JOHANNES KEPLER UNIVERSITY LINZ. Institute of Computational Mathematics

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1 JOHANNES KEPLER UNIVERSITY LINZ Institute of Computational Mathematics Robinson Stability of Parametric Constraint Systems via Variational Analysis Helmut Gfrerer Institute of Computational Mathematics, Johannes Kepler University Altenberger Str. 69, 4040 Linz, Austria Boris S. Mordukhovich Department of Mathematics, Wayne State University Detroit, MI 48202, USA and Peoples Friendship University of Russia Moscow , Russia NuMa-Report No August 2016 A 4040 LINZ, Altenbergerstraße 69, Austria

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3 Robinson Stability of Parametric Constraint Systems via Variational Analysis HELMUT GFRERER BORIS S. MORDUKHOVICH August 3, 2016 Abstract. This paper investigates a well-posedness property of parametric constraint systems named here Robinson stability. Based on advanced tools of variational analysis and generalized differentiation, we derive first-order and second-order conditions for this property under minimal constraint qualifications and establish relationships of Robinson stability with other well-posedness properties in variational analysis and optimization. The results obtained are applied to robust Lipschitzian stability of parametric variational systems. Key words. parametric constraint systems, Robinson stability, variational analysis, first-order and secondorder generalized differentiation, metric regularity and subregularity AMS subject classification. 49J53, 90C30, 90C31 1 Introduction and Discussion The main focus of this paper is on studying parametric constraint systems PCS of the type 1.1 gp,x C with x R n and p P, where x is the decision variable, and where p is the perturbation parameter belonging to a topological space P. In what follows we impose standard smoothness assumptions on g: P R n R l with respect to the decision variable and consider general constraint sets C R l, which are closed while not necessarily convex. Define the set-valued solution map Γ: P R n to 1.1 by 1.2 Γp := { x R n gp,x C } for all p P and fix the reference feasible pair p, x gphγ. The major attention below is paid to the following well-posedness property of PCS, which postulates the desired local behavior of the solution map 1.2. Definition 1.1 Robinson stability. We say that PCS 1.1 enjoys the ROBINSON STABILITY RS property at p, x with modulus κ 0 if there are neighborhoods U of x and V of p such that 1.3 dist x;γp κ dist gp,x;c for all p,x V U in terms of the usual point-to-set distance. The infimum over all such moduli κ is called the RS EXACT BOUND of 1.1 at p, x and is denoted by robg,c p, x. Institute of Computational Mathematics, Johannes Kepler University Linz, A-4040 Linz, Austria; helmut.gfrerer@jku.at Department of Mathematics, Wayne State University, Detroit, MI 48202, USA, and Peoples Friendship University of Russia, Moscow , Russia; boris@math.wayne.edu 1

4 Robinson [23] studied this property for closed convex cones C under the name of stability and proved that the following condition known now as the Robinson constraint qualification: int g p, x + x g p, xr n C is sufficient for RS in this case. Also, 1.4 is shown to be necessary for 1.3 if gp,x = gx p the case of canonical perturbations and P = R l or P is a neighborhood of 0 R l. Further results in this direction have been obtained in various publications see, e.g., [1, 2, 3, 4, 14] and the references therein, and in some of them condition 1.3 is called Robinson metric regularity of 1.2. In our opinion, the latter name is misleading since it contradicts the widely accepted notion of metric regularity in variational analysis [17, 24] meaning, for a given set-valued mapping F : Z Y between metric spaces and a given point z,ȳ gphf, that the following distance estimate 1.5 dist z;f 1 y κ dist y;fz for all z,y close to z,ȳ holds. Having in mind the weaker property of metric subregularity of F at z, ȳ, which corresponds to the validity of 1.5 with the fixed point y = ȳ therein, we can interpret the RS property 1.3 as the metric subregularity of the other mapping x gp,x C at every point x Γp close to x for every fixed parameter p P close to p with the uniform modulus κ. Another useful interpretation of 1.3 is as follows. Robinson defined in [23] the class of admissible perturbations of the system gx C at x as triples P, p,gp,x such that p P and g: P R n R l is partially differentiable with respect to x for all p P, is continuous together with x g at p, x, and satisfies g p,x = gx near x. It can be distilled from [23] that, in the case of convex cones C, the metric regularity of the mapping x gx C around x,0 is equivalent to the validity of 1.3 for all the admissible perturbations with some uniform modulus κ. However, the situation changes dramatically when we face realistic models with constraints on feasible perturbations. In such settings, which particularly include canonical perturbations with convex cones C while 0 bdp R l, the uniform subregularity viewpoint on Robinson stability is definitely useful. This approach naturally relates to a challenging issue of variational analysis on determining classes of perturbations under which the generally nonrobust property of metric subregularity is stable. The major goal of this paper is to obtain verifiable conditions on perturbation triples P, p, gp, x ensuring the validity of the RS estimate 1.3. The results obtained in this vein seem to be new not only for the case of general perturbations with nonconvex sets C, but even in the conventional settings where perturbations are canonical and C is a polyhedral convex cone. To achieve these results, we use powerful tools of first-order and second-order variational analysis and generalized differentiation, which are briefly reviewed in Section 2. The rest of the paper is organized as follows. Section 3 presents first-order results on the validity of Robinson stability and its relationships with some first-order constraint qualifications and Lagrange multipliers. In particular, a precise formula for calculating the exact stability bound robg,c p, x is derived under a new subamenability property of C. The main first-order conditions ensuring RS go far beyond metric regularity of x g p,x C while surely hold under its validity regardless the convexity of the set C. We further specify the obtained results in the settings where C is either convex or the union of finitely many convex polyhedra and also under more conventional constraint qualifications. Section 4 is devoted to second-order analysis of Robinson stability, which has never been previously done in the literature. We introduce new second-order quantities for closed sets and employ them to derive constructive second-order conditions to ensure Robinson stability of 1.1 in the case of general sets C with effective specifications for unions of convex polyhedra. As a by-product of 2

