Stability of Singular Solutions of Nonlinear Equations with Restricted Smoothness Assumptions

This work is concerned with conditions ensuring stability of a given solution of a system of nonlinear equations with respect to large (not asymptotically thin) classes of right-hand side perturbations. Our main focus is on those solutions that are in a sense singular, and hence, their stability properties are not guaranteed by “standard” inverse function-type theorems. In the twice differentiable case, these issues have received some attention in the existing literature. Moreover, a few results in this direction are known in the case when the first derivative is merely B-differentiable. Here, we further elaborate on a similar setting, but the main attention is paid to the case of piecewise smooth equations. Specifically, we study the effect of singularity of a solution for some active smooth selection on the overall stability properties, and we provide sufficient conditions ensuring the needed stability properties in the cases when such smooth selections may exist. Finally, an application to a piecewise smooth reformulation of complementarity problems is given.


Introduction
In this paper, we are concerned with stability issues for a given solution of a finite system of nonlinear equations in a finite number of variables. When the system is smooth, and the solution in question is nonsingular, by which we mean that the Jacobian of the system at this solution has full row rank, those stability issues are addressed in full by classical covering theorems; see [13] for a recent exposition of these results and their far-reaching extensions. In particular, such solutions are stable subject to any sufficiently small right-hand side perturbations, with the distance to solutions of the perturbed system being of the order of perturbation.
Elementary examples demonstrate that for singular solutions (and in particular for nonisolated solutions when the number of equations coincides with the number of variables), one cannot expect stability with respect to right-hand side perturbations restricted only by their size. Nevertheless, a natural question in this context is about conditions ensuring stability of a given solution with respect to large (not asymptotically thin) classes of right-hand side perturbations. In the twice differentiable case, these issues have received some attention in the existing literature: see, e.g., [20], where stability properties were established for critical solutions, a special type of singular solutions. Moreover, some stability results of this kind are known even in the absence of twice differentiability, in the case when the first derivative is merely B-differentiable [22]. Here, we further elaborate on the latter setting as well, but the main attention is paid to the case of piecewise smooth equations. Specifically, we study the effect of singularity of a solution for some active smooth selection of a piecewise smooth mapping on the overall stability properties for the equation in question. We provide sufficient conditions ensuring the desired stability properties with respect to large classes of perturbations in the cases when such smooth selections may exist.
In Sect. 2, we start with the problem setting and some preliminaries. Among others, we develop the needed directional tools for dealing with active smooth selections and a stability result for constrained equations in Theorem 2.1. The latter is not concerned with any singularity of solutions; its specificity is in restricted smoothness assumptions appropriate for the use of this result in the subsequent analysis. Section 3 contains a discussion of relations between error bounds, criticality of singular solutions for active smooth selections, and stability issues for piecewise smooth equations. In particular, employing Theorem 2.1, we specify some cases when nonsingularity of a solution with respect to some active smooth selection ensures stability with respect to large classes of perturbations. In Sect. 4, we develop stability properties of singular solutions for mappings whose first derivative fulfills some additional smoothness requirements, while Sect. 5 extends these results to the piecewise smooth case. An application of the obtained sufficient conditions for stability to piecewise smooth reformulations of complementarity problems is given in Sect. 6. Some words about our notation and terminology. By I, we denote the identity matrix of a size always clear from the context. For a given index set J , we write u J for the subvector of a vector u, with components u j , j ∈ J . Let ker M and imM stand for the null space and the range space of a matrix (linear operator) M, respectively. By coneU , we denote the conic hull of a set U ⊂ R p , i.e., the smallest convex cone containing U . A set U is called starlike with respect to a pointū ∈ U if tu + (1 − t)ū ∈ U for all u ∈ U and all t ∈ [0, 1]. For such a set, v ∈ R p is referred to as an excluded direction ifū + tv / ∈ U for all t > 0. Moreover, let the inner product ·, · and the norm · be Euclidean. We will use the same symbol · for a consistent matrix norm. We further define the open ball B(ū, ε) := {u ∈ R p | u −ū < ε} centered atū ∈ R p with radius ε > 0. For some fixedv ∈ R p and δ > 0, we define the sets In particular, C δ (v) is a closed convex cone, while the set K ε, δ (ū,v) is star-shaped atū. It results from shifting C δ (v) byū and by intersecting the shifted set with the ball B(ū, ε). Observe that C δ (0) = R p and K ε, δ (ū, 0) = B(ū, ε). Let dist(u, U ) := inf v∈U u − v stand for the distance from u to U ⊂ R p (with the convention that dist(u, ∅) := +∞), and The directional derivative of a mapping Φ : R p → R q atū ∈ R p in a direction v ∈ R p is understood in a standard way:

