Strongly stable C-stationary points for mathematical programs with complementarity constraints

In this paper we consider the class of mathematical programs with complementarity constraints (MPCC). Under an appropriate constraint qualification of Mangasarian–Fromovitz type we present a topological and an equivalent algebraic characterization of a strongly stable C-stationary point for MPCC. Strong stability refers to the local uniqueness, existence and continuous dependence of a solution for each sufficiently small perturbed problem where perturbations up to second order are allowed. This concept of strong stability was originally introduced by Kojima for standard nonlinear optimization; here, its generalization to MPCC demands a sophisticated technique which takes the disjunctive properties of the solution set of MPCC into account.


Introduction
We consider the class of mathematical programs with complementarity constraints (MPCC) given as where L = {1, . . . , l}, l ∈ N and the functions f : R n → R and r m , s m : R n → R, m ∈ L, are assumed to be twice continuously differentiable.
The new results in this paper are mainly related to complementarity constraints and can be easily extended to programs with additional finitely many equality or inequality constraints. Note that there is a huge variety of applications for MPCC, see e.g. [24,34].
The goal of this paper is to present necessary and sufficient conditions for the strong stability of a C-stationary point for MPCC. The concept of strong stability was introduced by Kojima [22] for standard nonlinear optimization programs and it refers to the local existence and uniqueness of a stationary point for each sufficiently small perturbed problem. There, the values of a perturbation and its derivatives up to second order are taken into consideration, but do not necessarily depend on real parameters. In particular, results on strong stability can be immediately applied whenever only sufficiently small linear and quadratic perturbations are allowed. Several results related to strong stability have been established, we refer e.g. to [5,9,15,21,31,32]. First results for a generalization to MPCC are given in [33] and [18] where the latter presents a characterization of strong stability of a C-stationary point under MPCC-LICQ.
There are several stationary concepts for MPCC and many related references, e.g. [6,7,19,28,35,36]. Note that e.g. M-and S-stationarity are stronger concepts concerning optimality conditions; in particular, C-stationarity does not exclude trivial first-order descend directions.
However, C-stationarity is related to certain geometric properties which are described by the so-called Morse-relations [13] and which refer to the topological changes of the feasible level set when the level varies. For sensitivity analysis and solution (homotopy) methods [30], it is important to know where and whether topological changes may arise. Such changes could be that a new connected component is born, or two connected components merge or, in general, the geometric shape (sphere, torus...) of the feasible level set changes. This geometric shape is in particular relevant for the possible number of different local minimizers. Moreover, one is interested in conditions under which these topological changes remain unchanged after small perturbations (stability). The classical (unconstrained) Morse Theory [25] and its generalizations to standard nonlinear optimization [13] show that such changes happen if and only if a level containing a stationary point is passed. Otherwise the feasible level sets remain homeomorphic (topologically identical). A corresponding result for MPCC was presented in [16]: here, topological changes happen exactly at levels that contain a C-stationary point. Therefore, strong stability of a C-stationary point refer to stability of geometric properties of MPCC which are important for sensitivity analysis and design of solutions methods. As a consequence, we might miss some of this topological changes if we consider strong stability (only) for M-or S-stationary points. On the other hand, when concerning optimality conditions, it remains an open and interesting question how to establish strong stability for M-or S-stationarity.
The adaptation to MPCC of Kojima's topological definition of strong stability is straightforward; the challenge is to find an algebraic characterization which is equivalent to this topological definition. Thus, the goal of this paper is to present such an algebraic characterization of a strongly stable C-stationary pointx of MPCC where we assume that: • MPCC-LICQ does not hold atx.
• A constraint qualification of Mangasarian-Fromovitz type holds atx.
As we will see, the disjunctive structure of MPCC implies the use of algebraic techniques which are different to those used in the standard nonlinear case. Moreover, we refer to our previous paper [11] where we characterized strong stability of C-stationary points for the particular case with n +1 active constraints; some of the results from that paper will be used here. We also refer to some related papers. Other stability results are established [2,4] (Lipschitz properties) and in [26,29] (Tilt stability); solutions methods are discussed e.g. in [8,12,20,23,30].
This paper is organized as follows. Section 2 contains some auxiliary results and notations. Section 3 summarizes some known results from standard nonlinear optimization and MPCC which are needed later. In Sect. 4 we introduce the crucial notation of a basic Lagrange vector. In Sect. 5 a necessary second order condition (Condition C * ) for the strong stability of a C-stationary point for MPCC is shown; moreover, several properties are proved in a series of preliminary lemmas. Section 6 contains the main results. Under two appropriate assumptions (A1 and A2), equivalent algebraic characterizations for the strong stability of a C-stationary point are presented. Finally, Sect. 7 delivers some final remarks.