5 the obtained results on Robinson stability, new second-order conditions for metric subregularity of constraint mappings are also derived in nonpolyhedral settings. Section 5 provides applications of the main results on Robinson stability to establish new firstorder and second-order conditions for robust Lipschitzian stability Lipschitz-like or Aubin property of solution maps in 1.2 with their specifications for parametric variational systems PVS. The latter systems reduce to PCS 1.1 with sets C represented as graphs of normal cone/subdifferential mappings in particular, parameter-dependent ones, which occur to be the most challenging for sensitivity analysis. The given numerical example shows that our results can be efficiently applied to such cases. In the concluding Section 6 we briefly summarize the obtained results for Robinson stability of PCS, present more discussions on its relationships with other well-posedness properties of PCS and PVS, and outline some topics of our future research. Throughout the paper we use standard notation of variational analysis and generalized differentiation see, e.g., [17, 24], except special symbols discussed in the text. 2 Preliminaries from Variational Analysis All the sets under consideration are supposed to be locally closed around the points in question without further mentioning. Given Ω R d and z Ω, recall first the standard constructions of variational analysis used in what follows see [17, 24]: The Bouligand-Severi contingent cone to Ω at z is: 2.1 T Ω z := { u R d t k 0, u k u with z +t k u k Ω for all k N }. The Fréchet regular normal cone to Ω at z is: 2.2 N Ω z := T Ω z = { v R d v,u 0 for all u T Ω z }. The Mordukhovich limiting normal cone to Ω at z is: 2.3 N Ω z := { v R d z k x, v k v with v k N Ω z k for all k N }. We will also employ the directional modification of 2.3 introduced recently by Gfrerer [?]. Given w R d, the limiting normal cone in direction w to Ω at z is 2.4 N Ω z;w := { v R d t k 0, w k w, v k v with v k N Ω z +t k w k }. The following calculus rule for limiting normals is largely used in the paper. It is an extension of the well-known result of variational analysis see, e.g., [17, Theorem 3.8] with replacing the metric regularity qualification condition by that of the imposed metric subregularity. Note that a similar result in somewhat different framework can be distilled from the proof of [12, Theorem 4.1]. Lemma 2.1 limiting normals to inverse images. Let f : R s R d be strictly differentiable at z f 1 C and such that the mapping z f z C is metrically subregular at z,0 with modulus κ 0. Then for every v N f 1 C z there exists some u N C f z κ v B R d satisfying v = f z u, where the sign indicates the matrix transposition. Proof. Denote Fz := f z C and pick v N f 1 C z = N F 1 0 z. Since the mapping F : R s R d is metrically subregular at z, 0 with modulus κ and closed-graph around this point, we can apply [11, Proposition 4.1] and get the inclusion 2.5 N F 1 ȳ z;w { v R s y κ v B R d with v,y N gphf z,0;w,0 }. 3

6 Using 2.5 with w = 0 yields the existence of u κ v B R l such that v, u N gphf z,0. The structure of the mapping F and elementary differentiation ensure the normal cone representation N gphf z,0 = { ξ,η R s R d η N C f z, ξ + f z η = 0 }, which therefore verifies the claimed statement of the lemma. Throughout the paper we systematically distinguish between metric regularity and subregularity assumptions. To illuminate the difference between these properties in the case of the underlying mapping F = f C, observe that F is metrically regular around z,0 if and only if the implication [ λ NC f z, f z λ = 0 ] = λ = holds. This is a direct consequence of the Mordukhovich criterion; see [24, Theorem 9.40]. On the other hand, it is shown in [9, Corollary 1] based on the results developed in [6] that the metric subregularity of F at z,0 is guaranteed by the condition that for all u 0 with f zu T C f x we have the implication [ 2.7 λ NC f z; f zu, f z λ = 0 ] = λ = 0. Next we introduce a new class of nice sets the properties of which extend the corresponding ones for amenable sets; see [24]. The difference is again in employing metric subregularity instead of metric regularity. Indeed, the qualification condition used in the definition of amenability [24, Definition 10.23] ensures the metric regularity of the mapping z qz Q below around z,0. Definition 2.2 subamenable sets. A set C R l is called SUBAMENABLE at z C if there exists a neighborhood W of z along with a C 1 -smooth mapping q: W R d for some d N and along with a closed convex set Q R d such that we have the representation C W = { z W qz Q } and the mapping z qz Q is metrically subregular at z,0. We say that C is STRONGLY SUB- AMENABLE at z,0 if this can be arranged with q of class C 2, and it is FULLY SUBAMENABLE at z,0 if in addition the set Q can be chosen as a convex polyhedron. Finally in this section, we formulate our standing assumptions on the mapping g: P R n R l in 1.1, which stay without further mentioning for the rest of the paper: There are neighborhoods U of x and V of p such that for each p V the mapping gp, is continuously differentiable on U and that both g and and its partial derivative x g are continuous at p, x. 3 First-Order Analysis of Robinson Stability We start this section with establishing relationships between Robinson stability and other important properties and constraint qualifications for parametric systems 1.1. Recall first from [10] that the partial metric regularity constraint qualification MSCQ holds for 1.1 at p, x with respect to x if there are neighborhoods U of x and V of p such that for every p V and every x Γp U the mapping gp, C is metrically subregular at x,0, i.e., there is a neighborhood U p,x of x and a constant κ p,x 0, possibly depending on p and x, for which 3.1 dist ξ ;Γp κ p,x dist gp,ξ ;C for all ξ U p,x. 4

7 Next we introduce a new property of PCS 1.1 at the reference point p, x that involves limiting normals and Lagrange multipliers. Given p,x gphγ and v R n, define the set of multipliers Λp,x,v := { λ N C gp,x x gp,x λ = v }. It follows from construction of Γ in 1.2 and Lemma 2.1 that the metric subregularity of the mapping gp, C at x,0 ensures the normal cone representation N Γp x x gp,x N C gp,x, and therefore the inclusion v N Γp x = Λp,x,v /0 holds. Definition 3.1 bounded multiplier property. We say that the PARTIAL BOUNDED MULTIPLIER PROPERTY BMP with respect to x is satisfied for system 1.1 at the point p, x gphγ with modulus κ 0 if there are neighborhoods V of p and U of x such that 3.2 Λp,x,v κ v B R n /0 for all p V, x Γp U, and v N Γp x. The infimum over all such moduli κ is denoted by bmp x g,c p, x. To emphasize what is behind Robinson stability, consider an important particular case of constraint systems in nonlinear programming NLP described by smooth equalities and inequalities g i p,x = 0 for i = 1,...,l E and g i p,x 0 for i = l E + 1,..., l E + l I = l. Such systems can be represented in the form of 1.1 as follows: 3.3 gp,x C := {0} l E R l I. It has been well recognized in nonlinear programming that, given p, x gph Γ, the metric regularity of the mapping g p, C around p, x in the setting of 3.3 is equivalent to the partial Mangasarian-Fromovitz constraint qualification MFCQ with respect to x at p, x, which in turn ensures the uniform boundedness of Lagrange multipliers around this point. Then the robustness of metric regularity allows us to conclude that the partial MFCQ implies the validity of both partial MSCQ and BMP for 3.3 at p, x with some modulus κ > 0. Let us now recall another classical constraint qualification ensuring both partial MSCQ and BMP for 3.3. Denote E := {1,...,l E }, I := {l I + 1,...,l} and for any p,x feasible to 3.3 consider the index set I p,x := {i I g i p,x = 0} of active inequalities and then put I + λ := {i I λ i > 0} where λ = λ 1,...,λ l. It is said that the partial constant rank constraint qualification CRCQ with respect to x holds at p, x gphγ if there are neighborhoods V of p and U of x such that for every subset J E I p, x the family of partial gradients { x g i p,x i J} has the same rank on V U. Proposition 3.2 MSCQ and BMP follow from CRCQ. Given p, x gph Γ for 3.3, the partial CRCQ at p, x implies that both partial MSCQ and BMP with respect to x holds at this point. Proof. Implication CRCQ= MSCQ for the partial versions under consideration can be deduced from [13, Proposition 2.5]. Let us verify that CRCQ= BMP. Assuming the contrary, find sequences p k,x k gphγ p, x and v k B R n g x p k,x k N C gp k,x k for which v k x gp k,x k N C gp k,x k kb Rn as k N. Choose a subset E E such that { x g i p, x i E } is a base of the span of the gradient family { x g i p, x i E}. The imposed partial CRCQ tells us that for all k sufficiently large the set { x g i p k,x k i E } is also a base of the span of { x g i p k,x k i E}, and hence the set Λ k := { λ N C gpk,x k v k = x gp k,x k λ, λ i = 0 for i E \ E } 5

