MPCC: Strong Stability of M-stationary Points

In this paper we study the class of mathematical programs with complementarity constraints MPCC. Under the Linear Independence constraint qualification MPCC-LICQ we state a topological as well as an equivalent algebraic characterization for the strong stability (in the sense of Kojima) of an M-stationary point for MPCC. By allowing perturbations of the describing functions up to second order, the concept of strong stability refers here to the local existence and uniqueness of an M-stationary point for any sufficiently small perturbed problem where this unique solution depends continuously on the perturbation. Finally, some relations to S- and C-stationarity are briefly discussed.


Introduction
In this paper we consider the following mathematical program with complementarity constraints (MPCC): P cc (f, r, s): min x∈M [r,s] f (x) (1.1) where L = {1, . . . , l}, l ∈ N and all describing functions f : R n → R and r m , s m : R n → R, m ∈ L, are assumed to be twice continuously differentiable. There are many applications where MPCC is used, we refer e.g. to [23,28]. The results of this paper refer to the particular structure imposed by the complementarity constraints. In order to keep the presentation as less technical as possible we did not include additional equality and inequality constraints which are usually used in standard nonlinear optimization. However, all results presented here can be straightforwardly extended to this more general case with these standard constraints.
In this paper we deal with the concept of strong stability to an M-stationary point for MPCC under the Linear Independence constraint qualification for MPCC (MPCC-LICQ). Recall that MPCC-LICQ holds at a feasible point if the gradients of the active constraints at this point are linearly independent.
The concept of strong stability of a stationary point for a standard nonlinear program was introduced by Kojima [21]. It refers to the local existence and uniqueness of a stationary point for each sufficiently small locally perturbed problem where perturbations up to second order (function values, first and second derivative) are allowed. Moreover, this unique stationary point of the perturbed problem depends continuously on the underlying perturbation. Obviously, these "well-posedness" features (existence, uniqueness and continuity on perturbation) play an important role in sensitivity analysis and parametric optimization. Besides the topological definition of strong stability, Kojima [21] proved an equivalent algebraic characterization using second order information and applying matrix analysis. This algebraic characterization can be used to decide whether or not a stationary point is strongly stable. We refer to related papers [20,25,26] on properties on strongly stable stationary points.
The goal of this paper is twofold. First, we introduce the strong stability of an M-stationary point for MPCC. Then, under MPCC-LICQ, we prove an equivalent algebraic characterization of this property. Note that since we only consider complementarity constraints, here, the constraint qualifications MPCC-LICQ and MPCC-MFCQ are identic. In general, they are different due to the presence of additional inequality constraints.
It is well-known that there exist several stationarity concepts for MPCC. Among them are A-, B-, C-, M-and S-stationarity; for an overview and discussion we refer to [29]. There exist already results on strongly stable C-stationary points for MPCC [4][5][6]17] where this particular interest is motivated by the fact that C-stationarity is related to changes of topological properties of the underlying problem [15,16]. However, in this paper we choose the stronger concept of M-stationarity which excludes more (but not all) decent directions than the concept of C-stationarity.
We would like to mention some other related papers: there are many references on stationarity concepts, optimality conditions, sensitivity and other properties for MPCC, see e.g. [1,3,10,18,24,27]. Moreover, we refer to some papers where solution methods for MPCC are discussed [2,8,11,19,22]. This paper is organized as follows. Section 2 provides some notations used later on. In Section 3 we recall the concept of strong stability of a stationary point for a standard nonlinear program. Section 4 contains the new results: We state a topological and, under MPCC-LICQ, an equivalent algebraic characterization of a strongly stable M-stationary point for MPCC. Some final remarks are given in Section 5.
Most of the presentation of basic notations in this section is taken from [5]. For w ∈ R n let w = (w 1 , . . . , w n ) and define Forx ∈ R n and δ > 0 the closed Euclidean ball centered atx with radius δ will be denoted 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 (R n , R m ) be the space of k−times continuously differentiable mappings with domain R n and codomain R m . For f ∈ C 2 (R n , R) denote the partial derivative of f at x ∈ R n with respect to x i by ∂f (x) ∂x i , i = 1, . . . , n. In addition, D x f (x) stands for its gradient taken as a row vector and D 2 x f (x) for its Hessian atx. For the strong stability we need a seminorm for functions. Let V ∈ V(x) andF ∈ C 2 (R n , R m ). Following [21], denote

