JOHANNES KEPLER UNIVERSITY LINZ Institute of Computational Mathematics New Verifiable Stationarity Concepts for a Class of Mathematical Programs with Disjunctive Constraints

In this paper we consider a sufficiently broad class of nonlinear mathematical programs with disjunctive constraints, which, e.g., include mathematical programs with complemetarity/vanishing constraints. We present an extension of the concept of Qstationarity as introduced in the recent paper [2]. Q-stationarity can be easily combined with the well-known notion of M-stationarity to obtain the stronger property of so-called QM -stationarity. We show how the property of QM -stationarity (and thus also of Mstationarity) can be efficiently verified for the considered problem class by computing Q-stationary solutions of a certain quadratic program. We consider further the situation that the point which is to be tested for QM -stationarity, is not known exactly, but is approximated by some convergent sequence, as it is usually the case when applying some numerical method.


Introduction
In this paper we consider the following mathematical program with disjunctive constraints (MPDC) min subject to where the mappings f : R n → R and F i : R n → R l i , i = 1, . . ., m D are assumed to be continuously differentiable and D j i ⊂ R l i , j = 1, . . ., K i , i = 1, . . ., m D are convex polyhedral sets.
Denoting m := m D i=1 l i , 1 we can rewrite the MPDC (1) in the form It is easy to see that D can also be written as the union of m D i=1 K i convex polyhedral sets by As an example for MPDC consider a mathematical program with complementarity constraints (MPCC) given by min subject to g i (x) ≤ 0, i = 1, . . .m I , with f : R n → R, g i : R n → R, i = 1, . . ., m I , h i : R n → R, i = 1, . . ., m E , G i , H i : R n → R, i = 1, . . ., m C .This problem fits into our setting (1) with m D = m C + 1, MPCC is known to be a difficult optimization problem, because, due to the complementarity constraints G i (x) ≥ 0, H i (x) ≥ 0, G i (x)H i (x) = 0, many of the standard constraint qualifications of nonlinear programming are violated at any feasible point.Hence it is likely that the usual Karush-Kuhn-Tucker conditions fail to hold at a local minimizer and various first-order optimality conditions such as Abadie (A-), Bouligand (B-), Clarke (C-), Mordukhovich (M-) and Strong (S-) stationarity conditions have been studied in the literature [6,9,17,19,20,24,25,26,27].
Another prominent example is the mathematical program with vanishing constraints (MPVC) min subject to with f : R n → R, g i : R n → R, i = 1, . . ., m I , h i : R n → R, i = 1, . . ., m E , G i , H i : R n → R, i = 1, . . ., m V .Again, the problem MPVC can be written in the form (1) with m D = m V + 1, F 1 , D 1 1 as in the case of MPCC and Similar as in the case of MPCC, many of the standard constraint qualifications of nonlinear programming can be violated at a local solution of ( 6) and a lot of stationarity concepts have been introduced.For a comprehensive overview for MPVC we refer to [15] and the references therein.
However, when we do not formulate MPCC or MPVC as a nonlinear program but as a disjunctive program MPDC, then first-order optimality conditions can be formulated which are valid under weak constraint qualifications.We know that a local minimizer is always Bstationary, which geometrically means that no feasible descent direction exists, or, in a dual formulation, that the negative gradient of the objective belongs to the regular normal cone of the feasible region, cf.[23,Theorem 6.12].The difficult task is now to estimate this regular normal cone.For this regular normal cone always a lower inclusion is available, which yields so-called S-stationarity conditions.For an upper estimate one can use the limiting normal cone which results in the so-called M-stationarity conditions.The notions of S-stationarity and M-stationarity have been introduced in [7] for general programs (3).S-stationarity always implies B-stationarity, but it requires some strong qualification condition on the constraints.On the other hand, M-stationarity requires only some weak constraint qualification but it does not preclude the existence of feasible descent directions.Further, it is not known in general how to efficiently verify the M-stationarity conditions, since the description of the limiting normal cone involves some combinatorial structure which is not known to be resolved without enumeration techniques.These difficulties in verifying M-stationarity have also some impact for numerical solution procedures.E.g., for many algorithms for MPCC it cannot be guaranteed that a limit point is M-stationary, cf.[18].