8 is a nonempty convex polyhedron having at least one extreme point. Let λ k denote an extreme point of Λ k meaning that the gradient family { x g i p k,x k i E I + λ k } is linearly independent. After passing to a subsequence if needed, suppose that the sequence λ k / λ k converges to some λ N C g p, x B R l. Since λ k > k and v k B R n, we obtain x gp k,x k λ k λ k = x k λ k < 1 k, k N, yielding x g p, x λ = 0. Since λi k = 0 for i E \E, we also have λ i = 0 for these indices, and so the family { x g i p, x i E I + λ} is linearly dependent. By I + λ I p, x due to λ N C g p, x and by λi k > 0 for each i I + λ when k is large, it follows that I + λ I + λ k. The partial CRCQ with respect to x at p, x ensures that the family { x g i p k,x k i E I + λ} is linearly dependent, and hence the family { x g i p k,x k i E I + λ k } is linearly dependent as well. This contradicts our choice of λ k and thus shows that the partial BMP with respect to x must hold at p, x. Following [14] and slightly adjusting the name, we say that x is a parametrically stable solution to 1.1 on P at p if for every neighborhood U of x there is some neighborhood V of p such that 3.4 Γp U /0 whenever p V. Now we are ready to establish relationships between Robinson stability of PCS 1.1 and the aforementioned properties and constraint qualifications. Theorem 3.3 first-order relationships for Robinson stability. Let p, x gph Γ in 1.2, and let κ 0. Consider the following statements for PCS 1.1 under the imposed standing assumptions: i Robinson stability holds for 1.1 at p, x. ii The point x is a parametrically stable solution to 1.1 on P at p, and the partial MSCQ together with the partial BMP are satisfied at p, x. Then i= ii with bmp x g,c p, x robg,c p, x. Conversely, if C is subamenable at g p, x, then ii= i and we have the the exact formula robg,c p, x = bmp x g,c p, x. Proof. To verify i= ii, find by i neighborhoods V and U such that the standing assumptions and the estimate 1.3 are satisfied with some modulus κ 0. Then the partial MSCQ with respect to x follows directly from definition 3.1. Furthermore, for every p,x V U gphγ the mapping M p : R n R l with M p ξ := gp,ξ C is metrically subregular at x,0 with the same modulus κ, and thus Lemma 2.1 tells us that for each p,x gphγ V U and each v N Γp x = N M 1 p 0 x there exists some λ v κb R l N C gp,x with v = x gp,x λ. This justifies 3.2 and the partial BMP with respect to x at p, x, and therefore bmp x g,c p, x robg,c p, x. To finish the proof of the claimed implication, it remains to show the parametric stability of x. Take any neighborhood Ũ of x and find the radius r > 0 with intb x;r Ũ. By the continuity of g there is a neighborhood Ṽ of p such that κ gp, x g p, x < r whenever p Ṽ. This yields dist x;γp κdist gp, x;c κ gp, x g p, x < r for all p V Ṽ, which shows therefore that /0 Γp intb x;r Γp Ũ for all p V Ṽ. Hence x is parametrically stable on P at p by definition 3.4, and we fully justify implication i= ii. To verify the converse implication ii= i, suppose that C is subamenable at g p, x. By ii take q, Q, and W according to Definition 2.2 and find neighborhoods V of p and U of x for which conditions 3.1, 3.2, and the standing assumptions are satisfied. Choosing κ > 0 so that 3.2 holds 6

9 and by shrinking W if necessary, suppose that q Q is metrically subregular with modulus κ C at every point z,0 with z q 1 Q W. Denote L := 2κ C 3 + x g p, x + 2, and let 0 < δ < min{1/κl,1} be arbitrarily fixed. Then choose some constant r z > 0 so that intb g p, x;2r z W and qz qz qz z z δ z z for all z,z intb g p, x;2r z. Shrinking U and V allows us to get x gp,x x g p, x δ and gp,x g p, x < r z for all p,x V U. Further, let r > 0 be such that B x;3r U and, by the parametric stability of x on P and by shrinking again V, we have Γp intb x;r /0 whenever p V. Fix now x, p intb x;r V and let ξ be a global solution to the optimization problem 3.5 minimize 1 2 ξ x 2 subject to ξ Γp. Such a global solution surely exists due to the closedness of Γp B x;3r. Then ξ x < 2r and hence ξ x < 3r yielding ξ intb x;3r U. This verifies the metric subregularity of gp, C at ξ,0. Applying now the necessary optimality condition in 3.5 from [18, Proposition 5.1] and then using Lemma 2.1 give us the inclusions ξ x N Γp ξ x gp, ξ N C gp, ξ, which show that Λp, ξ,x ξ /0, and thus there is a multiplier λ Λp, ξ,x ξ κ ξ x B R l by the imposed partial BMP. Moreover, we have gp, ξ intb g p, x;r z C W q 1 Q while concluding therefore that the mapping q Q is metrically subregular at gp, ξ,0. Using again Lemma 2.1 provides a multiplier µ N Q qgp, ξ λ κ C B R d with λ = qgp, ξ µ. Choose now z as a projection of the point gp,x on the set C. Since gp,x intbg p, x;r z and g p, x C, we get c g p, x < 2r z and thus λ, z gp, ξ = µ, qgp, ξ z gp, ξ µ,q z qgp, ξ + δ µ z gp, ξ. Using µ,q z qgp, ξ 0 by µ N Q qgp, ξ together with z gp, ξ z gp,x + gp,x gp, ξ 2 gp,x gp, ξ gives us the estimate Now taking into account the relationships gp,x gp, ξ x gp, ξ x ξ = λ, z gp, ξ 2κ C λ δ gp,x gp, ξ. 1 x g p, ξ +tx ξ x gp, ξ x ξ dt 0 x g p, ξ +tx ξ x gp, ξ x ξ dt x g p, ξ +tx ξ x g p, x + x gp, ξ x g p, x x ξ dt 2δ x ξ, 7