Problem Setting and Preliminaries
We consider the equation with a mapping Φ : R p → R q assumed to be piecewise smooth near a pointū ∈ R p . By this, we mean that there exists a finite collection of smooth (nearū) selection and Φ is continuous nearū. Here, a mapping is called smooth if it is continuously differentiable. Some results presented below will require stronger smoothness assumptions on the selection mappings. That said, one new feature of this work is that we would want to avoid invoking second derivatives of selection mappings, thus covering, in particular, the equations with merely locally Lipschitzian and directionally differentiable first derivatives [21,22,25]. In general, the collection {Φ 1 (u), . . . , Φ s (u)} of smooth selections associated with a piecewise smooth mapping Φ, and even the number s of these selections, is not uniquely defined. However, from this point on, we will assume that this collection is fixed. In other words, every piecewise smooth mapping is characterized by a given collection of smooth selections, and by a given rule defining selections active at every point.
Our main interest in this work is in characterization of those w ∈ R q for which one can guarantee that the equation has a solution close to a given solutionū of the unperturbed equation (3). When dealing with piecewise smooth mappings, the key role is played by the set that, for a given u ∈ R p , stands for the set of indices of all selection mappings active at u. By the continuity of Φ and its smooth selection mappings nearū, the set-valued mapping A(·) is evidently outer semicontinuous atū, i.e., A(u) ⊂ A(ū) holds for any u ∈ R p close enough toū. According to [15,Lemma 4.6.1] and under the stated assumptions, Φ is Lipschitzcontinuous nearū and B-differentiable atū. The latter concept stems from [28] and implies that Φ is directionally differentiable atū in any direction v ∈ R p and (see [15, as v → 0. Moreover, Φ (ū; ·) is everywhere continuous, and where (Φ j ) (ū) denotes the usual Jacobian of the smooth mapping Φ j atū, for j ∈ {1, . . . , s}.
Many considerations below will be directional by nature. To that end, for anȳ u ∈ R p and v ∈ R p , we define the index set which is nonempty by (7). Observe that A(ū, 0) = A(ū).
The following facts can be considered as generalizations of the results in [16, Proposition 2.1, Corollary 2.1]; they will serve as our key tools for dealing with piecewise smoothness.

Corollary 2.1 Under the assumptions of Proposition
Then, there exist ε > 0 and δ > 0 such that Some particular cases of Proposition 2.1 and Corollary 2.1 were used in [16] in order to demonstrate local attraction of piecewise Newton-type methods to solutions that are critical with respect to some active smooth selections. The idea here is to employ Proposition 2.1 and Corollary 2.1 for establishing stability properties of a solutionū of (3) from the corresponding properties for smooth selections Φ j , j ∈ A(ū,v), for a givenv.
We complete this section with a covering result that is not directly concerned with piecewise smoothness, but will be used below for deriving some conclusions concerning stability of (in some sense) nonsingular solutions of a piecewise smooth equation. The main specificity of this result is that the smoothness assumptions in it are still rather weak: Φ is supposed to be differentiable only at the solutionū of interest, and only with respect to a given set U ∈ R p containingū. According to the terminology in [6,Appendix II], this property means the existence of a linear operator J : R p → R q such that as v → 0 in such a way thatū + v stays in U . An interesting observation (not used in this paper though) is that this kind of differentiability allows Φ to be defined only on U rather than on an entire neighborhood ofū. If U is convex, and intU = ∅, there can be no more than one linear operator J with the specified properties, in which case we denote it in a standard way, as Φ (ū). This is the only reason for the assumption intU = ∅ in the following theorem. Theorem 2.1 Let U ⊂ R p be a convex set with intU = ∅, and let a continuous mapping Φ : U → R q be differentiable at some solutionū ∈ U of Eq. (3) with respect to U .
The presented result is strongly related to the one in [3, Thereom 3.1] where, however, the smoothness requirements are also stronger than those employed here.
Finally, we mention that Theorem 2.1 also generalizes the finite-dimensional result in [17,Theorem G], where differentiability is assumed only at a given point, but with respect to the entire space, and the derivative is assumed to be surjective, again allowing to takew = 0.
We complete the section by mentioning that if intΦ (ū)(U −ū) = ∅, Theorem 2.1 is vacuous (says nothing). This is the case when the first-order analysis does not allow to establish covering of non-thin sets. Then, some other tools relying on higher-order (generalized) derivatives have to be involved, as it is done, for example, in Sects. 4 and 5.

