HI. In the first case, we apply Lea 3.1 proved in the Appendix. Referring to the notation of this lea, we put I = and check the two assuptions ade there. As the only set I {1} with I I is given by I = {1} and then, as before, HI = HI, calness of HI follows fro that of HI. This shows the first assuption of Lea 3.1 to hold true. Concerning the second assuption, one has i = 1 and, hence, M reduces to the trivial constant ultifunction M α, β) R n which is cal. On the other hand, the second ultifunction introduced there reduces to M = HI, hence calness of M follows fro that of HI. As a consequence, Lea 3.1 yields calness of HI = H. Suarizing, the assertion of our proposition follows for the case = 1. Next assue that the Proposition holds true for all k and consider the case = k + 1. By assuption, the HI are cal at 0, x) for all I {1,...,k + 1}. In particular, the ultifunction M considered in the second assuption of Lea 3.1 and corresponding to the case #I = 1 is cal at 0, x). Moreover, the induction hypothesis ensures that also the ultifunctions {x F i x) = α i i I), F i x) α i i J I)} 13) are cal at 0, x) for all subsets I J and all J {1,...,k + 1} with #J = k. Since the ultifunction M considered in the second assuption of Lea 3.1 is of type 13) with J = {1,...,k + 1} { i }, it follows that M is cal at 0, 0, x). Suarizing, the second assuption of Lea 3.1 is always satisfied no atter how the index set I {1,...,k + 1} is chosen in the Lea. Therefore, it is enough to check the first assuption for its application. Now, choose an arbitrary I {1,...,k + 1}. We have to show that HI is cal at 0, x). IfI = {1,...,k + 1}, then H I = HI and calness of HI follows fro that of HI.If#I = k, then the only choice for the index set I considered in the first assuption of Lea 3.1 is I = {1,...,k + 1}. According to what we have shown just before, HI is cal, so we have shown that the HI are cal at 0, x) whenever #I k. Passing to the case #I = k 1 and recalling that the index set I considered in the first assuption of Lea 3.1 is always strictly larger than I, one derives calness of HI on the basis of what we have shown before due to #I > #I = k 1 which aounts to #I k. So, the first assuption of Lea 3.1 is satisfied again and we derive calness of HI whenever #I k 1. Proceeding this way until #I = 0, we get the desired calness at 0, x) for all subsets I {1,...,k + 1}. That the calness of the HI iplies the calness of the corresponding H I introduced in Proposition 3.1, is an iediate consequence of the calness definition and of the evident relations HI α, 0) = H I α). We ephasize that without considering subsystes, a result analogous to Proposition 3.2 cannot be obtained for a single constraint syste. For instance, for Fx) := x 2, x) one has that the equality syste F 1 x) = α 1, F 2 x) = α 2 is cal at 0, 0), whereas the inequality syste F 1 x) α 1, F 2 x) α 2 is not. The reason is that subsystes need not inherit calness for instance, the equality subsyste F 1 x) = α 1 fails to be cal at 0, 0)). We ay cobine Theore 3.3, Propositions 3.1 and 3.2 to get an assuption which copletely relies on constraint systes induced by F and thus can be considered to be a CQ weaker than surjectivity) for the apping F. Theore 3.4. In the setting of Theore 3.3 assue that 1. MFCQ is satisfied at x; 2. all perturbed equality subsystes {x F i x) = α i i I)} I {1,...,} are cal at 0, x). Then, the coderivative forula of Theore 3.3 holds true. Reark 3.1. If we consider the couple of constraint qualifications iposed in Theore 3.4 as a single one and give it the nae CQ, then the following holds true for the inequality syste Fx) 0: LICQ CQ MFCQ,