10 we obtain gp,x gp, ξ 2δ + x gp, ξ x ξ 3δ + x g p, x x ξ and x ξ 2 = x gp, ξ λ,x ξ λ,gp,x gp, ξ + λ gp,x gp, ξ x gp, ξ x ξ λ,gp,x z + λ, z gp, ξ + 2 λ δ x ξ λ dist gp,x;c + λ δ 2κ C 3δ + x g p, x + 2 x ξ κ x ξ distgp,x;c + δl x ξ due to 3.5 and δ < 1. Rearranging yields 1 δκl x ξ 2 κ x ξ dist qp,x;c, κ 1 δκl which implies that x ξ = distx;γp distgp,x;c and that the claimed Robinson stability holds with modulus κ/1 δ κl. This verifies implication ii= i. Finally, the arbitrary choice of δ > 0 close to zero allows us to conclude that the inequality robg,c p, x bmp x g,c p, x is satisfied. Remembering the opposite inequality derived above, we arrive at the equality robg,c p, x = bmp x g,c p, x and thus complete the proof of the theorem. Note that the results of Theorem 3.3 yield new formulas for calculating the exact bound infimum subreg of subregularity moduli of nonconvex mappings as follows. Corollary 3.4 calculating the exact subregularity bound. Let f : R n R l be continuously differentiable, and let C R l be subamenable at x f 1 C. If the mapping x Fx := f x C is metrically subregular at x,0, then the exact subregularity bound of F at x,0 is calculated by 3.6 subregf x,0 = limsup x f 1 C x sup inf { λ λ N C f x, f x λ = v } v N f 1 C x B R n = limsup inf { τ 0 N f 1 Cx B R n τ f x } N C f x BR l x f 1 C x = limsup inf { τ 0 N f 1 Cx B R n τd Fx,0B R l }, x f 1 C x where D Fx,0λ := f x λ if λ N C f x and D Fx,0λ := /0 otherwise. Proof. It follows from Theorem 3.3 when gp,x := f x. Indeed, in this case we have the relationships subregf x,0 = robg,c p, x = bmp x g,c p, x, where the latter quantity is calculated by using the normal cone representation for inverse images from Lemma 2.1 with taking into account the imposed subamenability of C and that the metric subregularity of F at x,0 implies this property for F at any x,0 with x F 1 0 close to x. The last formula in 3.6 corresponds to the result by Zheng and Ng [25, condition 3.6] obtained for convex-graph multifunctions, which in not the case of Corollary 3.4. Observe that the subregularity bound calculations in 3.6 are of a different type in comparison with known modulus estimates for subregularity see, e.g., Kruger [15] and the references therein, because 3.6 uses information at points x f 1 C near x while other formulas usually apply quantities at points x outside the set f 1 C. It seems to us that the subregularity modulus estimates of type [15] are restrictive for applications to Robinson stability interpreted as the uniform metric subregularity; see Section 1. Indeed, the latter property is robust for the class of perturbations under consideration while the usual subregularity is not. Since this issue is not detected by estimates of type [15], it restricts their robust applications. The next theorem is the main result of this section providing verifiable conditions for Robinson stability of PCS 1.1 involving the class of perturbation parameters P, p, gp, x under consideration. 8

11 For convenience of further applications we split the given system 1.1 into two parts i = 1,2: 3.7 gp,x = g 1 p,x,g 2 p,x C 1 C 2 = C,g i : P R n R l i,c i R l i,l 1 + l 2 = l in such a way that it is known in advance or easier to determine that RS holds for g 2 p,x C 2, while it is challenging to clarify this for the whole system gp,x C. It is particularly useful for the subsequent second-order analysis of RS and its applications to variational systems; see Sections 4,5. Since our parameter space P is general topological, we need a suitable differentiability notion for gp,x with respect to p. It can be done by using the following approximation scheme in the image space R l. Given any ζ : P R continuous at p, define the image derivative Im ζ D p g p, x of g in p at p, x as the closed cone generated by 0 and those v R l for which there is a sequence {p k } P with 0 < gp k, x g p, x < k 1, x gp k, x x g p, x < k 1, ζ p k ζ p < k 1, 3.8 gp k, x g p, x v = lim k gp k, x g p, x. If P is a metric space with metric ρ, the convergence of p k p can be ensured by letting ζ p := ρp, p. If P is a subset of a normed space and g is differentiable with respect to p at p, x, then we obviously have the inclusion with the contingent cone defined in 2.1 via the norm topology of P 3.9 p g p, xt P p Im ζ D p g p, x valid for any function ζ. Observe that inclusion 3.9 holds as equality with ζ p = p p if the operator p g p, x is injective while the inclusion may be strict otherwise. Theorem 3.5 first-order verification of Robinson stability for splitting systems. Given ζ : P R, assume that for every v Im ζ D p g p, x from 3.8 and every t k 0 there is u R n satisfying 3.10 [ lim inf dist g p, x +tk v + x g p, xu;c ] /t k = 0. k Suppose also that Robinson stability at p, x holds for the system g 2 p,x C 2 in 3.7 and that for every 0,0 v,u Im ζ D p g p, x R n with v + x g p, xu T C g p, x we have 3.11 [ λ = λ 1,λ 2 2 i=1 N Ci gi p, x;v i + x g i p, xu ], x g p, x λ = 0 = λ 1 = 0, where v = v 1,v 2 R l 1 R l 2. Then Robinson stability at p, x holds for the whole system 3.7. Proof. Assuming on the contrary that Robinson stability fails for 3.7 at p, x, for any κ > 0 and any neighborhoods U of x and V of p we find p,x V U such that 3.12 dist x;γp > κdist gp,x;c. Our goal is to show by several steps that 3.12 eventually contradicts the imposed assumption 3.11 by using first-order necessary optimality conditions in a certain nonsmooth optimization problem under the metric subregularity constraint qualification. First observe that, since Robinson stability at p, x holds for the system g 2 p,x C 2, we get R > 0 together with a neighborhood V of p and a positive constant κ 2 satisfying 3.13 dist x;γ 2 p κ 2 dist g 2 p,x;c 2 for all p V, x intb x;r, 9