Error Bounds, Criticality, and Stability Issues
According to [20,Theorem 2], in the case of a smooth mapping Φ, criticality of a solutionū of (3) can be understood as the violation of the local Lipschitzian error bound as u ∈ R p tends toū. Moreover, as demonstrated in [20,Proposition1 and Theorem 4], critical solutions possess special stability properties that are necessarily missing for noncritical singular solutions.
However, even in the simplest nonsmooth cases (see Example 3.1), and even for the particular case of piecewise smooth reformulations of complementarity systems (see [4]), solutions violating strict complementarity (i.e., those with A(ū) not being a singleton) can be stable subject to wide classes of perturbations, even if the local Lipschitzian error bound is satisfied. This suggests that the absence of the error bound is perhaps not an adequate understanding of criticality when A(ū) is not a singleton.
Moreover, if we think of critical solutions as those possessing special stability properties, and/or attraction properties for Newton-type methods [19], perhaps the criticality concept should be left for smooth equations altogether, while for piecewise smooth ones, its impact should be regarded as arising through active smooth selections. To begin with, the discussion below suggests to consider a solution of the piecewise smooth equation as singular if it is singular for at least one active selection.
Suppose that for some sequences {w k } ⊂ R q \{0} and {u k } ⊂ R p it holds that {w k } → 0, {u k } →ū, and u k solves (4) with w = w k for all k. Then, there exists j ∈ A(ū) such that u k solves the equation with w = w k for infinitely many k, and passing to subsequences, we may suppose that it holds for all k. For every k, let u k stand for a projection of u k onto (Φ j ) −1 (0). Then, assuming that Φ j is strictly differentiable atū with respect to (Φ j ) −1 (0) (as defined in [20]), and thatū is a noncritical solution of the equation similarly to [20, Proposition 1] we find that the sequence {(w k , u k − u k )/ w k } is bounded, and any accumulation point (d, v) of this sequence satisfies d = 0 and the equality Ifū is a singular solution of Eq. (16), i.e., rank(Φ j ) (ū) < q, this may only hold for special d = 0, namely, for those in im(Φ j ) (ū). Therefore, an active smooth selection with the specified properties can give rise to stability of the solutionū of Eq. (3) subject to the right-hand side perturbations approaching 0 tangentially to the thin subset im(Φ j ) (ū) of R q only. In particular, ifū is a singular noncritical solution of Eq. (16) for all j ∈ A(ū), then the solutionū can be stable subject to special right-hand side perturbations of Eq. (3) only, forming an asymptotically thin subset of R q . Therefore, when attempting to characterize the lack of stability, the case of interest is whenū is a singular solution of (16) for at least one j ∈ A(ū). It is evident that, say, this always holds ifū is a nonisolated solution of (3) and p = q. However, even in the presence of such j,ū can be a nonsingular (and hence noncritical) solution of (16) for some other j ∈ A(ū), and the existence of such selections may of course have a positive impact on stability properties.
One may ask whether the existence of j ∈ A(ū) such thatū is a nonsingular (and hence noncritical) solution of (16) necessarily implies stability with respect to large classes of perturbations. Without further assumptions on a smooth selection in question, the answer is evidently negative, as this selection can be inactive at any nearby point, and hence, locally redundant, i.e., can be removed from the list of smooth selections without changing Φ nearū.
Moreover, even in the absence of redundant selections, the existence of a "nonsingular" active selection may have no strong positive effect on stability properties.
This Φ is piecewise smooth, with the specified smooth selection functions Φ 1 , Φ 2 , and Φ 3 . Any solutionū of (3) , w belongs to a set that is asymptotically thin at 0.