In the recent paper [2] we derived another upper estimate for the regular normal cone yielding so-called Q-stationarity conditions.Q-stationarity can be easily combined with Mstationarity to obtain so-called Q M stationarity which is stronger than M-stationarity.For the disjunctive formulations of the problems MPCC and MPVC the Q-and Q M -stationarity conditions have been worked out in detail in [2].In this paper we extend this approach to the general problem MPDC.We show that under a qualification condition which ensures S-stationarity of local minimizers, Q-stationarity and S-stationarity are equivalent.Further we prove that under some weak constraint qualification every local minimizer of MPDC is a Q M -stationary solution and we provide an efficient algorithm for verifying Q M -stationarity of some feasible point.More exactly, this algorithm either proves the existence of some feasible descent direction, i.e. the point is not B-stationary, or it computes multipliers fulfilling the Q M -stationarity condition.To this end we consider quadratic programs with disjunctive constraints (QPDC), i.e., the objective function f in MPDC is a convex quadratic function and the mappings F i , i = 1, . . ., m D are linear.We propose a basic algorithm for QPDC, which either returns a Q-stationary point or proves that the problem is unbounded.Further we show that M-stationarity for MPDC is related with Q-stationarity of some QPDC and the combination of the two parts yields the algorithm for verifying Q M -stationarity.
Our approach is well suited to the MPDC (1) when all the numbers K i , i = 1, . . ., m D are small or of moderate size.Our disjunctive structure is not induced by integral variables like, e.g., in [16].It is also not related to the approach of considering the convex hull of a family of convex sets like in [1,4].
The outline of the paper is as follows.In Section 2 we recall some basic definitions from variational analysis and discuss various stationarity concepts.In Section 3 we introduce the concepts of Q-and Q M -stationarity for general optimization problems.These concepts are worked out in more detail for MPDC in Section 4. In Section 5 we consider quadratic programs with disjunctive linear constraints.We present a basic algorithm for solving such problems, which either return a Q-stationary solution or prove that the problem is not bounded below.In the next section we demonstrate how this basic algorithm can be applied to a certain quadratic program with disjunctive linear constraints in order to verify M-stationarity or Q M -staionarity of a point or to compute a descent direction.In the last Section 7 we present some results for numerical methods for solving MPDC which prevent convergence to non M-stationary and non Q M -stationary points.
Our notation is fairly standard.In Euclidean space R n we denote by • and •, • the Euclidean norm and scalar product, respectively, whereas we denote by u ∞ := max{|u i | | i = 1, . . ., n} the maximum norm.The closed ball around some point x with radius r is denoted by B(x, r).Given some cone Q ⊂ R n , we denote by Q • := {q * ∈ R n | q * , q ≤ 0∀q ∈ Q} its polar cone.By d(x, A) := inf{ x − y | y ∈ A} we refer to the usual distance of some point x to a set A. We denote by 0 + C the recession cone of a convex set C.

Preliminaries
For the reader's convenience we start with several notions from variational analysis.Given a set Ω ⊂ R n and a point z ∈ Ω, the cone is called the (Bouligand/Severi) tangent/contingent cone to Ω at z.The (Fréchet) regular normal cone to Ω at z ∈ Ω can be equivalently defined either by where z Ω →z means that z → z with z ∈ Ω, or as the dual/polar to the contingent cone, i.e., by N Ω (z) := T Ω (z) • .
For convenience, we put NΩ (z) := ∅ for z / ∈ Ω.Further, the (Mordukhovich) limiting/basic normal cone to Ω at z ∈ Ω is given by If Ω is convex, then both the regular and the limiting normal cones coincide with the normal cone in the sense of convex analysis.Therefore we will use in this case the notation N Ω .Consider now the general mathematical program where denote the feasible region of the program (7).Then a necessary condition for a point x ∈ Ω being locally optimal is ∇f (x), u ≥ 0 ∀u ∈ T Ω (x), (9) which is the same as cf. [23,Theorem 6.12].The main task of applying this first-order optimality condition now is the computation of the regular normal cone N Ω (x) which is very difficult for nonconvex D.
We always have the inclusion but equality will hold in (11) and In order to state an upper estimate for the regular normal cone N Ω (x) we need some constraint qualification.
1. We say that the generalized Abadie constraint qualification (GACQ) holds at x if where T lin Ω (x) := {u ∈ R n | ∇F (x)u ∈ T D (F (x))} denotes the linearized cone.
2. We say that the generalized Guignard constraint qualification (GGCQ) holds at x if Obviously GGCQ is weaker than GACQ, but GACQ is easier to verify by using some advanced tools of variational analysis.E.g., if the mapping x ⇒ F (x) − D is metrically subregular at (x, 0) then GACQ is fulfilled at x, cf.[14, Proposition 1].Tools for verifying metric subregularity of constraint systems can be found e.g. in [11].
Note that we always have N T D (F (x)) (0) ⊂ N D (F (x)), see [23,Proposition 6.27].However, if D is the union of finitely many convex polyhedral sets, then equality holds.This is due to the fact that by the assumption on D there is some neighborhood Let us mention that metric subregularity of the constraint mapping x ⇒ F (x) − D at (x, 0) does not only imply GACQ and consequently GGCQ, but also metric subregularity of the mapping u ⇒ ∇F (x)u − T D (F (x)) at (0, 0) with the same modulus, see [10,Proposition 2.1].