12 where Γ 2 p := {x g 2 p,x C 2 }. The standing assumptions allow us to claim that for every p V the mapping gp, is continuously differentiable on intb x;r and then to construct a sequence of neighborhoods V k V k 1 with V 0 := V together with positive radii R k min{r/2,1/k} such that x gp,x x g p, x k 1 whenever p,x V k intb x;r k. Furthermore, for each k N there exist a neighborhood V k V k and a radius r k R k /4 for which { R k gp,x g p, x < min 4k 2 max{κ 2,1}, 1 } k as p,x V k intb x;r k. There is no loss of generality to suppose that ζ p ζ p 1/k for all p V k. According to 3.12 we select p k,x k V k intb x;r k satisfying dist x k ;Γp k > k + κ 2 dist gp k,x k ;C and by using 3.13 find x k Γ 2 p k such that x k x k κ 2 dist g 2 p k,x k ;C 2 κ2 g 2 p k,x k g 2 p, x < R k 4k 2 and hence x k x x k x k + x k x < R k /4k 2 +R k /4 R k /2. Since x gp k,x x g p, x + 1/k L := x g p, x + 1 for all x intb x;r k, we conclude that gp k, is Lipschitz continuous on intb x;r k with the modulus L defined above, and therefore dist g 1 p k, x k ;C 1 = dist gpk, x k ;C dist gp k,x k ;C + L x k x k Lκ 2 + 1dist gp k,x k ;C. Further, it follows that dist x k ;Γp k distx k ;Γp k x k x k, and thus we arrive at the estimates dist x k ;Γp k dist x k ;Γp k x k x k > k + κ 2 dist gp k,x k ;C 3.14 x k x k k dist gp k,x k ;C k Lκ dist gp k, x k ;C. Now for any fixed k N define the positive number 3.15 σ k := Lκ k 2 dist gp k, x k ;C and let x k,ȳ k be an optimal solution to the problem of minimizing 3.16 φ k x,y := y + σ k x x k 2 subject to g 1 p k,x + y C 1, g 2 p k,x C 2. Take ỹ k such that g 1 p k, x k +ỹ k C 1 and ỹ k = distg 1 p k, x k ;C 1 = distgp k, x k ;C. Since x k,ỹ k is a feasible solution to 3.16, we get 3.17 ȳ k φ k x k,ȳ k φ k x k,ỹ k = dist gp k, x k ;C. It follows that ȳ k 0 since otherwise x k Γp k while implying that Lκ dist xk k 2 ;Γp k 2 = σk dist x k ;Γp k 2 distgp k, x k ;C σ k x k x k 2 = φ k x k,0 dist gp k, x k ;C, 10

13 which contradicts the last inequality in We have furthermore by the choice of σ k in 3.15 that σ k x k x k 2 = Lκ k 2 dist gp k, x k ;C x k x k 2 φ k x k,ȳ k dist gp k, x k ;C yielding in turn the following estimates for all k N: 3.18 x k x k k Lκ dist gp k, x k ;C k dist gp k,x k ;C k gp k,x k g p, x < R k 4k. Thus we get x k x x k x k + x k x < R k /4k + R k /2 3 4 R k and gp k, x k g p, x gp k,x k g p, x + gp k, x k gp k,x k R k 4k 2 max{κ 2,1} + L x k x k + x k x k R k 1 4k k max{κ 2,1} + L + L 0 as k. k Letting now t k := x k x + gp k, x g p, x for k N and passing to a subsequence if necessary allows us to claim that the sequence of x k x,gp k, x g p, x/t k converges to some ū, v. Then ū, v 0,0 with v Im ζ D p g p, x by 3.8. It shows furthermore that gp k, x k g p, x +t k v + x g p, xū lim = lim k t k gp k, x k gpk, x + x g p, x x k x 3.19 = lim k t k [ x g p k, x + ξ x k x x g p, x ] x k xdξ 3.20 = lim k 1 0 t k k gp k, x k gp k, x +t k x g p, xū t k lim k x k x kt k = 0. Our next step is to prove that the solution ȳ k to the optimization problem 3.16 satisfies ȳ k lim = 0. k t k Assume on the contrary that there is ε > 0 such that after passing to some subsequence we have 3.21 ȳ k εt k, k N. For every k sufficiently large find jk k with t k u R jk such that lim k jk = and therefore gp k, x +t k u gp k, x +t k x g p, xu = 1 0 t k x gp k, x + ξt k u x g p, x udξ t k u jk, which implies by 3.10 that liminf k distgp k, x +t k u;c/t k = 0. After passing to a subsequence we can assume that distgp k, x +t k u;c < t k /k for all k N and then get the conditions x k x + t k u κ 2 t k /k and g 2 p k, x k C 2 for some x k. This tells us that dist gp k, x k ;C = dist g 1 p k, x k ;C Lκ2 t k /k, k N. Picking ŷ k with g 1 p k, x k + ŷ k C 1 and distg 1 p k, x k ;C 1 = ŷ k, get by φ k x k,ȳ k φ k x k,ŷ k that 3.22 ȳ k ŷ k + σ k 2 xk x k, x k x k + x k x k Lκ 2 t k /k + 2σ k x k x k x k x k + σ k x k x k 2 11

14 and then deduce from 3.18 the relationships Lκ σ k x k x k k 2 dist gp k, x k ;C k dist gpk, x k ;C 3.23 Lκ = Lκ k 0 as k. Using x k x k x k x + x k x t k u + κ 2 /k + 1 and combining 3.17 and 3.22 yield ȳ k min { 1 + Lκ 2 t k /k + 2σ k x k x k x k x k + σ k x k x k 2,dist gp k, x k ;C } t k k Lκ u + 2 κ 2 k min { σ k x k x k 2,dist gp k, x k ;C } t k k Lκ u + 2 κ 2 k σ k dist gp k, x k ;C x k x k t k k Lκ u + 3 κ 2 k + 4, which contradicts 3.21 and thus justifies Since gp k, x k + ȳ k,0 C, the results obtained in 3.19, 3.20 and definition 2.1 of the contingent cone allow us to conclude that 3.24 v + x g p, xū T C g p, x. Let us next show that the constraint mapping G k x,y := g 1 p k,x + y C 1,g 2 p k,x C 2 of R program 3.16 is metrically subregular at x k,y k,0. Indeed, pick x,y intb x k ; k 4max{κ 2 L,1} R l 1 and find ξ Γ 2 p k with ξ x κ 2 distg 2 p k,x;c 2. Since g 2 p k, x k C 2, it gives us ξ x κ 2 g 2 p k,x g 2 p k, x k κ 2 L x x k < R k 4, and consequently we have ξ intb x;r k and g 1 p k,ξ g 1 p k,x L ξ x. Consider now a solution ϑ R l 1 to the following optimization problem: minimize ϑ y subject to g 1 p k,ξ + ϑ C 1. Then 0 G k ξ, ϑ and, with ϑ C 1 g 1 p k,x+y such that ϑ = distg 1 p k,x+y;c 1, we get ϑ y g 1 p k,x + y + ϑ g 1 p k,ξ y L ξ x + ϑ. Thus it verifies the metric subregularity of G k at x k,y k,0 by ξ x + ϑ y dist g 1 p k,x + y;c 1 + L + 1 ξ x L + 1κ dist g 1 p k,x + y;c 1 + dist g2 p k,x;c 2 2 L + 1κ dist 0;G k x,y. Since metric subregularity is a constraint qualification MSCQ for NLPs, we apply to problem 3.16 the well-recognized necessary optimality conditions via limiting normals at x k,ȳ k cf. [18, 24]: there exist multipliers λ 1 k N C 1 g 1 p k, x k + ȳ k and λ 2 k N C 2 g 2 p k, x k such that σ k x k x k + x g 1 p k, x k λ 1 k + x g 2 p k, x k λ 2 k = 0 and ȳ k ȳ k + λ 1 k = 0. Remembering by Theorem 3.3 that Robinson stability of 3.7 implies the partial BMP with respect to x at the corresponding points, for each k sufficiently large we can choose the multiplier λ 2 k satisfying λ 2 k κ σ k x k x k + x g 1 p k, x k λ 1 k. 12