Preliminary notations and results
In this section we describe some basic notations which will be used later. Main parts of this description are taken from our previous paper [11,Sect. 2]. For p ∈ N and w ∈ R p define If E ⊂ R n is a linear subspace and A is an n × n symmetric matrix, then A is called Letx ∈ R n and δ > 0. The Euclidean norm ofx will be denoted by x , the closed Euclidean ball centered atx with radius δ by B(x, δ) and the Euclidean sphere centered atx with radius δ by S(x, δ). We abbreviate the sentence "V is a neighborhood ofx" by letting V(x) to be the set of all neighborhoods ofx. This allows us to write the aforementioned sentence as "V ∈ V(x)".
Let C k (A 1 , A 2 ) be the space of k−times continuously differentiable mappings with domain A 1 ⊂ R n and codomain A 2 ⊂ R m . Let f ∈ C 2 (R n , R),x ∈ R n and v ∈ R n , with v = 1. As usual, ∂ f (x) ∂ x i and ∂ f (x) ∂v , denote the partial derivative with respect to x i and the directional derivative with respect to v, respectively, of the function f atx.
In addition, D x f (x) stands for its gradient taken as a row vector and D 2 x f (x) for its Hessian.
Moreover, for ⊂ R n let bd denote the boundary of and denote the set of its extreme points. For defining strong stability we need a seminorm for functions. Let V ∈ V(x) and F ∈ C 2 (R n , R m ). Following [22], let (1) There exist V ∈ V(x) and U ∈ U V (F) such that for all F ∈ U the set V contains exactly one solution to the equation F(x) = 0, which we denote byx(F).
The previous theorem is similar to upcoming Theorems 3.1 and 3.2 which present characterizations for strong stability of stationary and C-stationary points, respectively. We end this section by presenting a property of the Clarke subdifferential of min-type functions.
where the latter denotes the set of all (n, n)-matrices whose ith row belongs to the set Proof It is a straightforward consequence of Propositions 2.3.1, 2.3.12 and 2.6.2 in [3].