Strong Stability for Standard Nonlinear Programs
In this section we recall the concept of a strongly stable stationary point for a standard nonlinear program. This result was introduced by Kojima [21] and we will use it in the next section for the algebraic characterization of the strong stability of an M-stationary point for a mathematical program with complementarity constraint.
Consider a standard nonlinear program with finitely many equality and inequality constraints given as P sn (f, h, g) : min with finite index sets I and J as well as functions 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 . For a feasible pointx ∈ M sn [h, g] let the index set of active inequality constraints be given as For two standard nonlinear programs P 1 , P 2 we write P 1 = P 2 if they are defined by the same functions f , h i , i ∈ I and g j , j ∈ J . Moreover, if two problems P 1 = P sn (f 1 , h 1 , g 1 ) and P 2 = P sn (f 2 , h 2 , g 2 ) have the same number of equality and inequality constraints, then the subtraction of these problems is defined by the following expression Recall that a pointx ∈ R n is called a stationary point for P = P sn (f, h, g) if there exist λ i ∈ R, i ∈ I and μ j ∈ R, j ∈ J such that where 0 denotes the origin (column vector) in R n and is the Lagrange function for P . Let the set of stationary points for P be denoted by (P ) and the set of (λ, μ) satisfying (3.1) by L(P ,x). Recall also that the Linear Independence constraint qualification (LICQ) is said to hold atx ∈ M sn [h, g] if the vectors are linearly independent. If LICQ holds at a local minimizerx for P , thenx is also a stationary point for P . Note that we can generically assume that LICQ holds at any feasible point [13].
As already mentioned, the concept of strong stability of a stationary point was introduced by Kojima [21] and it is related to the maximum modulus of the differences between the values of the functions, and their first and second order derivatives. For this concept we use the following seminorm: givenx ∈ R n , V ∈ V(x) and P = P sn (f, h, g), we define where the right-hand-side is obtained from (2.1) by choosingF = (f, h, g). Then, for P = P sn (f ,h,ḡ) and δ > 0 define where P andP have the same number of equality and inequality constraints. Now, we recall Kojima's definition of a strongly stable stationary point.
there exists a real numberδ > 0 such that for all δ ∈ (0,δ] there exists a real number ε > 0 such that for every P ∈ B B(x,δ) (P , ε) it holds that The set of strongly stable stationary points forP is denoted by S (P ).
Note that in the previous definition strong stability is stated as a topological property which cannot be straightforwardly checked in general. In this regard, Kojima [21] did not only provide the concept of a strongly stable stationary point, but also an equivalent algebraic characterization of it under the well-known Mangasarian-Fromovitz constraint qualification. In the remainder of this section we recall this characterization for the case that LICQ holds at the considered point. For that we assume: •P = P sn (f ,h,ḡ) with Lagrange functionL sn andx ∈ (P ).
• For each index setJ with Let C(J ) ∈ R n×(n−|I |−|J |) be a matrix whose columns form a basis of the space The latter means that C(J ) is a so-called basis matrix of Tx(h,ḡ,J ). Note that, by Sylvester's law, the number does not depend on the particular choice of C(J ). Now, we recall Kojima's result.
Proposition 3.1 [21] Assume that LICQ holds atx ∈ S (P ). Then, the following two conditions are equivalent.

μ) C(J ) is constant and nonvanishing for allJ satisfying (3.2).
Remark 3. 1 We use the convention that det C(J ) T We mention that under the Linear Independence constraint qualification, Kojima's concept of strong stability is equivalent to Robinson's concept of strong regularity for generalized equations [25]; see e.g. [12]. In the following section we will use the results presented in the current section in the context of mathematical programs with complementarity constraints.