The concept of metric subregularity has the drawback that, in general, it is not stable under small perturbations.It is well known that the stronger property of metric regularity is robust.Definition 3. A multifunction Ψ : R n ⇒ R m is called metrically regular near a point (x, ȳ) of its graph gph Ψ with modulus κ > 0, if there are neighborhoods U of x and V of ȳ such that The infimum of the moduli κ for which the property of metric regularity holds is denoted by reg Ψ(x, ȳ).
In the following proposition we gather some well-known properties of metric regularity: Proposition 2. Let x ∈ F −1 (D) where F : R n → R m is continuously differentiable and D is the union of finitely many convex polyhedral sets and consider the multifunctions x ⇒ Ψ(x) := F (x) − D and u ⇒ DΨ(x)(u) := ∇F (x)u − T D (F (x)).Then Moreover for every κ > reg Ψ(x, ȳ) there is a neighborhood W of x such that for all x ∈ W the mapping u ⇒ ∇F (x)u − T D (F (x)) is metrically regular near (0, 0) with modulus κ, and Proof.The statement follows from [23,Exercise 9.44] together with the facts that by our assumption on D condition (17) holds and that T D (F (x)) is a cone.
We now recall some well known stationarity concepts based on the considerations above.
Definition 4. Let x be feasible for the program (7).
(ii) We say that x is S-stationary, if S-and M-stationarity have been introduced in [7] as a generalization of these notions for MPCC.Using the inclusion (5) it immediately follows, that S-stationarity implies Bstationarity.However the reverse implication only holds true under some additional condition on the constraints, e.g.under the assumptions of Theorem 1.A B-stationary point is M-stationary under the assumptions of Proposition 1.However, the inclusion N Ω (x) ⊂ ∇F (x) T N D (F (x)) can be strict, implying that a M-stationary point x needs not to be Bstationary.Hence M-stationarity does eventually not preclude the existence of feasible descent directions, i.e. directions u ∈ T Ω (x) with ∇f (x), u < 0.
3 On Q-and Q M -stationarity In this section we consider an extension of the concept of Q-stationarity as introduced in the recent paper [2].Q-stationarity is based on the following simple observation.
Consider the general program (7), assume that GGCQ holds at the point x ∈ Ω and assume that we are given If we further assume that (F (x and by taking into account, that by [2, Lemma 1] we have for arbitrary sets S 1 , S 2 ⊂ R m , we obtain Here we use the convention that for for sets S 1 , . . ., S K ⊂ R m we set K i=l S i = R m for l > K.It is an easy consequence of (11), that equality holds in this inclusion, provided ∇F (x Hence we have shown the following theorem.
Theorem 2. Assume that GGCQ holds at x ∈ Ω and assume that Q then Further, if then equality holds in (20) and Remark 1. Condition ( 19) is e.g.fulfilled, if for each i = 1, . . ., K either there is a direction The proper choice of Q 1 , . . ., Q K is crucial in order that (20) provides a good estimate for the regular normal cone.It is obvious that we want to choose the cones Q i , i = 1, . . ., K as large as possible in order that the inclusion (20 because then equation ( 21) holds whenever ∇F (x) has full rank.We now show that (21) holds not only under this full rank condition but also under some weaker assumption.
Theorem 3. Assume that GGCQ holds at x ∈ Ω and assume that we are given convex cones (22) and Then In particular, (23) holds if there is a subspace such that (13) holds.
Proof.The statement follows from Theorem 2 if we can show that (21) holds.Consider Hence x * ∈ ∇F (x) T N D (F (x)) and ( 21) is verified.In order to show the last assertion note that from (24) we conclude L ⊂ Q i and consequently it follows that (23) holds.
The following definition is motivated by Theorem 2.
Definition 5. Let x be feasible for the program (7) and let Q 1 , . . ., Q K be convex cones contained in T D (F (x)) fulfilling (19).
Note that this definition is an extension of the definition of Q-and Q M -stationarity in [2], where only the case K = 2 was considered.
The following corollary is an immediate consequence of the definitions and Theorem 2.
Corollary 2. Assume that GGCQ is fulfilled at the point x feasible for (7).Further assume that we are given convex cones 21) is fulfilled, then x is S-stationary and consequently B-stationary.