9 R. Henrion et al. / Nonlinear Analysis ) where MFCQ and LICQ refer to the Mangasarian Froovitz and Linear Independence constraint qualifications, respectively, with the latter being the sae as the surjectivity condition iposed in Theore 3.1. Indeed, the second iplication being evident, suppose that Fx) 0 satisfies LICQ at x. Then, all gradients { F i x)} i=1,..., and trivially all subsets of gradients are linearly independent. But linear independence of a set of gradients iplies the Aubin property and, hence, calness for the corresponding set of equations. Consequently, CQ follows fro LICQ. Suarizing, CQ is soething in between LICQ and MFCQ and it sees that it is closely related to the constant rank constraint qualification CRCQ see [11]). At the end of this section, we provide a useful and easy to check constraint qualification ensuring condition 2. in Theore 3.4. Proposition 3.3. Assue that at x the following full rank constraint qualification is satisfied: rank { F i x)} i I = in {n, #I} I {1,...,}. 14) Then, the ultifunctions HI introduced in Proposition 3.2 are cal at 0, x) for all I {1,...,}. Proof. Choose an arbitrary I {1,...,}. Consider first the case that #I n. Then, by 14), the set of gradients { F i x)} i I is linearly independent. Consequently, HI is cal at 0, x). Now, if #I > n, then select an arbitrary J I with #J = n.by14), the set of gradients { F i x)} i J is linearly independent, hence HJ 0) = { x} by the inverse function theore. Since F x) = 0 and HI 0) HJ 0), it follows that HI 0) = HJ 0). Moreover, according to what has been entioned before, HJ is cal at 0, x). Consequently, there are constants L,ε > 0 such that dx, HJ 0)) L α x HJ α) B ε x) α B ε 0). Fro here it follows with HI α) HJ α), where α is the subvector of α according to the index set J I, that dx, HI 0)) = dx, HJ 0)) L α L α x HI α) B ε x) α B ε 0). This, however, aounts to calness of HI at 0, x). As an illustration, we revisit Exaple 3.3. Because this exaple is nonlinear, we cannot take for granted that assuption 2. of Theore 3.3 will be autoatically satisfied as we could in Exaples 3.4 and 3.5. On the other hand, the four constraint gradients in this exaple, though linearly dependent in R 3 satisfy the full rank constraint qualification 14). Indeed, any of the 4 triples that can be selected fro the original set of gradients happens to be a linearly independent set. Therefore, the second assuption of Theore 3.4 is satisfied by virtue of Proposition 3.3. Since the Mangasarian Froovitz Constraint Qualification is easily seen to be fulfilled at x, we ay apply the co-derivative forula of Theore 3.3. Doing so would yield the sae result as in the linearized exaples discussed before Beyond calness Before we address the issue of coputing the co-derivative in the event that the calness condition is violated, we provide an instructive counterexaple to Theore 3.3 showing that the provided forula does not hold when the calness condition is dropped. Exaple 3.6. Let F x 1, x 2 ) := x 2,ϕx 1 ) x 2 ), where ϕt) := t 5 sin1/t) for t 0 and ϕ0) := 0. Since ϕ is twice continuously differentiable, so is F and one has F 1 x 1, x 2 ) = 0, 1), F 2 x 1, x 2 ) = ϕ x 1 ), 1 ). 15) We choose x := 0, 0). Then, F x) = 0, 0) and, taking into account that ϕ 0) = 0, it holds that F 1 x) = F 2 x) = 0, 1). This eans that both gradients are positively linearly independent i.e., MFCQ is satisfied). Of course they are linearly dependent, thus preventing us fro applying 2). Suarizing, all assuptions of Theore 3.3 are fulfilled except calness this could be easily checked directly, but we shall see it as a consequence of the conclusion of that theore being violated). We choose v := 0 and v := F 1 x) + F 2 x) iplying that v N C x) in view of MFCQ). Forally applying Theore 3.3 would yield the inclusion D N C x, v) 0) T F x) D 2 0, λ ) 0). λ 0, T F x)= v Given the fact that D 2 0, λ ) 0) = R 2, regardless of the value of λ 0 see 3)), and taking into account 16), we end up at the inclusion D N C x, v) 0) {0} R. 17) 16)