Strong Stability for M-stationary Points
This section contains our main result which is the algebraic characterization of strong stability of M-stationary points for MPCC. Here, we return to the mathematical program P = P cc (f, r, s) defined in (1.1). The equality and subtraction of problems are defined analogously to those in Section 3. Moreover, forx ∈ M[r, s] we define the active index sets: We recall the definition of an M-stationary point for P .
is the MPCC-Lagrange function for P . We denote the set of M-stationary points by M (P ) and the set of Lagrange vectors (ρ, σ ) ∈ R 2l satisfying (4.1)-(4.3) by L M (P ,x).
We also recall that the Linear Independence constraint qualification for P (MPCC- are linearly independent. It is well-known that any local minimizer for P at which MPCC-LICQ holds is also an M-stationary point for P , see e. g. [3,Theorem 3.9].
In the remainder of this paper we assume that: For (ρ,σ ) we define the sets In the previous section we recalled the concept of strong stability of a stationary point for standard nonlinear programs. Now, we generalize it to M-stationary points for MPCC. Analogously to Section 3, we define an appropriate seminorm. Given V ∈ V(x) and P = P cc (f, r, s), let P V = (f, r, s) V , where the right-hand-side is obtained from (2.1) by choosingF = (f, r, s). For δ > 0 let Denote for a given V ∈ V(x) the set of all neighborhoods ofP by W V (P ).

Definition 4.2 A pointx ∈ M (P )
is called strongly stable if there existsδ > 0 such that for all δ ∈ (0,δ] there exists ε > 0 such that for P ∈ B B(x,δ) (P , ε) it holds that The set of strongly stable M-stationary points forP is denoted by MS (P ).
It is not hard to see that both (ε, 0) T and (0, ε) T are M-stationary points, see Fig. 1b. Consequently, 0 is not a strongly stable M-stationary point.
For ε > 0 sufficiently small, consider the perturbation After a straightforward calculation, it follows that 0 is the only point satisfying (4.2) in Definition 4.1, see Fig. 2b. However, it is not hard to see that 0 is not an M-stationary point for the perturbed problem. Thus, 0 is not a strongly stable M-stationary point forP .
Examples 4.1 and 4.2 provide some hints for obtaining necessary conditions for strong stability. These will be used when proving the forthcoming Theorem 4.1. Since the proof of the following lemma uses only continuity arguments, we present a short proof of its first statement and skip the proof of the second one.

Lemma 4.1 Assume that MPCC-LICQ holds atx ∈ M (P ). Then, there exist V ∈ V(x) and W ∈ W V (P ) such that for all P ∈ W and all x ∈ V ∩ M (P ) we have that
(1) MPCC-LICQ holds at x for P .
(2) The uniquely determined (ρ, σ ) ∈ L M (P , x) satisfies that Proof (1). By a continuity argument on the derivatives of the active constraints, it follows that there exists a compact V ∈ V(x) such that MPCC-LICQ holds for all x ∈ V ∩ M[r,s]. Now, suppose contrarily to (1) that there exist sequences P k with P k −P V → 0 and x k ∈ V ∩ M (P k ) such that MPCC-LICQ does not hold at x k for P k . Hence, for all k ∈ N there exists (α k , β k ) ∈ R 2l with (α k , β k ) = 1 such that By the compactness of V and (α k , β k ) = 1, after perhaps reducing to appropriate subsequences, assume that x k → x 0 for some x 0 ∈ V ∩ M[r,s] and that (α k , β k ) → (ᾱ,β) with (ᾱ,β) = 1. Now, letting k → +∞ in (4.4) and (4.5) we get that MPCC-LICQ does not hold at x 0 and, therefore, a contradiction to the fact that MPCC-LICQ holds for all Next, we present an auxiliary result and a necessary condition for strong stability.
Proof It follows immediately from Steps 1 and 2 in the proof of [13, Theorem 2.7.2].