We know that under the assumptions of Proposition 1 every B-stationary point x for the problem ( 7) is both M-stationary and Q-stationary with respect to every collection of cones Comparing this relation with the definition of Q M -stationarity we see that Q M -stationarity with respect to Q 1 , . . ., Q K is stronger than the simultaneous fulfillment of M-stationarity and Q-stationarity with respect to Q 1 , . . ., Q K .We refer to [2, Example 2] for an example which shows that Q M -stationarity is strictly stronger than M-stationarity.However, to ensure Q M -stationarity of a B-stationary point x, some additional assumption has to be fulfilled.
Lemma 1.Let x be B-stationary for the program (7) and assume that the assumptions of Proposition 1 are fulfilled at x.Further assume that for every λ ∈ N T D (F (x)) (0) there exists a convex cone Proof.From the definition of B-stationarity and ( 16) we deduce the existence of λ (19).Similar to the derivation of Theorem 2 we obtain and the lemma is proved.
Lemma 2. Let x be feasible for (7) and assume that T D (F (x)) is the union of finitely many closed convex cones C 1 , . . ., C p .Then for every λ ∈ N T D (F (x)) (0) there is some ī ∈ {1, . . ., p} . By the definition of the limiting normal cone there are −→ 0 and λ k → λ with By passing to a subsequence if necessary we can assume that there is an index ī such that t k ∈ Cī for all k and we obtain If T D (F (x)) is the union of finitely many convex polyhedral cones C 1 , . . ., C p , then the mapping u ⇒ ∇F (x)u − T D (F (x)) is a polyhedral multifunction and thus metrically subregular at (0, 0) by Robinson's result [21].Further we know that for any convex polyhedral cone Hence we obtain the following corollary.
Corollary 3. Assume that x is B-stationary for the program (7), that GGCQ is fulfilled at x and that T D (F (x)) is the union of finitely many convex polyhedral cones.Then there is a convex polyhedral cone

Application to MPDC
It is clear that Q-stationarity is not a very strong optimality condition for every choice of Q 1 , . . ., Q K ⊂ T D (F (x)).As mentioned above the fulfillment of ( 22) is desirable.For the general problem (7) it can be impossible to choose the cones Q 1 , . . ., Q K such that (22) holds.If T D (F (x)) is the union of finitely many convex cones C 1 , . . ., C p then we obviously have However, to consider Q-stationarity with respect to C 1 , . . ., C p is in general not a feasible approach because p is often very large.We will now work out that the concepts of Q-and Q M -stationarity are tailored for the MPDC (1).In what follows let D and F be given by (2).Given a point y = (y 1 , . . ., y m D ) ∈ D, we denote by the indices of sets D j i which contain y i .Further we choose for each i = 1, . . ., m D some index set J i (y) ⊂ A i (y) such that Obviously the choice J i (y) = A i (y) is feasible but for practical reasons it is better to choose then we will not include j 2 in J i (y).Such a situation can occur e.g. in case of MPVC when Since for every i = 1, . . ., m D the set D i is the union of finitely many convex polyhedral sets, for every tangent direction t ∈ T D i (y i ) we have y i + αt ∈ D i for all α > 0 sufficiently small.Hence we can apply [13, Proposition 1] to obtain with D(ν) given by ( 4), and We will apply this setting in particular to points y = F (x) with x feasible for MPDC.
Lemma 3. Let x be feasible for the MPDC (1) and assume that we are given K elements ν 1 , . . ., ν K ∈ J (F (x)) such that Then for each l = 1, . . ., K the cone Proof.Obviously for every l = 1, . . ., K the cone Q l is convex and polyhedral because it is the product of convex polyhedral cones.This implies ∇F (x (27).By taking into account (27) the last assertion follows from Definition 6.Let x be feasible for the MPDC (1) and let index sets J i (F (x)) ⊂ A i (x), i = 1, . . ., m D fulfilling (26) be given.Further we denote by Q(x) the collection of all elements (ν 1 , . . ., ν K ) with ν l ∈ J (F (x)) = m D i=1 J i (F (x)), l = 1, . . ., K such that (28) holds.
1. We say that Definition 6 is an extension of the definition of Q-and Q M -stationarity made for MPCC and MPVC in [2].Note that the number K appearing in the definition of Q(x) is not fixed.Denoting K min (x) the minimal number K such that (ν 1 , . . ., ν K ) ∈ Q(x), we obviously have We see from (27) that the tangent cone T D (F (x)) is the union of the |J (F (x))| = m D i=1 |J i (F (x))| convex polyhedral cones T D(ν) (y).Hence the minimal number K min (x) is much smaller than the number of components of the tangent cone, except when all or nearly all sets J i (F (x)) have cardinality 1.Further it is clear that for every ν 1 ∈ J (F (x)) and every K ≥ K min (x) we can find ν 2 , . . ., ν K ∈ J (F (x)) such that (ν 1 , . . ., ν K ) ∈ Q(x).