Auxiliary results for standard nonlinear programs and for MPCC
In this section we present some auxiliary results and definitions that are mainly taken from [11,Sects. 3 and 4]. The exception is the forthcoming Lemma 3.2 which, to our knowledge, is new although its proof is essentially an adaptation of the proof of [22,Theorem 7.2]. Let P = P sn ( f , h, g) denote the standard nonlinear program where the index sets I and J are finite, f ∈ C 2 (R n , R), h i ∈ C 2 (R n , R), i ∈ I and g j ∈ C 2 (R n , R), j ∈ J . We say that two nonlinear programs P 1 and P 2 are equal (P 1 = P 2 ) if they are defined by the same functions f , is the Lagrange function for P. The set of stationary points for P is denoted by (P). The set of (λ, μ) such that (3.1) holds is denoted by L(P,x).
It is well-known that the following constraint qualifications relate local minimizers to stationary points: • The Linear Independence constraint qualification (LICQ) holds atx ∈ M sn [h, g] if the vectors are linearly independent and there exists v ∈ R n such that It is well known that LICQ implies MFCQ and that if MFCQ holds at a local minimizer x for P, thenx is a stationary point for P.
Since we deal with stationary points under sufficiently small perturbations, we recall the concept of a strongly stable stationary point introduced by Kojima in [22]. For this we need a seminorm for functions. Given V ∈ V(x) and P = P sn ( f , h, g), we define where ( f , h, g) V is obtained by takingF = ( f , h, g) in (2.1). LetP = P sn (f ,h,ḡ) and δ > 0 be fixed and where P andP have the same number of equality and inequality constraints; the set of all neighborhoods ofP is denoted by W V (P). Now, we recall Kojima's [22] definition of a strongly stable stationary point and a convenient characterization of it.
The set of strongly stable stationary points forP is denoted by S (P). (1) There exist V ∈ V(x) and W ∈ W V (P) such that for all P ∈ W the set (P) ∩ V contains exactly one element, which we denote byx(P). (2) The mappingx : W → V , P →x(P) is continuous.
Under MFCQ, the set L(P,x) remains contained in a certain compact set after any sufficiently small perturbations (see e.g. [22,Lemma 7.4]). Moreover, it holds that (λ,μ) ∈ ext L(P,x) if and only if the gradients are linearly independent. By the latter fact together with a continuity argument, the next result readily follows. Lemma 3.1 Assume that M FC Q holds atx. Then, there exist V ∈ V(x) and W ∈ W V (P) such that for all P ∈ W , x ∈ V ∩ (P) and all (λ, μ) ∈ ext L(P, x) there exists (λ,μ) ∈ ext L(P,x) such that In addition, for (λ, μ) and (λ,μ) it holds that λ i ·λ i > 0, i ∈ I * (λ).
In the remainder of this section, we assume that the vectors D xhi (x), i ∈ I are linearly independent. By Carathéodory's theorem, the latter ensures that ext L(P,x) = ∅ wheneverx ∈ (P). Forx ∈ (P) and (λ,μ) ∈ L(P,x) let The next lemma relates a second order condition to the existence of two stationary points nearx after a sufficiently small perturbation ofP. Lemma 3.2 Assume that LICQ does not hold atx ∈ (P). If for some (λ,μ) ∈ ext L(P,x) the condition does not hold, then there exist sequences P k → P, x 1,k , x 2,k →x with x 1,k = x 2,k and x 1,k , x 2,k ∈ (P k ) such that LICQ holds at x 1,k , x 2,k and that if Proof The main idea of the proof is given in the "only if" part of [22,Theorem 7.2]. There, the condition MFCQ is only needed to ensure that L(P,x) is bounded and to express its elements as a convex combination of its extreme points. Afterwards, a vector (λ,μ) ∈ ext L(P,x) is fixed andP perturbed sufficiently small in such ways that LICQ holds atx ∈ (P)\ S (P) and L(P,x) = {(λ,μ)}, whenever (3.2) does not hold. Thus, by applying [22,Theorem 4.2], the desired result follows.
The novelty of the latter result consists in its independence from the condition MFCQ. As we already mentioned, this condition is necessary for strong stability. However, in our MPCC setting it is worth studying auxiliary standard nonlinear programs whose stationary points do not fulfill MFCQ.
In the remainder of this section we turn our attention to MPCC and recall now that P = P cc ( f , r , s) is a problem with the objective function f and the feasible set M[r , s] as given in (1.1) where f ∈ C 2 (R n , R) and r m , s m ∈ C 2 (R n , R), m ∈ L. Analogously to the standard nonlinear program, we say that two MPCCs are equal if they are defined by the same functions ( f , r , s). Moreover, forx ∈ M[r , s] we define the active index sets: Concerning the number of active constraints for P atx ∈ M[r , s] define Remark 3. 1 To simplify notation, we use the same letters that were used for defining sets for standard nonlinear programs, now for defining analogous sets for MPCCs. From now on, we assume that P = P cc ( f , r , s) andP = P cc (f ,r ,s) are two MPCCs with the same number of complementarity constraint. In addition, we use auxiliary standard nonlinear programs that we denote by the superscript "aux", for instance P aux ,P aux , P aux,1 , etc.
is the MPCC-Lagrange function for P. The set of FJC points for P is denoted by F (P).
The motivation for defining FJC points comes from the fact that for a local minimizer x for P it holds thatx ∈ F (P), see [33,Lemma 1]. Now, we recall the definitions of C-MFCQ and C-stationarity. Note that C-MFCQ is called MFC in [10,11,17,18,34].

Definition 3.3 We say that C-MFCQ holds atx ∈ M[r , s] if the vectors
are linearly independent for any choice of λ m ∈ [0, 1], m ∈ I rs (x).