15 It follows from 3.23 and from λk 1 = 1 due to 3.25 that the sequence of λ k 1,λ k 2 is bounded and thus its subsequence converges to some λ = λ 1,λ 2 with λ 1 = 1. By taking 3.24 into account, we conclude that λ N C g p, x; v + x g p, xū = 2 i=1 N C i g i p, x; v i + x g i p, xū. Using finally 3.25 together with 3.23 tells us that x g p, x λ = x g 1 p, x λ 1 + x g 2 p, x λ 2 = 0, which contradicts the assumed condition 3.11 and thus completes the proof of the theorem. Next we present several consequences of Theorem 3.5 referring the reader to Sections 4 and 5 for further applications. Let us first formulate a version of the theorem without splitting system 1.1 into two subsystems, i.e., with l 2 = 0 in 3.7. Corollary 3.6 verification of Robinson stability without splitting. Let l 2 = 0 in the framework of Theorem 3.5, i.e., in addition to condition 3.10 the implication [ 3.26 λ NC g p, x;v + x g p, xu, x g p, x λ = 0 ] = λ = 0 holds for every 0,0 v,u Im ζ D p g p, x R n with v + x g p, xu T C g p, x. Then system 1.1 enjoys the Robinson stability property at p, x. It is easy to check that all the assumptions of Corollary 3.6 are satisfied under the metric regularity of the underlying mapping x g p,x C around the reference point x,0. Corollary 3.7 Robinson stability from metric regularity. If the mapping g p, C is metrically regular around x,0, then Robinson stability holds for system 1.1 at p, x. Proof. Recall that g p, C is metrically regular around x,0 if and only if the mapping u g p, x + x g p, xu C is metrically regular around 0,0, cf. [4, Corollary 3F.5]. Then for any v R l and any sequence t k 0 we can find u k with g p, x + t k v + x g p, xu k C and u k κ distg p, x + t k v C t k v whenever k N is sufficiently large. Hence the sequence {u k /t k } is bounded and its subsequence {u ki /t ki } converges to some u R n, and so dist g p, x +t k v + x g p, xu;c dist lim inf lim k t k i g p, x +tki v + x g p, x u k i t ki ;C t ki = 0. Thus assumption 3.10 is satisfied in this setting. The validity of implication 3.26 follows immediately from the metric regularity characterization in 2.6. To conclude this section, we discuss some important settings where the major assumptions of Theorem 3.5 are satisfied without imposing metric regularity of the mapping g p, C. Remark 3.8 assumption verification. i Let C be a union of finitely many convex polyhedra C 1,...,C m, and let the triple z,u,v fulfill the condition v + g p, xz,u T C g p, x. Then there is t > 0 with g p, x +t v + g p, xq,u C for all t [0, t]; in particular, condition 3.10 is satisfied for every sequence t k 0. Indeed, in this case any tangent direction w T C ȳ belongs to the contingent cone of one of the sets C i, and hence there exists t > 0 for which ȳ +tw C i C whenever t [0, t]. ii If C is convex, then the inclusion λ 1,λ 2 2 i=1 N C i g i p, x;v i + x g i p, xu appearing in 3.11 is equivalent to the conditions: v i + x g i p, xu T C gi p, x, λ i N C gi p, x, λ i,v i + x g i p, xu = 0, i = 1,2. This follows directly from [8, Lemma 2.1]. 13

16 4 Second-Order Conditions for Robinson Stability and Subregularity This section is devoted to deriving verifiable second-order conditions for Robinson stability of PCS, which has never been done in the literature. Our results below take into account the curvatures of the constraint set C and the parameter set P. Given a closed subset Ω R s, a point z Ω, a direction v T Ω z and a multiplier λ R s, we introduce the following directional upper curvature and lower curvature of Ω, respectively: { λ,v v } 4.1 χ Ω λ, z;v := limsup 0 < τ < ε, v v < ε, z + τv Ω, ε 0 τ { λ,v v χ Ω λ, z;v := liminf 0 < τ < ε, v v < ε,dist λ;n Ω z + τv } 4.2 < ε. ε 0 τ Note that both χ Ω λ, z;v and χ Ω λ, z;v can have values ± and that χ Ω λ, z;v = if λ N Ω z;v. Recall [1] that for Ω R s the second-order tangent set to Ω at z Ω in direction v T Ω z is TΩ z;v 2 Ω z τv := limsup. 1 τ 0 2 τ2 Proposition 4.1 upper curvature via second-order tangent set. We have the relationship χ Ω λ, z;v 1 } { λ,w 2 sup w TΩ z;v 2. Proof. Letting η := sup{ λ,w w TΩ 2 z;v}, observe that η = if T Ω 2 z;v = /0, and thus the statement is trivial in this case. When TΩ 2 z;v /0, for an arbitrarily fixed δ > 0 consider w T Ω 2 z;v satisfying λ,w η δ if η < and λ,w 1/δ if η =. Then we can find sequences τ k 0 and z k Ω z with 2vk v/τ k w, where v k := z k z/τ k. Since v k v, for any ε > 0 there is k N such that τ k < ε, v k v < ε, and 2 λ,v k v/τ k λ,w ε. By z k = z +t k v k Ω we conclude that { λ,v v sup 0 < τ < ε, v v ε, z + τv Ω} λ,v k v/τ k 1 λ,w ε, τ 2 which therefore completes the proof of the proposition. The next important result provides explicit evaluations for the upper and lower curvatures of sets under a certain subamenability. In fact, the first statement of the following theorem holds for strongly subamenable sets from Definition 2.2, while the second statement covers fully subamenable sets if the image set Q in the representation below is just one convex polyhedron instead of their finite unions. Theorem 4.2 upper and lower curvatures of set under subamenability. Let Ω := {z R s qz Q}, where q : R s R p is twice differentiable at z Ω, Q R p is a closed set, and where the mapping q Q is metrically subregular at z,0. Given v T Ω z, we have the assertions: i If Q is convex, λ N Ω z, and λ,v = 0, then χ Ω λ, z;v 1 { 2 sup 2 } 4.3 µ,q zv,v µ N Q q z, λ = q z µ. ii If λ N Ω z;v and Q is the union of finitely many convex polyhedra, then there is a vector µ N Q q z; q zv such that λ = q z µ and that 4.4 χ Ω λ, z;v = µ,q zv,v. 14