We allow K to be greater than K min (x) for numerical reasons primarily.Recall that for testing Q-stationarity with respect to (ν 1 , . . ., ν K ), we have to check for all l = 1, . . ., K whether −∇f (x) ∈ ∇F (x) T Q • l , or equivalently, that u = 0 is a solution of the linear optimization program min ∇f (x), u subject to . Theoretically the treatment of degenerated linear constraints is not a big problem but the numerical practice tells us the contrary.In [3] we have implemented an algorithm for solving MPVC based on Q-stationarity and the degeneracy of the linear constraints was the reason when the algorithm crashed.The possibility of choosing K > K min (x) gives us more flexibility to avoid linear programs with degenerated constraints.
The following theorem follows from Corollaries 2, 3, Theorem 3 and the considerations above.
Theorem 4. Let x be feasible for the MPDC (1) and assume that GGCQ is fulfilled at x.

On quadratic programs with disjunctive constraints
In this section we consider the special case of quadratic programs with disjunctive constraints (QPDC) min where B is a positive semidefinite n × n matrix, d ∈ R n , A i , i = 1, . . ., m D are l i × n matrices and D j i ⊂ R l i , i = 1, . . ., m D , j = 1, . . ., K j are convex polyhedral sets, i.e., QPDC is a special case of MPDC with f (x) = q(x) and In what follows we denote by A the m × n matrix where m := m D i=1 l i .We start our analysis with the following preparatory lemma.

Lemma 4. Assume that the convex quadratic program
is feasible, where B is some symmetric positive semidefinite n × n matrix, d ∈ R n , A is a m × n matrix and C ⊂ R m is a convex polyhedral set.Then either there exists a direction w satisfying Bw = 0, or the program (32) has a global solution x.
Proof.Assume that for every w with Bw = 0, Aw ∈ 0 + C we have d T w ≥ 0, i.e. 0 is a global solution of the program Since C is a convex polyhedral set, its recession cone 0 + C is a convex polyhedral cone and so is {0} n × 0 + C as well.Hence and from the first-order optimality condition −d ∈ N S (0) we derive the existence of multipliers The convex polyhedral set C is the sum of the convex hull Σ of its extreme points and its recession cone.Hence for every x feasible for (32) there is some c 1 ∈ Σ and some c 2 ∈ 0 + C such that Ax = c 1 + c 2 and, by taking into account µ T C c 2 ≤ 0, we obtain 1 2 x The set Σ is compact and we conclude that the objective of (32) is bounded below on the feasible domain is finite and there remains to show that the infimum is attained.Consider some sequence x k 2 is bounded as well and we can conclude also the boundedness of d T x k .By passing to a subsequence we can assume that the sequence (B 1/2 x k , d T x k ) converges to some (z, β) and it follows that α = 1 2 z 2 + β.Since C is a convex polyhedral set, it follows by applying [22,Theorem 19.3] twice, that the sets A −1 C and {(B 1/2 u, d T u) | u ∈ A −1 C} are convex and polyhedral.Since convex polyhedral sets are closed, it follows that (z, In what follows we assume that we have at hand an algorithm for solving (32), which either computes a global solution x or a descent direction w fulfilling (33).Such an algorithm is e.g. the active set method as described in [8], where we have to rewrite the constraints equivalently in the form A T a i , x ≤ b i , i = 1, . . ., p using the representation of C as the intersection of finitely many half-spaces, C = {c | a i , c ≤ b i , i = 1, . . ., p}.
Consider now the following algorithm.
Algorithm 1 (Basic algorithm for QPDC).Input: starting point x 1 feasible for the QPDC (31).1.) Set the iteration counter k := 1. 2.) Select (ν k,1 , . . ., ν k,K ) ∈ Q(x k ) and consider for l = 1, . . ., K the quadratic programs If one of these programs is unbounded below, stop the algorithm and return the current iterate x k together with ν := ν k,l and the descent direction w fulfilling (33).Otherwise let x k,l , l = 1, . . ., K denote the global solutions of (QP k,l ).3.) If q(x k ) = q(x k,l ), l = 1, . . ., K, stop the algorithm and return x k together with ν := ν k,1 .4.) Choose l ∈ {1, . . ., K} with q(x k,l ) < q(x k ), set x k+1 = x k,l , increase the iteration counter k := k + 1 and go to step 2.) Note that the iterate x k is feasible for every quadratic subproblem (QP k,l ).Further note that the number K will also depend on x k .Theorem 5. Algorithm 1 terminates after a finite number of iterations either with some feasible point and some descent direction w indicating that QPDC is unbounded below or with some Q-stationary solution.