One case when "nonsingular" active selection gives rise to stability with respect to large classes of perturbations is specified in the following result.
Proof Without loss of generality, we may suppose that ε > 0 and δ > 0 are taken as in the assertion of Corollary 2.1 (as the smaller are those quantities, the more restrictive is the assertions of the proposition being proven). Then, (10) The needed result now follows, e.g., from [4, Corollary 2.1, Remark 2.1] applied with Φ := Φ j and K := C δ (v), or from Theorem 2.1, applied with the same Φ, with U := K ε, δ (ū,v), and with Then,ū = 0 is the unique solution of Eq. (3), with A(0) = {1, 2}. Moreover, this u is the unique nonsingular solution of Eq. (16) with j = 1. Thus, Proposition 3.1 is applicable withv = −1 and j = 1 and yields the existence of a solution u(w) of Eq. (4) with w ≤ 0 close enough to 0, satisfying the estimate (11). And indeed, for every w ≤ 0, Eq. (4) has the unique solution u(w) = w. In other words, the "nonsingular" selection Φ 1 allows to cover all w ≤ 0, and with the specified estimate.
Observe that at the same time, the "singular" selection Φ 2 allows to cover all w ∈ [0, 1], and the latter is justified (at least for w ≥ 0 small enough) by Proposition 5.1 (b), applied withv = 1 and j = 2.
Proposition 3.1 is also applicable at the solutionū = 0 in Example 3.1, withv = −1 and j = 1. At the same time, it is not applicable at the solutionū = 0 in Example 3.2. Indeed, for directionsv withv 1 , and not only it is not a singleton, but it also involves a "singular" selection Φ 3 .
We next discuss the case when A(ū, ·) is not necessarily a singleton. From continuity of Φ (ū; ·) it follows that the set-valued mapping A(ū, ·) is everywhere outer semicontinuous, i.e., A(ū, v) ⊂ A(ū,v) holds for any v ∈ R p close enough to a givenv ∈ R p .
The argument below is related to the one used in [16, Section 2.3] for somewhat different purposes. Suppose that there exist j 1 , Then, for any real t close enough to 0, it holds that A(ū,v), implying, in particular, that A(ū,v + t v) cannot contain both indices j 1 and j 2 simultaneously. Continuing this procedure withv replaced bȳ v + t v, we end up with somev such that either A(ū,v) is a singleton, or (Φ j ) (ū) coincide with the same matrix J for all j ∈ A(ū,v). In the latter case, by (1)-(2), (6), and by outer semicontinuity of A(ū, ·) atv, we conclude that for any ε > 0 and a sufficiently small δ > 0, the mapping Φ is differentiable atū with respect to the set K ε, δ (ū,v), with the derivative Φ (ū) = J . Moreover, thisv can be taken arbitrarily close to the original one, and if, say, for the originalv it holds that rank(Φ j ) (ū) = q for all j ∈ A(ū,v), then in the former case, the assumptions of Proposition 3.1 are satisfied with the newly constructedv and some j , while in the latter case, Φ (ū) is nonsingular, and then Theorem 2.1 is applicable with U := K ε, δ (ū,v), and with w = Φ (ū)v, yielding the same assertion as the one of Proposition 3.1. Therefore, the following generalization of Proposition 3.1 holds. Proposition 3.2 Let Φ : R p → R q be piecewise smooth near a given solutionū of (3). Letv ∈ R p be such that for some q × p matrix J it holds that rankJ = q and (Φ j ) (ū) = J for all j ∈ A(ū,v).
Considerations above lead, in particular, to the following conclusion: ifū is a nonsingular solution of (16) for all j ∈ A(ū), then it is stable subject to large classes of perturbations. If p = q, the assumption thatū is a nonsingular solution of (16) for all j ∈ A(ū) (combined with some additional requirements, including the requirement that the signs of all determinants of (Φ j ) (ū), j ∈ A(ū), are the same) appears in [15,Theorem 4.6.5 (b)] providing necessary and sufficient conditions for Φ to be a local Lipschitzian homeomorphism. However, here we are interested in much weaker covering properties, not concerned with local injectivity of Φ (e.g., like for Φ(u) = |u|).