Definition 3.4
The set of all (ρ, σ ) ∈ R 2l with (3.3), (3.4) and μ 0 = 1 is denoted by L(P,x) and is called the set of Lagrange vectors for P atx. The pointx The set of C-stationary points for P is denoted by C (P).
In order to present the concept of a strongly stable C-stationary point for MPCC we introduce a seminorm analogously as above. Given V ∈ V(x) and P = P cc ( f , r , s), we define The set of all neighborhoods ofP is denoted by W V (P) and the set of neighborhoods of (r ,s) by U V (r ,s). Now, we present the definition of a strongly stable C-stationary point.
Definition 3.5 [18] A pointx ∈ C (P) is called strongly stable if there existsδ > 0 such that for all δ ∈ (0,δ] there exists ε > 0 such that for every P ∈ B B(x,δ) (P, ε) it holds that The set of strongly stable C-stationary points forP is denoted by S (P).
Furthermore, we have the following characterizations of the strong stability of a Cstationary point.

Theorem 3.2 [10, Lemma 2.5 and Theorem 4.5] The pointx ∈ C (P) is strongly stable if and only if the following condition hold:
(1) There exist V ∈ V(x) and W ∈ W V (P) such that for all P ∈ W the set C (P)∩V contains exactly one element, sayx(P).
We terminate this section by presenting a brief discussion about the relationship between MFCQ, MPCC-MFCQ and C-MFCQ. Note that C-MFCQ appeared (probably) first in [17] in the context of topological stability of the feasible set of MPCC.

Remark 3.2
Consider for a moment a standard nonlinear program as defined in the beginning of this section and a feasible pointx ∈ M sn [h,ḡ]. Then, the following conditions are equivalent: (c) There exists a compact set K 2 ⊂ R |I |+|J | which contains the set of Lagrange vectors for any sufficiently small perturbed problem and x nearx [22,Lemma 7.4].
Note that (ii) in Lemma 3.3 and (b) in Remark 3.2 are dual formulations of C-MFCQ and MFCQ, respectively. Moreover, the properties (ii) and (iii) in Lemma 3.3 are analogous to (b) and (c) in Remark 3.2, respectively. That is the reason why C-MFCQ is called a constraint qualification of Mangasarian-Fromovitz-type.

Remark 3.3
Now, we consider an MPCC, which might have standard constraints, and MPCC-MFCQ [33]. If the problem under consideration has no standard constraints, then MPCC-LICQ and MPCC-MFCQ are equivalent. Analogously to (ii) in Lemma 3.3, one obtains the dual formulation of MPCC-LICQ which is obviously related to the so-called weak stationarity [33]. Roughly speaking, MPCC-MFCQ relates to weak stationarity in the same way as C-MFCQ relates to C-stationarity. Moreover, MPCC-MFCQ implies C-MFCQ. Since we deal with C-stationarity, C-MFCQ is the appropriate constraint qualification in the context of this paper.

Basic Lagrange vectors
In [22,Theorem 7.2], the concept of extreme points of a convex set plays an essential role. However, in our MPCC setting the set L(P,x) is, in general, not convex and, therefore, this concept cannot be applied. In the following, we consider instead the concept of a basic Lagrange vector which is crucial for necessary and sufficient conditions for strong stability. Throughout this section, we do not always assume that C-MFCQ holds atx.
The set of basic Lagrange vectors is denoted by L 0 (P,x).
are linearly independent.
Note that a basic Lagrange vector is an extreme point (vertex) in case that L(P,x) is a convex polyhedron. Furthermore, we refer again to [11] where we considered the particular case with n + 1 active constraints. There, the definition of a basic Lagrange vector becomes much simpler [11,Definition 5.4]. The latter is equivalent to Definition 4.1 whenever the assumptions in [11] are fulfilled. The next result states that L(P,x) is the union of certain polyhedrons whose extreme points belong to L 0 (P,x).