17 Proof. To verify i, consider sequences τ k 0 and v k v with z +τ k v k Ω, k N, such that λ,v k v /τ k χ Ω λ, z;v and take µ N Q z with q z µ = λ. Then q z + τ k v k = q z + τ k q zv k τ2 k 2 q zv,v + oτ 2 k and µ,q z + τ kv k q z 0 by the convexity of Q thus yielding λ,v k v = µ, q xv k 1 τ k τ k 2 2 µ,q zv,v + oτ2 k. This readily justifies 4.3 by definition 4.2 of the upper curvature. To proceed with the verification of ii, take sequences τ k 0, v k v and λ k λ as k with λ k N Ω z + τ k v k for all k N such that λ,v k v /τ k χ Ω λ, z;v. It follows from Lemma 2.1 by the assumed metric subregularity that there is κ > 0 such that for each k sufficiently large we can find µ k N Q q z + τ k v k κ λ k B R p with λ k = q z + τ k v k µ k. Hence the sequence of µ k is bounded and its subsequence converges to some µ N Q q z; q zv satisfying λ = q z µ. Then [7, Lemma 3.4] allows us to find d k Q with d k q z + τ k v k τ 3 k such that µ N Q d k. Remembering that Q is the union of the convex polyhedra Q 1,...,Q m having the representations Q i = {d R p a i j,d b i j, j = 1,...,m i } for i = 1,...,m, we get N Q d k = i J k N Qi d k with J k := { i {1,...,m} d k Q i }. Since J k /0 for each k N, there is a subsequence {k} and some index î such that î J k along this subsequence. By passing to a subsequence again no relabeling, we can suppose that the index sets Jîd k := { j aî j,d = bî j } reduces to a constant set J for all k N. Employing now the Generalized Farkas Lemma from [1, Proposition 2.201] gives us a constant β 0 such that for every k there are numbers ν k j 0 as j J for which µ k = aî j ν k j and j J ν k j β µ k. j J Thus the sequences {ν k j } for all j J are bounded, and the passage to a subsequence tells us that for every j J the sequence of ν k j converges to some ν j as k. Then ν j 0, µ = j J aî j ν j, and µ N Qîd k for all k. Hence µ,q z d k 0 and, since we also have µ N Qîq z, it follows that µ,q z d k = 0. Furthermore, taking into account the representations d k q z = q z + τ k v k q z + oτ k 2 = τ k q zv k τ2 k 2 q zv,v + oτ k 2 allows us finally to arrive at the relationships λ,v = µ, q zv = lim k µ,d k q z τ k = 0, τ 2 k λ,v k v = µ, q xv k = 1 τ k τ k 2 2 µ,q zv,v + oτ k 2 2, τ k which complete the proof of the theorem by recalling definition 4.2 of the lower curvature. The next theorem is the major result of this section. For simplicity we restrict ourselves to the case where the parameter space P is finite-dimensional. 15

18 Theorem 4.3 second-order verification of Robinson stability. Consider the splitting system 3.7, where P R m in our standing assumptions. Suppose also that for every w T P p and for every sequence τ k 0 there exists u R n with 4.5 lim inf k dist g p, x + τ k g p, xw,u;c /τ k = 0, that Robinson stability holds at p, x for the system g 2 p,x C 2 in 3.7, and that for every triple w,u,λ R m R n R l satisfying the conditions 4.6 0,0 w,u, w T P p, g p, xw,u T C g p, x, 4.7 λ N C g p, x; g p, xw,u, x g p, x λ = 0, λ,g p, xw,u,w,u + χ P p g p, x λ, p;w 4.8 χ C λ,g p, x; g p, xw,u we have λ 1 = 0, where λ = λ 1,λ 2 R l 1 R l 2. Then the Robinson stability property at p, x also holds for the whole system gp,x C in 3.7. Proof. Assuming on the contrary that Robinson stability fails at p, x for the system gp,x C in 3.7 and taking ζ p := p p, we proceed as in the proof of Theorem 3.5 and find sequences x k, y k, and λ k := λ 1 k,λ 2 k such that the limit λ = λ 1,λ 2 = lim k λ k satisfies the relationships λ k N C c k, λ 1 k = y k y k, xg p, x λ = 0, λ 1 = 1, where c k := gp k, x k +y k,0 C. For each k N choosing R k as above, define 0 < τ k := p k p + x k x 1 k R k < 2 k and suppose by passing to a subsequence that the sequence {p k p, x k x/τ k } converges to some z := w,ũ 0,0. Let us show that the triple w,ũ,λ satisfies conditions , which contradicts the assumption of the theorem due to λ 1 0. We obviously have w T P p and lim k gp k, x g p, x/τ k = p g p, x w =: ṽ together with gp k, x k g p, x lim = g p, x z = ṽ + x g p, xũ. k τ k Now we proceed as in the proof of Theorem 3.5 to verify Using the same arguments as in Theorem 3.5 with replacing v, ū, and t k by ṽ,ũ, and τ k, respectively, implies that y k /τ k 0. Further, by setting s k := c k g p, x/τ k we obtain lim s gpk, x k g p, x k = lim k k τ k + y k,0 = g p, x z, τ k which gives us g p, x z T C g p, x and λ N C g p, x; g p, x z. Hence 4.6 and 4.7 are fulfilled, and it remains to justify 4.8. By passing to a subsequence, suppose the validity of 4.9 λ,c k g p, x τ k g p, x z τ 2 k = λ,s k g p, x z 1 χ τ C λ,g p, x; g p, x z k k for all k N. By 1 = λ 1 k,y k/ y k we also get 1 2 λ 1,y k / y k = λ,c k gp k, x k y k and λ,c k gp k, x k 0 16

19 when k is sufficiently large. Hence 4.9 yields the estimate λ,gp k, x k g p, x τ k g p, x z τ 2 k which implies, by passing to a subsequence if necessary, that χ C λ,g p, x; g p, x z 1 k, λ,g p, x z, z + λ, g p, x x k x τ k ũ, p k p τ k w 2 τk 2 χ C λ,g p, x; g p, x z k for all k. Setting w k := p k p/τ k as k N, we have λ, p g p, xp k p τ k w τ 2 k = λ, pg p, xw k w τ k = pg p, x λ,w k w τ k χ P p g p, x λ,g p, x; w + 1 k when k is sufficiently large. Taking into account that λ, g p, x x k x τ k ũ = x g p, x λ, x k x τ k ũ = 0, this gives us the estimate λ,g p, x z, z + χ P p g p, x λ,g p, x; w 3 χ C λ,g p, x; g p, x z k. It shows by passing to the limit that the triple w,ũ,λ satisfies 4.8. This contradicts the assumptions of the theorem and thus completes the proof. Let us now present a consequence of Theorem 4.3 for an important special case of PCS 1.1, where C is the union of finitely many convex polyhedra, and where the parameter space 4.10 P := { p R m hi p 0 for i = 1,...,l P } is described by smooth functions h = h 1,...,h lp : R m R l P. Suppose for simplicity that h p = 0. Corollary 4.4 second-order conditions for Robinson stability of PCS defined by unions of convex polyhedra. Consider PCS 1.1, where P R m is defined by 4.10, and where the mappings g : R m R n R l and h : R m R l P are twice differentiable at p, x gphγ and p P, respectively. Assume that the set C is the union of finitely many convex polyhedra and that the mapping h R l P is metrically subregular at p,0. Suppose also that for every w R m with h pw 0 there is u R n with g p, xw,u T C g p, x and that for every triple w,u,λ satisfying ,0 w,u, h pw 0, g p, xw,u T C g p, x, λ N C g p, x; g p, xw,u, x g p, x λ = 0 there exists µ R l P + such that p g p, x λ = h p µ and λ,g p, xw,u,w,u 2 µ,h pw,w < 0. Then the Robinson stability property holds for system 1.1 at the point p, x. 17