Stability of Singular Solutions in the Case of B-Differentiable Derivatives
In this section, we consider a single smooth selection. To that end, we skip the index j ∈ {1, . . . , s} and consider the mapping Φ satisfying the following assumptions for a givenū ∈ R p : Assumption 1 Φ is differentiable nearū, with its Jacobian Φ being continuous atū.
Assumption 2 Π is the projector in R q onto some complementary linear subspace of imΦ (ū), parallel to imΦ (ū).

Assumption 3
The mapping ΠΦ is Lipschitz-continuous nearū and directionally differentiable atū in every direction.
If we were to restrict ourselves in Sect. 5 to piecewise smooth mappings with twice differentiable smooth selections, then instead of Theorems 4.1 and 4.2 presented in the current section, we might refer in Sect. 5 to some earlier results of this kind known for the twice differentiable case [8,20]. However, the setting adopted here allows for more generality, and Theorem 4.2 is a new result of independent interest and importance.
Observe that Assumption 3 certainly holds if Φ is itself Lipschitz-continuous near u and directionally differentiable atū in every direction. As demonstrated in [29], Assumption 3 implies that the mapping ΠΦ is B-differentiable atū. Taking into account Assumption 2, the latter means that For any v ∈ R p , define the matrix (linear operator) Following [21,22], we will be saying that Φ is 2-regular atū in the directionv ∈ R p if rankΨ (ū, Π;v) = q.
One can easily see that 2-regularity is indeed a directional condition, i.e., if it holds with somev, then it also holds withv replaced by tv for every t > 0. Observe that condition (18) holds with anyv (includingv = 0) provided the regularity condition rankΦ (ū) = q (19) holds. However, here we are mostly interested in the cases whenū is a singular solution of Eq. (3), by which in the setting of this section we mean precisely that (19) is violated.
In the twice differentiable case, the 2-regularity construction dates back at least to [7]; see also [1] for a systematic use of this construction in nonlinear analysis and optimization theory.
As forū = 0, we have that Φ (0) = 0, implying, in particular, that (Φ ) (0; v)v = 0 for, say, v = (1, 0) or v = (0, 1), and Φ is 2-regular at 0 in any such direction v. Therefore, Theorem 4.1 is applicable, ensuring that for every w close enough to zero, Eq. (4) has a solution whose distance to 0 is estimated from above as O(|w| 1/2 ) as w → 0. And indeed, for every w, Eq. (4) has a solution Observe that the assumptions of Theorem 4.1 are only sufficient but not necessary for its assertion to hold. To see this, consider, e.g., p = 2, q = 1, Φ(u) = u 1 u 2 , and u = 0. Then, Φ is not 2-regular atū in the directionv = 0, while the assertion of Theorem 4.1 is valid with thisv.

Remark 4.1
It can be seen from the proof of Theorem 4.2 that constant N > 0 in it can actually be chosen independently of ε > 0 and δ > 0. Observe that this is not true for Theorem 4.1, where it may be needed to infinitely increase N as δ → 0. This can be demonstrated by considering again p = 2, q = 1, Φ(u) = u 1 u 2 , andū = 0, but withv = (±1, 0) orv = (0, ±1).

Remark 4.2
Ifū is a nonsingular solution, i.e., the regularity condition (19) holds, then necessarily Π = 0, and Theorem 4.1 is applicable with anyv ∈ ker Φ (ū), includinḡ v = 0, with the estimate (20) taking the form This gives the classical covering (hemiregularity, semiregularity) result for regular mappings, with the additional restriction of solutions to K ε, δ (ū,v) (making sense whenv = 0). The same result follows, for instance, from any of Propositions 3.1 or 3.2, applied with A(ū) being a singleton. Observe that Theorem 4.2 is not applicable in the nonsingular case.
The other extreme case is when Φ (ū) = 0 (as in both Examples 4.1 and 4.2). Then, necessarily Π = I, and the estimate (20) in Theorem 4.1 takes the form while the estimate (21) in Theorem 4.2 takes the form The intermediate cases of singularity (when (19) is violated but Φ (ū) = 0) give freedom in choosing a complementary subspace in Assumption 2, and hence, in choosing Π . Observe that different Π may give rise to different directions (Π Φ ) (ū;v)v in Theorem 4.2, and for each of these directions (21) holds with some N > 0, ε > 0, and δ > 0, also possibly depending on Π .
That said, if we strengthen Assumption 3 by assuming that Φ is itself directionally differentiable atū in every direction, then the norm of (Π Φ ) (ū;v)v = Π(Φ ) (ū;v)v is separated from zero by its value for Π being the orthogonal projector onto (imΦ (ū)) ⊥ . Then, from (24) and (42) it can be seen that one can actually take the same N > 0 for all Π , compensating the dependence of N on Π by taking δ > 0 small enough. Furthermore, under the strengthened Assumption 3, condition (Π Φ ) (ū;v)v = 0 in Theorem 4.2 can be written in the form (Φ ) (ū;v)v / ∈ imΦ (ū) independent of Π , and the union of (Π Φ ) (ū;v)v over all possible Π coincides with (Φ ) (ū;v)v + imΦ (ū). Hence, the latter is contained in the interior of the cone of feasible directions to the corresponding union W ε, δ (ū,v) of the sets K ε, δ (0, (ΠΦ ) (ū;v)v) at 0, and In particular, if rankΦ (ū) = q − 1, then imΦ (ū) is a hyperplane in R q , while the set cone((Φ ) (ū;v)v + imΦ (ū)) is the entire half-space (the one containing (Φ ) (ū;v)v) associated with this hyperplane with excluded nonzero points belonging to imΦ (ū). In other words, the set W ε, δ (ū,v) is starlike with respect to 0, with excluded directions necessarily belonging to the half-space associated with the hyperplane imΦ (ū) (to the one not containing (Φ ) (ū;v)v). This set that according to (44) is covered by Φ on K ε, δ (ū,v), is large (half-ball asymptotically), and cannot be expected to be larger, even in the twice differentiable case.
We emphasize that the observations regarding taking the union of sets K ε, δ (0, (ΠΦ ) (ū;v)v) over possible values of Π appear to be new even in the twice differentiable case. This is demonstrated by the following example.