Lemma 4.2 For any I ⊂ Irs(x) define the polyhedron
Then, the following holds Proof The first equality follows from the definition of L(P,x); the second one from the first one and by Lemma 4.1.
Now, we provide a characterization of the existence of basic Lagrange vectors and a necessary condition for the strong stability of a C-stationary point.
are linearly independent.
Proof By Lemma 4.1, if L 0 (P,x) = ∅, then the vectors in (4.2) are linearly independent. Now, assume the latter condition. By Lemma 4.2, for some I ⊂ Irs(x) it holds that L(P,x, I ) = ∅. Since the vectors in (4.2) are linearly independent, application of [1, Proposition 3.3.1] to L(P,x, I ) yields ext L(P,x, I ) = ∅. By using Lemma 4.2 again, we obtain L 0 (P,x) = ∅.
Proof It is a straightforward consequence of [10, Theorem 5.5] and Lemma 4.3.
In the following theorem C-MFCQ is assumed and Lemma 4.2 is strengthened. Moreover, a result analogous to Lemma 3.1 follows immediately.

A necessary condition for strong stability
In the remainder of this paper letx ∈ C (P) be our point under consideration and assume that MPCC-LICQ does not hold atx ∈ M[r , s]. As already mentioned in Sect. 1, strong stability under MPCC-LICQ is completely described in [18]. In this section we present a necessary second order condition (Condition C * ) for the strong stability of a C-stationary point. We define for (ρ,σ ) ∈ L(P,x) the sets . and Note that the set Tx (r ,s,ρ,σ ) is a so-called tangent space, see e.g. [34]. The next result is obvious and therefore its proof is skipped.
Lemma 5.1 Assume that for some (ρ,σ ) ∈ L(P,x) and some sets I , J ⊂ L it holds that I * (ρ) ∪ Ir (x) ⊂ I , I * (σ ) ∪ Is(x) ⊂ J and that the vectors Dr i (x), i ∈ I , Ds j (x), j ∈ J are linearly independent. Let the vectors ξ q ∈ R n , q ∈ Q form an orthonormal basis of the subspace where Q is an appropriate index set. Then, there exist V ∈ V(x) and functionŝ for v, w ∈ S(0, 1).
The next lemma presents a first necessary condition for strong stability.
Proof Suppose contrarily that N 0 (P,x) ≤ n. A contradiction easily follows by noting that N 0 (P,x) is an upper bound for the left hand side of (5.2).
Since MPCC-LICQ does not hold atx, there exists (ᾱ,β) ∈ R 2l \{0} such that In the following lemmas we assume thatN (P,x) = 1. As mentioned in [11], this assumption implies the following characterization of the set L(P,x).
The next two lemmas relate the strong stability ofx to L 0 (P,x) and the signs of some components of (ᾱ,β).

Remark 5.1
We observe in the previous proof that the conditionᾱ i 0 ·β j 0 > 0 is needed when N 0 (P,x) = n + 1. If N 0 (P,x) ≤ n, then it is sufficient to assumeᾱ i 0 ·β j 0 = 0 in Lemma 5.6.
Next, we consider a case with exactly one basic Lagrange vector.
Proof We distinguish two cases.

Remark 5.2
According to the proof of Lemma 5.7, we can delete the words "orx ∈ S (P)" in Lemma 5.7 (since Case 1 in this proof is not possible).
The following corollary completes the preparation for the forthcoming Theorem 5.1.
form a basis of the subspace generated by Since MPCC-LICQ does not hold atx, it follows that Iρ\I = ∅. Fix ε > 0 sufficiently small, choose arbitrarily i ∈ Iρ\I and let Therefore, without loss of generality assume now thatN (P,x) = 1. By Corollary 5.4, we obtain (5.17) for all i 1 , i 2 ∈ Iρ. Now, fix i 0 ∈ Iρ and consider the following auxiliary standard nonlinear program Analogously to the proof of Lemma 5.5, application of Lemma 3.2 yields a contradiction tox ∈ S (P). Therefore, (ρ,σ ) fulfills Condition C * .