10 1222 R. Henrion et al. / Nonlinear Analysis ) To see that this is wrong, consider the sequences x k := 1/ kπ), 0), v k := ) ) F 1 x k + F 2 x k. Then taking into account that ϕx k 1 ) = 0 and, thus, F x k) = 0, 0), it follows that x k x, x k C, v k v, v k N C x k ). Here, the last relation relies on the fact that MFCQ is an open property and and pertains to hold at x k close to x. Furtherore, we observe that ϕ x k) 1 0, which, as a consequence of 15), iplies F x k) is surjective; in fact, F x k) is even a regular ) atrix. This allows to apply 2) at x k,v k : D N C x k,v k ) 0) = T F x k) D 2 0, 1, 1)) 0) = T F x k) R 2 = R 2, where the last equality follows fro the fact that, rank T F x k) = rank F x k) = 2. Exploiting the robustness property of the co-derivative, we let k and thus derive R 2 D N C x, v) 0) Lisup k therefore D N C x, v) 0) = R 2, contradicting 17). D N C x k,v k ) 0) = R 2, We see then that by dropping the calness condition, one can no longer expect the forula of Theore 3.3 to hold true. Nevertheless, the forula ay serve as a part of calculating the co-derivative in a ore eleentary according to its basic definition) aggregation process. More precisely, by introducing the set C := {x bd C F x) is surjective}, we have for any v R n see Section 2), D N C x, v) v ) = Lisup ˆD NC x,v) v ) x x,v v,x C,v N C x),v v = Lisup x x,v v,x int C,v N C x),v v { Lisup ˆD NC x,v) v ) x x,x C,v v,v N C x),v v { { ˆD NC x,v) v ) Lisup ˆD NC x,v) v ) x x,x C\C,v v,v N C x),v v }{{} P v ) } ={0} P v ) Lisup D N C x,v) v ). x x,x C,v v,v N C x),v v }{{} R v ) Here, the first ter is trivial and follows easily fro the definition of the Fréchet coderivative evaluated in the interior of the feasible set, whereas the last equality results fro the robustness outer seicontinuity) of the co-derivative. In this way, we have subdivided the coputation of the co-derivative into a pathological part P v ), where Fréchet coderivatives have to be calculated and aggregated in an eleentary way, and a regular part R v ), where we ay exploit forula 2) in the aggregation process. It is iportant to observe that the pathological part is sall in the following sense [12], Th. 2.1): If the MFCQ is satisfied everywhere in the feasible set C, then the subset of points around which the feasible set ay be locally described by a regular constraint syste i.e., with surjective Jacobian Fx)) is open and dense in the boundary of C.Inthe following, we derive soe upper estiates of the regular part. Proposition 3.4. Let C = {x R n Fx) 0}, where F C 2 R n, R ). Consider soe x C with F x) = 0 and v N C x). Assue that MFCQ is satisfied at x. Then, for all v R n, v { ) R ) [ λ i 2 F i x) v + Lisup T F x) D x x,x C 0, λ ) F x) v )]}. λ 0, F x) λ= v i=1 } } Proof. Let v R n and x R v ) be arbitrarily given. By definition, there are sequences x k x, x k C, v k v, v k N C x k ), v k v, x k x, x k D N C x k,v k ) v ) k.

11 R. Henrion et al. / Nonlinear Analysis ) By the definition of C, F x k ) is surjective. Then, applying the transforation forula 2), we obtain ) x k λ k i 2 F i x k ) v k + T F x k ) D F xk ),λ k) F x k ) v ) k, i=1 where λ k 0 is the unique solution of the equation Fx k )λ = v k. Therefore, we ay write ) x = k λ k i 2 F i x k ) v + k w k i=1 for soe w k T F x k ) D F xk ),λ k) F x k ) v ) k. 18) { MFCQ being satisfied at x guarantees boundedness of the sequence λ k} see, e.g., Prop in [6]). Therefore, upon passing to a subsequence, which we do not relabel, we ay assue that λ k μ for soe μ 0. It follows that w k w for ) w := x μ i 2 F i x) v. 19) i=1 Moreover, Fx k )λ k = v k entails that F x) μ = v. 20) We clai that, for k large enough, D F xk ),λ k) F x k ) v k ) D NR 0,μ) F x) v ). 21) According to 3), we ay write D F xk ),λ k) F x k ) v k ) = A k 1 Ak, where if λ k i F i x k ) v k 0 A k {0} if λ i = k i = 0 and F i x k ) v < k 0 R + if λ k i = 0 and F i x k ) v > k 0 R if F i x k ) v = 0. k Siilarly, D 0,μ) F x) v ) = A 1 A, where if μ i F i x) v 0 {0} if μ A i = i = 0 and F i x) v < 0 R + if μ i = 0 and F i x) v > 0 R if F i x) v = 0. In order to verify 21) it is enough to show that A k i A i for all large enough k N and all i {1,...,}. Let i be an arbitrary such index. We proceed by case distinction. If F i x) v = 0, then A i = R and A k i A i holds trivially. If F i x) v < 0, then F i x k ) v < k 0 for all large enough k. In particular, Ak i {0}. If, additionally, μ i = 0, then A i ={0} showing again the inclusion to hold. Otherwise, if μ i > 0, then λ k i > 0 for all large enough and, so, A k i = which iplies the inclusion a third tie. A siilar reasoning applies to the reaining case of F i x) v > 0 upon replacing {0} by R +. This finally proves 21). Multiplying 21) fro the left by T F x k ), we ay infer fro 18) that w k T F x k ) D 0,μ) F x) v ). In other words, [ w Lisup T F x) D x x,x C 0,μ) F x) v )]