Theorem 4.1 Assume that MPCC-LICQ holds atx ∈ MS (P ).
Then, the following conditions hold: Proof Letx ∈ MS (P ) and recall that, consequently, (4.3) in Definition 4.1 holds for (ρ, σ ) = (ρ,σ ). Now, suppose contrarily that there exists an index m 0 ∈ Irs(x) such that one of the following conditions is fulfilled: After perhaps adding sufficiently small positive constants to the functionsr m , m ∈ Iρ \{m 0 } ands m , m ∈ Iσ \ {m 0 }, assume that Iρ ∩ Iσ = {m 0 }. Next, we will perform several sufficiently small perturbations off . Since C ∞ (R n , R) is a dense subset of C 2 (R n , R) [7, Theorem 2.4], there exists f 1 ∈ C ∞ (R n , R) sufficiently close tof such that , . Analogously, we obtain functions r 1 ∈ C ∞ (R n , R l ) and s 1 ∈ C ∞ (R n , R l ) sufficiently close tor ands, respectively. Note that after performing this perturbation the Lagrange vector atx for P 1 = P cc (f 1 , r 1 , s 1 ) remains the same.
Moreover, by MPCC-LICQ, for our local consideration we perform a coordinate transformation by choosing as new coordinates a basis of R n which contains the gradients of the active constraints atx; see e.g. [14,16,17] where such a coordinate transformation is called standard diffeomorphism. As a consequence, we can restrict ourselves to the following situation: f 1 (0, 0) = 0 and the problem P 1 becomes Note that now it holds that Next, we distinguish two cases.
Case 1:ρ =σ = 0. A contradiction to this case follows by observing that the two C-stationary points obtained in the proof of Case 2 in [17, Theorem 3.1] are also M-stationary f completeness we provide here the details of this proof. By Lemma 4.2, we get for some v 11 , v 12 , v 22 ∈ C ∞ (R 2 , R). By possibly adding sufficiently small quadratic terms to f 1 we assume v 11 (x 1 , x 2 ) = 0 and v 22 (x 1 , x 2 ) = 0. Then, an appropriate local coordinate transformation leaves the feasible set of P 1 unchanged while the objective function becomes where c 1 , c 2 ∈ R \ {0} and v ∈ C ∞ (R 2 , R). For ε > 0 sufficiently small let and consider the problem P ε which is obtained by substituting f 2 by f ε . Note that , Hence, we obtain two different points (ε, 0) T , (0, ε) T ∈ M (P ε ) which contradicts thatx ∈ MS (P ). Case 2:ρ < 0,σ = 0. Choose f 2 ∈ C ∞ (R 2 , R) such that f 1 − f 2 is a small multiple of x 2 2 and that Let P 2 denote the problem that results from substituting f 1 by f 2 in P 1 . Furthermore, for ε ∈ R with consider the perturbation f ε of f 2 given by and let P ε be the resulting perturbed problem. Choose V ∈ V(0, 0) and W ∈ W V (P 2 ) as in Lemma 4.1. We will get a contradiction tox ∈ MS (P ) by showing that for any ε such that P ε ∈ W and that (4.6) holds. Suppose contrarily that there exists (x ε 1 , x ε 2 ) T ∈ M (P ε ) ∩ V with corresponding Lagrange vector (ρ ε , σ ε ). By Lemma 4.1, it follows that ρ ε < 0 and, by M-stationarity, that σ ε = 0. Therefore, we have Now, consider the mapping G ∈ C 2 (R 2 × R, R 2 ) given as with t ∈ R. Since D x G(0, 0, 0) is nonsingular, the Implicit Function Theorem yields a uniquely determined and continuously differentiable solutionx 1 (t),x 2 (t) for t near the origin with By (4.8) and (4.9) we get x ε 2 =x 2 (ε) and a straightforward calculation provides ∂x 2 (0) ∂t Hence, by the previous expression and (4.6), we obtain ∂x 2 (0) ∂t ε < 0 and, therefore x ε 2 < 0. This contradicts the feasibility of (x ε 1 , x ε 2 ) T and completes the proof of this case. Note that the proof of the caseρ = 0,σ < 0 runs analogously.
Obviously, Examples 4.1 and 4.2 correspond in the previous proof to Cases 1 and 2, respectively. In particular, Figs. 1 and 2 illustrate the local changes of the set of M-stationary points after a sufficiently small translation of the graph off .
In Theorem 4.1 we presented a necessary condition for the strong stability of an Mstationary point. Our goal is now to state an equivalent algebraic characterization for it. For this, following [27], we consider a relaxed program forP atx which we denote here byP rel . Moreover, we consider the perturbation P rel ofP rel which is a standard nonlinear program given as P rel : min where the functions f , r and s are assumed to be nearf ,r ands, respectively.
Remark 4.1 LetP = P cc (f ,r,s) and consider its perturbation P = P cc (f, r, s). Note that P rel does not depend only on the functions that describe P , but also on those describingP as well as on the pointx.
The following auxiliary result relates locally the M-stationary points of a perturbed MPCC to the stationary points of the corresponding perturbed relaxed program.