Proof.If Algorithm 1 terminates in step 2.) the output is a feasible point together with some descent direction showing that QPDC is unbounded below.If the algorithm does not terminate in step 2.) the computed sequence of function values q(x k ) is strictly decreasing.Moreover, denoting ν k := ν k−1,l where l is the index chosen in step 4., we see that for each k ≥ 2 the point x k is global minimizer of the problem min q(x) subject to Ax ∈ D(ν k ).
This shows that all the vectors ν k must be pairwise different and since there is only a finite number of possible choices for ν k , the algorithm must stop in step 3.).We will now show that the final iterate x k is Q-stationary with respect to (ν k,1 , . . ., ν k,K ).Since for each l = 1, . . ., K the point x k is a global minimizer of the subproblem (Q k,l ), it also satisfies the first order optimality condition This shows Q-stationarity of x k and the theorem is proved.

On verifying Q M -stationarity for MPDC
The following theorem is crucial for the verification of M-stationarity.Theorem 6. (i) Let x be feasible for the general program (7).If there exists a B-stationary solution of the program then x is M-stationary.
(ii) Let x be B-stationary for the MPDC (1) and assume that GGCQ holds at x. Then the program (35) has a global solution.
Proof.(ii) Consider for arbitrarily fixed ν ∈ J (F (x)) the convex quadratic program min Assuming that this quadratic program does not have a solution, by Lemma 4 we could find a direction (w u , w v ) satisfying This implies ) and ∇f (x), w u < 0 and thus, together with GGCQ, −∇f (x) ∈ (T lin Ω (x)) • = N D (F (x)) contradicting our assumption that x is B-stationary for (1).Hence the quadratic program (36) must possess some global solution (u ν , v ν ).By choosing ν ∈ J (F (x)) such that ∇f (x), We now want to apply Algorithm 1 to the problem (35).Note that the point (0, 0) is feasible for (35) and therefore we can start Algorithm 1 with (u 1 , v 1 ) = (0, 0).Corollary 4. Let x be feasible for the MPDC (1) and apply Algorithm 1 to the QPDC (35).If the algorithm returns an iterate together with some descent direction indicating that (35) is unbounded below and if GGCQ is fulfilled at x, then x is not B-stationary.On the other hand, if the algorithm returns a Q-stationary solution, then x is M-stationary.
Proof.Observe that in case when Algorithm 1 returns a Q-stationary solution, by Theorem 4(ii) this solution is B-stationary because the Jocobian of the constraints (∇F (x) . . .I) obviously has full rank.Now the statement follows from Theorem 6.
We now want to analyze how the output of Algorithm 1 can be further utilized.Recalling that T D (F (x)) has the disjunctive structure we define for y = (y 1 , . . ., y m D ) ∈ T D (F (x)) the index sets Further we choose for each i = 1, . . ., m D some index set and set Note that we always have J T D (y) ⊂ J (F (x)).
In order to verify Q-stationarity for the problem (35) at some feasible point (u, v), we have to consider the set Q T D (u, v) consisting of all (ν 1 , . . ., ν K ) with ν l ∈ J T D (∇F (x)u + v), l = 1, . . ., K such that At the k-th iterate (u k , v k ) we have to choose (ν k,1 , . . ., ν k,K ) ∈ Q T D (u k , v k ) and then for each l = 1, . . ., K we must analyze the convex quadratic program If for some l ∈ {1, . . ., K} this quadratic program is unbounded below then Algorithm 1 returns the index ν := ν k, l together with a descent direction (w u , w v ) fulfilling, as argued in the proof of Theorem 6(ii), Therefore w u constitutes a feasible descent direction, provided GACQ holds at x, i.e., for every α > 0 sufficiently small the projection of x + αw u on the feasible set F −1 (D) yields a point with a smaller objective function value than x.If GACQ also holds for the constraint F (x) ∈ D(ν) at x, then we can also project the point x + αw u on F −1 (D(ν)) in order to reduce the objective function.Now assume that the final iterate (u k , v k ) of Algorithm 1 is Q-stationary for (35) and consequently x is M-stationary for the MPDC (1).Setting λ := −v k , the first order optimality conditions for the quadratic programs (QP k,l ) result in 1 is the index vector returned from Algorithm 1.Now choosing ν 2 , . . ., ν K such that (ν, ν 2 , . . ., ν K ) ∈ Q(x) we can simply check by testing −∇f (x) ∈ N D(ν l ) (F (x)), l = 2, . . ., K, whether x is Q M stationary or x is not B-stationary.
Further we have the following corollary.