12 1224 R. Henrion et al. / Nonlinear Analysis ) which entails via 19) and 20) that ) [ x μ i 2 F i x) v + Lisup T F x) D i=1 x x,x C 0,μ) F x) v )] { ) [ λ i 2 F i x) v + Lisup T F x) D λ 0, F x) λ= v i=1 x x,x C 0, λ ) F x) v )]}. If the apping x T F x) D 0, λ ) F x) v ) happens to be outer seicontinuous, then one could replace the Lisup in the forula of the last Proposition by the liiting expression T F x) D 0, λ ) F x) v ) and would be back to the forula of Theore 3.3. Unfortunately, this outer seicontinuity does not hold true in general without the calness condition of Theore 3.3), such that the Lisup ay becoe strictly larger see Exaple 3.6). Nevertheless, we ay verify outer seicontinuity in a special situation: Proposition 3.5. In the setting of Proposition 3.4, suppose that F i x) v 0 i = 1,...,. Then, for any λ 0 with F x) λ = v, one has that [ Lisup T F x) D x x,x C 0, λ ) F x) v )] T F x) D 0, λ ) F x) v ). Proof. Consider an arbitrary w in the left-hand side of the asserted inclusion. Accordingly, there are sequences x k x and w k w such that x k C and w k = T F x k ) α k for certain α k D 0, λ ) F x) v ). Recalling the representation of the set D 0, λ ) F x) v ) provided in the proof of Proposition 3.4 with λ replaced by μ), we see that α k R +. Now an arguent already used in the proof of Proposition 3.4 based on MFCQ being satisfied at x) allows to derive boundedness of the sequence {α k }. Hence, α k l α for soe subsequence and soe α 0. It follows that w = T F x) α. Since α D 0, λ ) F x) v ) by closedness of this latter set, we are done. Corollary 3.4. In the setting of Proposition 3.4 and under the additional assuption of Proposition 3.5, it holds true that and R v ) = if v 0 R v ) con { T F i x) i I } if v = 0, where con denotes the convex conic hull and I := {i F i x) v > 0}. Proof. If v 0, then in the representation F x) λ = v there ust be at least one i such that λ i > 0. Since, by assuption in Proposition 3.5 one also has that F i x) v 0, it follows fro 3) that D 0, λ ) F x) v ) =. As a consequence of Propositions 3.4 and 3.5, R v ) =. If, in contrast, v = 0, then also λ = 0 by MFCQ being satisfied at x), hence see 3) again), D 0, λ ) F x) v ) = A 1 A, where either A i ={0} if F i x) v < 0) or A i = R + if F i x) v > 0). This proves the second inclusion along with Propositions 3.4 and 3.5. Unfortunately, it sees to be difficult to find siilar eaningful upper estiates for R v ) if the assuption of Proposition 3.5 is violated. Siilarly, it reains an open question if there is a general characterization of the previously entioned contribution P v ) along the pathological set C \ C.

13 R. Henrion et al. / Nonlinear Analysis ) Appendix Lea 3.1. Fix an arbitrary I {1,...,}. Referring back to the ultifunctions HI introduced in Proposition 3.2, assue that 1. For all I I with I I {1,...,} the HI are cal at 0, x). 2. For soe i I I the ultifunctions { { Mα, β) := x R n Fi x) = α i i I ) }, F j x) β j j {1,...,} I { i })), Mt) := { x R n F i x) = } t are cal at 0, 0, x) and 0, x), respectively. Then, HI is cal at 0, x). Proof. Assue that HI fails to be cal at 0, x). Then, by 1), there is a sequence x k x such that [ dx k, HI 0)) >k F i x k ) + Fj x k ) ] ). 