20 Proof. To verify this result, we apply Theorem 4.3 with l 2 = 0 and l 1 = l. Since T P p {w h pw 0}, the imposed assumptions imply that for every w T P p there is u R n with g p, xw,u T C g p, x and that condition 4.5 holds because C is the union of finitely many convex polyhedra; see Remark 3.8i. In order to apply Theorem 4.3, it now suffices to show that there is no triple w,u,λ fulfilling with λ 0. To proceed, consider any triple w, u, λ satisfying conditions 4.6 and 4.8 with λ 0. Then w,u,λ also satisfies 4.11 and 4.12, and thus there is µ R l P + fulfilling p g p, x λ = h p µ and From [8, Lemma 2.1] we deduce that 0 = λ, g p, xq,u, which implies together with x g p, x λ = 0 that 0 = λ, p g p, xw = p g p, x λ,w = h p µ,w. Furthermore, it follows from Theorem 4.2i that χ P p g p, x λ, p;w µ,h pw,w. Applying now Theorem 4.2ii with Ω = Q = C and qz = z yields χ C λ,g p, x; g p, xw,u = 0, and then from 4.13 we obtain the relationships λ,g p, xw,u,w,u + χ P p g p, x λ, p;w λ,g p, xw,u,w,u µ,h pw,w < 0 = χ C λ,g p, x; g p, xw,u, which show that conditions and λ 0 cannot hold simultaneously. The following instructive example illustrates the efficiency of the obtained first-order and secondorder verification conditions for Robinson stability of a general class of PCS with the splitting structure 3.7. In this example we employ the second-order conditions from Corollary 4.4 to verify Robinson stability of the system g 2 p,x C 2 in 3.7 and then deduce Robinson stability of the whole system gp,x C in 3.7 from the first-order Theorem 3.5. Example 4.5 implementation of the verification procedure for Robinson stability. Define the functions f i : R 3 R 2 R for i = 1,2,3 by f 1 p,x := x 2 1/2x 2 1 p 1 + p 2 x 2, f 2 p,x := x 2 1/2x 2 1 p 2 + p 1 x 1, f 3 p,x := x 1 + x p3 and consider the system of parameterized nonlinear inequalities f i p,x 0 with the parameter space P := { p R 2 hp := p1 p 2 + 3/2p } R and the reference pair p, x = 0,0. This system can be written as a PCS 1.1 with g = f 1, f 2, f 3 and C = R 3. It is convenient to represent 1.1 in the splitting form 3.7 with g 1 := f 3, g 2 := f 1, f 2, C 1 = R, and C 2 = R 2 for which the results of Theorem 3.5 and Corollary 4.4 can be applied. To proceed, consider the mapping g: R 2 R 2 R 2 defined by gp 1, p 2,x := g 2 p 1, p 2,0,x and the system gp,x C with the parameter space P = h 1 R and the reference pair p, x = 0,0. The mapping h R is metrically subregular at p,0 since it is actually metrically regular around this point due to the validity of MFCQ therein. It is easy to see furthermore that for every w R 2 satisfying h pw = w 1 w 2 0 the system u2 w g p, xw, u = 1 T u 2 w R 2 2 g p, x = R 2 18

21 has the unique solution u = 0,w 2. By Remark 3.8ii the conditions in 4.11 and 4.12 amount to w 1,w 2,u 1,u 2 0,0,0,0, w 1 w 2 0, u 2 w 1 0, u 2 w 2 0, 0,0 λ 1,λ 2 R 2 +, λ 1 u 2 w 1 = λ 2 u 2 w 2 = 0, λ 1 0, 1 + λ 2 0,1 = 0 yielding in turn λ 1 = λ 2 > 0 and u 2 = w 2 = w 1. Hence for every triple w,u,λ satisfying 4.11 and 4.12 we have p g p, x λ = h p µ together with the equalities µ = λ 1 and 2 λ, g p, xw,u,w,u 2 µ,h pw,w = λ 1 u w 2 u 2 + λ 2 u w 1 u 1 3µw 2 1 = λ 1 2u 2 1 2w 1 u 1 w 2 1 < 0. Thus Corollary 4.4 tells us that Robinson stability holds for the system gp,x R 2 at p, x. Since g 2 p 1, p 2, p 3,x = gp 1, p 2,x for all p 1, p 2, p 3 R 3 and x R 2, it follows that Robinson stability holds also for the system g 2 p,x R 2 at the initial pair p, x. Now we apply Theorem 3.5 with ζ = to system 3.7 splitting above. Since for every p P we have gp, x g p, x = p 1, p 2, p 3 and p 1 + p p2 1, it follows that Im ζ D p g p, x = { v R 3 v 1 + v 2 0 }. Then for every v Im ζ D p g p, x the element u = v 3,v 2 satisfies the conditions v + x g p, xu = v 1 v 2,0,0 T C g p, x, and thus 3.10 holds because C is polyhedral; see Remark 3.8i. By observing that x g p, x λ = λ 3,λ 1 +λ 2 = 0 yields λ 1 = λ 3 = 0, we deduce from Theorem 3.5 that the Robinson stability property is fulfilled for system 3.7 at p, x. Note that MFCQ fails to hold for the system g p, 0 at x. The obtained results on Robinson stability in Theorem 4.3 allow us to derive new second-order conditions for metric subregularity of constraint systems. Earlier results of this type have been known only in some particular settings: in the case where the constraint set C is the union of finitely many convex polyhedra [8] and for subdifferential systems that can be written in a constraint form [5]. Corollary 4.6 second-order conditions for metric subregularity of constraint systems. Consider the constraint system gx C, where C is an arbitrary closed set, and where g : R n R l is twice differentiable at x g 1 C. Assume that for every pair u,λ satisfying 0 u, g xu T C g p, x, λ N C g x; g xu, g xu λ = 0, λ,g xu,u χ C λ,g x; g xu we have λ = 0. Then the mapping g C is metrically subregular at x,0. Proof. Follows directly from Theorem 4.3 with gp,x = gx. 5 Applications to Parametric Variational Systems In this section we first show that the obtained results on Robinson stability of PCS 1.1 allow us to establish new verifiable conditions for robust Lipschitzian stability of their solution maps 1.2. By the latter we understand, in the case where P is a metric space equipped with the metric ρ,, the 19