Stability of Singular Solutions in the Piecewise Smooth Case
We now get back to the piecewise smooth setting. However, in contrast to Sect. 3, we consider the stability of special singular solutions and apply results from Sect. 4. Proposition 5.1 Let Φ : R p → R q be piecewise smooth near a given solutionū of (3). Letv ∈ R p be such that Φ (ū;v) = 0, A(ū,v) = { j }, and the selection mapping Φ j satisfies Assumptions 1-3 (with Φ substituted by Φ j ) and is 2-regular atū in the directionv.
Proof Again, without loss of generality, we assume that ε > 0 and δ > 0 are taken as in the assertion of Corollary 2.1. Then, choose N > 0 and ε > 0 according to Theorem 4.1 for assertion (a), and N > 0, ε > 0, and δ > 0, according to Theorem 4.2 for assertion (b), applied with Φ substituted by Φ j . With these choices, the needed assertions become evident, taking into account that according to (10), Φ(u) = Φ j (u) for all u ∈ K ε, δ (ū,v), and also employing Remark 4.2.

Remark 5.1
According to the discussion in Remark 4.2, if in the last assertion of Propo- is an open half-space, and this large set is contained in the cone of feasible directions to W ε, δ (ū,v) at 0.
Examples illustrating Proposition 5.1 can be easily constructed, for instance, on the basis of examples in Sect. 4. We note that all these examples can be modified by adding to functions involved any differentiable terms small of order higher than 2 at 0, without changing the conclusions concerned with application of Theorems 4.1-4.2 and Proposition 5.1, but making direct explicit solution of Eq. (4) difficult or even impossible, or even making (4) unsolvable for w away from zero.

Applications to the Nonlinear Complementarity Problem
In this section we apply the results obtained above to a reformulation of the nonlinear complementarity problem (NCP) with a smooth mapping F : R n → R n . One traditional approach to tackle NCP (46) consists of reformulating it as Eq. (3) with Φ : R n × R n → R n × R n (thus setting p = q = 2n) defined by where u = (x, y) with a slack variable y, and the min-operation is applied componentwise. Involving slack variable y in this reformulation is needed exclusively to tackle, by means of Proposition 5.1, additive constant perturbations of F, rather than generic perturbations of Φ, which would be of unclear meaning. Let s := 2 n , and fix any one-to-one mapping j → I ( j) from {1, . . . , s} to the set of all pairwise different subsets of {1, . . . , n} (including ∅ and the entire {1, . . . , n}). The mapping Φ defined in (47) is piecewise smooth, and the components of the corresponding smooth selection mappings Φ j : R p → R p for j ∈ {1, . . . , s} are given by Then, the set of indices of selection mappings that are active at u, defined according to (5), takes the form where For any J ⊂ I = , we will use the notation \J := I = \J .

Conclusions
We presented new results on stability of a given solution of a piecewise smooth equation with respect to large (not asymptotically thin) classes of right-hand side perturbations. Apart from piecewise (rather than true) smoothness, we allow the first derivatives of the smooth selections to be merely B-differentiable and do not require them to be twice differentiable. The main case of interest in this work is when the solution in question is singular for some smooth selection active at it, and hence, its stability properties cannot be tackled by standard analysis tools. Some applications of the results obtained to complementarity problems are also discussed. appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.