22) + i I j {1,...,} I ) Suppose there is soe index j {1,...,} I and soe subsequence x kl with F j xkl 0. Put I := I { j }. Due to )) H I 0) HI 0) and to x kl HI F xkl one would arrive fro 22) at ) ) [ )] dx kl, HI 0)) >k Fi l xkl + Fj xkl, + i I j {1,...,} I a contradiction with assuption 1. Hence, there is soe k 0 such that F j x k ) < 0 k k 0 j {1,...,} I. 23) Together with 22), this iplies that dx k, HI 0)) >k F i x k ). i I We clai the existence of soe ρ>0 and k 1 k 0 such that i I F i x k ) >ρ F i x k ) k k 1, 25) where i refers to assuption 2. Indeed, otherwise there was a subsequence x kl such that ) ) Fi xkl l 1 Fi xkl. i I In the following, we lead this relation to a contradiction. Now, justified by x M0), where M is defined in assuption 2, we ay select for any l soe y l M0) such that dx kl, M0)) = xkl y l. The assued calness at 0, x) of M entails the existence of soe L1 > 0 such that ) dx kl, M0)) L1 Fi xkl l 0 which in turn iplies that y l x. Consequently, for all large enough l, ) ) Fi xkl = Fi xkl Fi y l ) L2 xkl y l where L 2 is soe Lipschitz odulus of F i near x. Now, referring to the ultifunction M defined in assuption 2, we observe by virtue of 23) that, for all large enough l, x kl M α l), ) 0, where α l) ) i := F i xkl for i I. Now, the assued calness at 0, x) of M leads to dx kl, M0, 0)) L 3 α l) ) ) = L Fi 3 xkl l 1 L Fi 3 xkl i I l 1 L 3 L xkl 2 y l = l 1 L 3 L 2 dx kl, M0)), 24)

14 1226 R. Henrion et al. / Nonlinear Analysis ) for all large enough l. If also l > L 3 L 2, then dx kl, M0, 0)) < dx kl, M0)). Now, justified by x M0, 0), we ay select z l M0, 0) such that dx kl, M0, 0)) = xkl z l l. 26) ) It follows fro 26) that z l M0), whence Fi z l ) 0. Recalling that F i xkl < 0 for large enough l see 23)), one would find in case of F i z l ) > [ ] 0 soe z on the line segent x kl, ) z l with Fi z = 0 and xkl z < xkl z l yielding a contradiction with 26) due to z M0). Therefore, Fi z l ) < 0 and, hence, one ay invoke the definition of M to infer fro z l M0, 0) that z l HI 0) for large enough l. Now, 23) and 24) provide, for large enough l that k l [ )] ) F i x k ) + Fj xkl = k Fi + l xkl < dxkl, HI 0)) xkl z l = dxkl, M0, 0)), i I j {1,...,} I {i }) i I a contradiction with the assued calness at 0, 0, x) of M. This contradiction proves the desired relation 25). Using this, we ay continue 24) as ) 1 dx k, HI 0)) >k F i x k ) + ρ F i x k ) ρ + 1 i I ρ + 1 i I ρ > k F i x k ) ρ + 1 i I {i } ρ = k [ F i x k ) + Fj x k ) ] ρ i I {i } j {1,...,} I {i }) k k 1, where in the last relation, we exploited again 23). Put I := I { i }. Due to HI 0) HI 0) we end up at the relation ρ [ dx k, HI 0)) >k F i x k ) + Fj x k ) ] ) k k ρ i I j {1,...,} I This, however, is in contradiction with the assued calness at 0, x) of HI see assuption 1.) Hence, we have finally led to a contradiction our initial assuption that HI fails to be cal at 0, x). References [1] B.S. Mordukhovich, Variational Analysis and Generalized Differentiation, Vol. 1: Basic Theory, Vol. 2: Applications, Springer, Berlin, [2] B.S. Mordukhovich, J. Outrata, On second-order subdifferentials and their applications, SIAM Journal on Optiization ) [3] B.S. Mordukhovich, J. Outrata, Coderivative analysis of quasi-variational inequalities with applications to stability and optiization, SIAM Journal on Optiization ) [4] R. Henrion, W. Röisch, On M-stationary points for a stochastic equilibriu proble under equilibriu constraints in electricity spot arket odeling, Applications of Matheatics ) [5] J.V. Outrata, D. Sun, On the coderivative of the projection operator onto the second-order cone, Set-Valued Analysis, in press doi: /s x). [6] J.F. Bonnans, A. Shapiro, Perturbation Analysis of Optiization Probles, Springer, New York, [7] R.T. Rockafellar, R.J.-B. Wets, Variational Analysis, Springer, Berlin, [8] A.L. Dontchev, R.T. Rockafellar, Characterization of strong regularity for variational inequalities over polyhedral convex sets, SIAM Journal on Optiization ) [9] R. Henrion, J. Outrata, Calness of constraint systes with applications, Matheatical Prograing ) [10] J. Heerda, B. Kuer, Characterization of Calness for Banach space appings, Huboldt University Berlin, Institute of Matheatics, Preprint 06-26, [11] F. Facchinei, J.-S. Pang, Finite-diensional Variational Inequalities and Copleentarity Probles, Springer, New York, [12] R. Henrion, On constraint qualifications, Journal of Optiization Theory and Applications )