Numerical aspects
In practice the point x which should be checked for M-stationarity and Q M -stationarity, respectively, often is not known exactly.E.g., x can be the limit point of a sequence generated by some numerical method for solving MPDC.Hence let us assume that we are given some point x close to x and we want to state some rules when we can consider x as approximately M-stationary or Q M -stationary.Let us assume that the convex polyhedral sets D j i have the representation where without loss of generality we assume a i,j l = 1.We use here the following approach.
T j i (x, ) .
In the first step of Algorithm 2 we want to estimate the tangent cone T D (F (x)).In fact, to calculate T D (F (x)) we need not to know the point F (x), we only need the index sets of constraints active at x and these index sets are approximated by -active constraints.Note that whenever Ãi (x, ) = Ãi (x, 0) = A i (F (x)) and Ĩj i (x, ) = Ĩj i (x, 0), i = 1, . . ., m D , j ∈ A i (F (x)) this approach yields the exact tangent cones T D j i (F (x)) = T j i (x, ) for all i = 1, . . ., m D , j ∈ A i (F (x)).To be consistent with the notation of Section 4 we make the convention that in this case the index vector ν computed in step 2.) belongs to J (x) and also, whenever we determine ν 2 , . . .ν K is step 4.), we have (ν, ν 2 , . . ., ν K ) ∈ Q(x).The regularization term σ 2 u 2 in QP DC(x, , σ) forces the objective to be strictly convex and therefore Algorithm 1 will always terminate with a Q-stationary solution.Further note that the verification of (38) requires the solution of K − 1 linear optimization problems.
The following theorem justifies Algorithm 2. Im the sequel we denote by M(x) (M sub (x)) the set of all ν ∈ J (x) such that the mapping F (•) − D(ν) is metrically regular near (x, 0) (metrically subregular at (x, 0)).Theorem 7. Let x be feasible for the MPDC (1) and assume that ∇f and ∇F are Lipschitz near x.Consider sequences x t → x, t ↓ 0, σ t ↓ 0 and η t ↓ 0 with and let (ũ t , ṽt ), νt and eventually ν t,2 . . ., ν t,Kt and lt denote the output of Algorithm 2 with input data (x t , t , σ t , η t ).
(i) For all t sufficiently large and for all i ∈ {1, . . ., m D } we have (ii) Assume that the mapping x ⇒ F (x) − D is metrically regular near (x, 0).
(a) If x is B-stationary then for all t sufficiently large the point x t is accepted as approximately M-stationary and approximately Q M -stationary.
(b) If for infinitely many t the point x t is accepted as approximately M-stationary then x is M-stationary.
(c) If for infinitely many t the point x t is accepted as approximately Q M -stationary and {ν t , ν t,2 , . . ., ν t,Kt } ⊂ M(x) then x is Q M -stationary.
(d) For every t sufficiently large such that the point x t is not accepted as approximately M-stationary and νt ∈ M sub (x) we have min{f (e) For every t sufficiently large such that the point x t is not accepted as approximately Q M -stationary and ν t, lt ∈ M sub (x) we have min{f Proof.(i) Let R > 0 be chosen such that f , F and their derivatives are Lipschitz on B(x, R) with constant L. It is easy to see that we can choose > 0 such that for all i ∈ {1, . . ., m D } we have Ãi (x, ) = Ãi (x, 0) = A i (F (x)) and such that for every j ∈ A i (F (x)) we have Ĩj i (x, ) = Ĩj i (x, 0).Consider t with x t − x < R, L x t − x < t < /2 and fix i ∈ {1, . . ., m D }.For every j ∈ A i (F (x)) we have showing Ãi (x t , t ) = A i (F (x)).Now fix j ∈ A i (F (x)) and let l ∈ Ĩj i (x, 0), i.e. a i,j l , F i (x) = b i,j l .By taking into account a i,j l = 1 we obtain showing l ∈ Ĩj i (x t , t ).Hence Ĩj i (x t , t ) = Ĩ(x, 0).Because of our assumptions we have x t − x < R and L x t − x < t < /2 for all t sufficiently large and this proves (39).
(ii) In view of Proposition 2 we can choose κ large enough such that the mappings are metrically regular near (x, 0) with modulus κ.By eventually shrinking R we can assume that for every x ∈ B(x, R) the mappings u ⇒ ∇F (x)u − T D (F (x)), u ⇒ ∇F (x)u − T D(ν) (F (x)), ν ∈ M(x) are metrically regular near (0, 0) with modulus κ + 1.