Remark 5.3
By [11, Lemma 5.2, Theorem 5.14], it follows that the statement of the latter theorem holds independently from C-MFCQ.
For (ρ,σ ) ∈ L(P,x) define The next result is straightforward but crucial for the characterization of strong stability.
Theorem 6.1 Ifx ∈ S (P), then at least one of the following conditions hold: (1) There exists (ρ 0 ,σ 0 ) ∈ L 0 (P,x) such that, after possibly interchanging constraints, it holds that Iσ 0 = ∅ and thatσ 0 Proof It is a direct consequence of Theorem 5.1.
We will characterize strong stability when (1) or (2) in Theorem 6.1 holds. Next, we present two preliminary results.  Fix an arbitrary m 0 ∈ Iρ. Obviously, the condition holds only when t = 0. Then, by Lemma 5.3, we obtain .
The remainder of the proof is given in eight steps.
This completes the proof.
Next, we provide an example in which the previous theorem is used. Note that in this case the Hessian is not positive definite but negative definite on the corresponding tangent space. Example 6.1 Let n = 5,x = 0 and consider the problemP given by The set of Lagrange vectors atx is and the elements of L 0 (P,x) are those listed in the following Moreover, for any (ρ,σ ) ∈ L 0 (P,x) it holds that cc (x,ρ,σ ) = diag(0, 0, 0, 0, −2), Tx (r ,s,ρ,σ ) = span{(0, 0, 0, 0, 1)} and, therefore, (ρ,σ ) fulfills Condition C * . By Theorem 6.2, it follows that 0 ∈ S (P).
Now, we provide a characterization of strong stability of a C-stationary point when N 0 (P,x) ≤ n. We point out that in the following corollary both A1 and A2 appear as necessary conditions for strong stability (and not as assumptions throughout this section).
Proof (ii) ⇒ (i). Assume without loss of generality that Irs(x) = L and for a problem P with the same number of constraints asP consider the mapping F P,I : R n+2l → R n+2l given by where (τ, ζ ) ∈ R 2l and I ⊂ Irs(x) is chosen as in (ii). Consider the sets Note that where A is a diagonal matrix with entry 1 in |I | rows and entry −1 in |L\I | rows. By C-MFCQ, the set S I (P,x) is bounded. Hence, analogously to (6.1), there exist V ∈ V(x) and W ∈ W V (P) such that Note that V and W can be further shrunk. Consequently,x ∈ S (P).
Step 5 Finally, analogously to Step  This completes the proof.
In the following example we apply Theorem 6.3.  To verify the remaining of condition (ii) in Theorem 6.3, for each pair of basic Lagrange vectors we list the corresponding (ᾱ,β) ∈ R 2l in the following table. We terminate this section by presenting a result about the convexity of L(P,x), which easily follows from Theorems 6.2 and 6.3. Corollary 6.3 Ifx ∈ S (P), then L(P,x) = conv L 0 (P,x).

Final remarks
In this paper we considered mathematical programs with complementarity constraints (MPCC) and presented a topological as well as an equivalent algebraic characterization of the strong stability of a C-stationary point of MPCC. Strong stability refers to the local uniqueness, existence and continuous dependence of a solution for each sufficiently small perturbed problem where perturbations up to second order are allowed. Moreover, a second order necessary condition, which we called Condition C * , was presented.
Since an MPCC has a more combinatorial structure than a standard nonlinear optimization problem, more sophisticated algebraic techniques were necessarily applied to establish our results. For example, we characterized the set of Lagrange vectors and defined the set of basic Lagrange vectors, which we denoted by L(P,x) and L 0 (P,x), respectively. As mentioned in the beginning of Sect. 4, in [22,Theorem 7.2], the concept of extreme points plays an essential role. However, in our setting the set L(P,x) is, in general, not convex and, consequently, this concept cannot be directly employed in the characterization of strongly stable C-stationary points. Our solution to this issue was to define and characterize the set L 0 (P,x), which plays a role in our results similarly to that of the set of extreme points in [22,Theorem 7.2].
Moreover, note that Condition C * plays a crucial role in Theorems 6.2 and 6.3, but not in [18,Theorem 3.1] where MPCC-LICQ holds. Hence, our results differ from those in [18] not just in the fulfillment of MPCC-LICQ, but in the matter of the second order Condition C * . Furthermore, the set L(P,x) is always a singleton in [18], whereas in our context, that need not to be the case, see for instance Examples 6.1 and 6.2. Instead, under Assumptions A1 and A2, we showed that the convexity of L(P,x) is necessary for the strong stability of C-stationary points.
Finally, we recall that there are several other concepts of stationarity for MPCC such as A-, B-, M-and S-stationarity; we refer e.g. to [34,36] for an overview on relations among them. The characterization of strong stability of these types of stationary points are topic of current research.