Lemma 4.3 Assume that MPCC-LICQ holds atx
for all V ⊂ V and all P ∈ W .
Proof (i) ⇒ (ii). By Theorem 4.1, we obtain (4.14) According to Lemma 4.3, there exist V ∈ V(x) and W ∈ W V (P ) such that (4.10) holds for all V ⊂ V and all P ∈ W . By (i) and (4.10), it follows that and, therefore,x ∈ S (P rel ).
The main result of this paper is an algebraic characterization for the strong stability of an M-stationary point by taking first and second order information of (f ,r,s) into account. For this, we need a notation that is analogous to that at the end of Section 3. For each index setsJ 1 ,J 2 with let C(J 1 ,J 2 ) ∈ R n×(n−|J 1 |−|J 2 |) be a matrix whose columns form a basis of the space is constant and nonvanishing for allJ 1 ,J 2 satisfying (4.16).
Proof A moment of reflexion shows that after reordering constraints the Lagrange function forP rel isL cc . Moreover, the family of all matrices C(J ) withJ satisfying (  Moreover, the matrices C(J 1 ,J 2 ) with Tx(r,s,J 1 , and a straightforward calculation yields (4.17). Thus, by Theorem 4.3, it follows thatx ∈ MS (P ). However, note that if we swap x 2 and x 3 in the constraints ofP , thenx is not a strongly stable M-stationary point for the resulting problem.

Final Remarks
We conclude this paper by presenting some final remarks.
Remark 5.1 Each M-stationary point is also a C-stationary one, but not vice versa; this is nicely illustrated in [17, Example 2.1 and Remark 4.2]. As shown in [16] as a generalization of classical Morse theory, the topological structure of the feasible level set of MPCC is changing if and only if a level is passed that contains a C-stationary point. Therefore, this topological structure (e.g. the number of connected components of the feasible set) is changing when the level of an M-stationary point is passed. Such changes play a crucial role for numerical solution methods via global optimization or homotopy methods.
Obviously, an S-stationary point is also M-stationary but not vice versa. By Theorem 4.2, the strong stability of an M-stationary pointx implies thatx is also a strongly stable Sstationary point where the latter is defined analogously. Moreover, ifx is a local minimizer, thenx is an S-stationary point. This property is closely related to the discussion in [9] where S-stationarity is considered as the most appropriate stationarity concept for both optimality conditions and sensitivity analysis.

Remark 5.3
Since we assumed that MPCC-LICQ holds at the pointx ∈ M (P ) under consideration, the corresponding Lagrange vector (ρ,σ ) is uniquely determined. A moment of reflection shows that, ifx ∈ MS (P ), then the properties of strong stability (existence, uniqueness and continuous dependence on the perturbation) is not only satisfied for x but also for the pair (x, (ρ, σ )).
Funding Open access funding provided by University of Bergen (incl Haukeland University Hospital).
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