Without loss of generality we can assume that x t ∈ B(x, R) and (39) holds for all t implying that T D j i (F (x)) = T j i (x, t ) holds for all i = 1, . . ., m D , j ∈ A i (F (x)).In fact then the problem QP DC(x t , t , σ t ) is the same as min To prove the statement that x t is also accepted as approximately Q M -stationary for all t sufficiently large we can proceed in a similar way.Assume on the contrary that x is B-stationary but for infinitely many t the point x t is not accepted as approximately Q M -stationary.For those t let w t denote some element fulfilling ∇F (x t )w t ∈ T D(ν t, lt ) ⊂ T D (F (x)), w t ∞ ≤ 1 and ∇f (x t ), w t ≤ −η t .Then, similar as before we can find ŵt ∈ ∇F (x) −1 T D (F (x)) such that ŵt − w t ≤ κ ∇F (x) − ∇F (x t ) w t ≤ κL √ n x t − x and for large t we obtain ∇f (x), ŵt ≤ ∇f (x t ), w t + ∇f (x) − ∇f (x t ) w t + ∇f (x) ŵt − w t ≤ −η t + L √ n(1 + κ ∇f (x) ) x t − x < 0 contradicting B-stationarity of x.(b) By passing to a subsequence we can assume that for all t the point x t is accepted as approximately M-stationary and hence σ t u t ≤ η t → 0. By (43) we have that the sequence λ t ∈ N T D (F (x)) (0) is uniformly bounded and by passing to a subsequence once more we can assume that it converges to some λ ∈ N T D (F (x)) (0).By [23,Proposition 6.27] we have λ ∈ N D (F (x)) and together with 0 = lim t→∞ ∇f (x t ) + ∇F (x t ) T λ t = ∇f (x) + ∇F (x) T λ M-stationarity of x is established.(c) By passing to a subsequence we can assume that for all t the point x t is accepted as approximately Q M -stationary and {ν t , ν t,2 , . . ., ν t,Kt } ⊂ M(x).Hence for all t the point x t is also accepted as M-stationary and by passing to a subsequence and arguing as in (b) we can assume that λ t converges to some λ ∈ N D (F (x)) fulfilling ∇f (x) + ∇F (x) T λ = 0. Since the set M(x) is finite, by passing to a subsequence once more we can assume that there is a number K and elements ν, ν 2 , . . ., ν K such that K t = K , νt = ν and ν t,l = ν l , l = 2, . . ., K holds for all t.Since we assume that (39) holds we have (ν, ν 2 , . . ., ν K ) ∈ Q(x) and we will now show that x is Q M -stationary with respect to (ν, ν 2 , . . ., ν K ).Since (ũ t , ṽt ) also solves (41), it follows that λ t = −v t ∈ N T D(ν) (F (x)) (∇F (x t )ũ t + ṽt ) ⊂ N D(ν) (F (x)) and thus λ ∈ N D (F (x)) ∩ N D(ν) (F (x)) implying −∇f (x) ∈ ∇F (x) T N D (F (x)) ∩ T D(ν) (F (x)) • .

1 2 xT
B x + d T x = α follows.This shows that x is a global minimizer for (32).
accept x as approximately Q M -stationary.Otherwise consider the nonlinear programming problem min f (x) subject to F (x) ∈ D(ν l) (18)ect to ∇F (x t )u + v ∈ T D (F (x)).The point (ũ t , ṽt ) is Q-stationary for this program and thus also S-stationary by Theorem 4(ii) and the full rank property of the matrix (∇F (x t ) ...I).Hence there is a multiplierλ t ∈ N T D (F (x)) (∇F (x t )ũ t + ṽt ) ⊂ N T D (F (x)) (0) fulfilling ṽt + λ t = 0, ∇f (x t ) + σ t ũt + ∇F (x t ) T λ t = 0 and we conclude ṽt = λ t ≤ (κ + 1) ∇f (x t ) + σ t ũt(40)from(18).By Q-stationarity of (ũ t , ṽt ) we know that (ũ t , ṽt ) is the unique solution of the strictly convex quadratic programmin ∇f (x t ), u + σ t 2 u 2 + 1 2 v 2 subject to ∇F (x t )u + v ∈ T D(ν t ) (F (x)).(41)Foreveryα≥ 0 the point α(ũ t , ṽt ) is feasible for this quadratic program and thus α = 1 is solution of min Assume on the contrary that x is B-stationary but for infinitely many t the point x t is not accepted as approximately M-stationary and hence ũt ≥ η t /σ t .This implies L x t − x and by the metric regularity of u ⇒ ∇F (x)u − T D (F (x)) near (0, 0) we can find ût ∈ ∇F (x) −1 T D (F (x)) with Our choice of the parameters σ t , η t together with (42) ensures that for t sufficiently large we have t ũt + L x t − x + ∇f (x) ût − ũt ũt≤ −η t + L x t − x + ∇f (x) κ 2(κ + 1) f (x t ) σ t η t + L x t − x < 0which contradicts B-stationarity of x.Hence for all t sufficiently large the point x t must be accepted as approximately M-stationary.