Topological Approach to Mathematical Programs with Switching Constraints

We study mathematical programs with switching constraints (for short, MPSC) from the topological perspective. Two basic theorems from Morse theory are proved. Outside the W-stationary point set, continuous deformation of lower level sets can be performed. However, when passing a W-stationary level, the topology of the lower level set changes via the attachment of a w-dimensional cell. The dimension w equals the W-index of the nondegenerate W-stationary point. The W-index depends on both the number of negative eigenvalues of the restricted Lagrangian’s Hessian and the number of bi-active switching constraints. As a consequence, we show the mountain pass theorem for MPSC. Additionally, we address the question if the assumption on the nondegeneracy of W-stationary points is too restrictive in the context of MPSC. It turns out that all W-stationary points are generically nondegenerate. Besides, we examine the gap between nondegeneracy and strong stability of W-stationary points. A complete characterization of strong stability for W-stationary points by means of first and second order information of the MPSC defining functions under linear independence constraint qualification is provided. In particular, no bi-active Lagrange multipliers of a strongly stable W-stationary point can vanish.


Introduction
We consider the following mathematical program with switching constraints: with M[h, g, F 1 , F 2 ] = x ∈ R n h i (x) = 0, i ∈ I, g j (x) ≥ 0, j ∈ J, F 1,m (x) · F 2,m (x) = 0, m = 1, . . . , k , where f ∈ C 2 (R n , R), h ∈ C 2 (R n , R |I | ), g ∈ C 2 (R n , R |J | ), F 1 , F 2 ∈ C 2 (R n , R k ). For simplicity, we write M for M[h, g, F 1 , F 2 ] if no confusion is possible. MPSC has been introduced in [15] by arguing that switching structures, which demand at most one control to be active at any time instance, appear rather frequently in optimal control. Discretization of such optimal control problems naturally leads to MPSC. In [15], various stationarity concepts for MPSC were suggested, such as W-, M-, and S-stationarity. Under suitable MPSC-tailored constraint qualifications, necessary optimality conditions were derived. Numerical aspects of MPSC were handled in [11], where different relaxation schemes for switching constraints are discussed and compared. The goal of this paper is the study of MPSC from the topological point of view as it has been pioneered by H. Th. Jongen for nonlinear programming in [6] and popularized in [7]. Subsequent studies in this direction are summarized in [18], where, in particular, mathematical programs with complementarity constraints (MPCC) and mathematical programs with vanishing constraints (MPVC) are considered. Their feasible sets are given by means of the same defining functions as above: The main question here is how the topological properties of MPSC lower level sets change as the level a ∈ R varies. For MPSC, it turns out that the concept of W-stationarity is adequate in order to describe these changes. In particular, within this context, we present two basic theorems from Morse theory, cf. [7,16]. First, we show that, for a < b, the set M a is a strong deformation retract of M b if the set M b a = {x ∈ M | a ≤ f (x) ≤ b} does not contain W-stationary points, see Theorem 4(a). As a consequence, the homotopy type of the lower level sets M a and M b are equal. Second, if M b a contains exactly one nondegenerate W-stationary point, then M b is shown to be homotopy-equivalent to M a with a w-cell attached, see Theorem 4(b). Here, the dimension w is the so-called W-index. It depends on both the number of negative eigenvalues of the restricted Lagrangian's Hessian and the number of bi-active switching constraints. The latter fact constitutes the main difference to the cases where feasible set is described by complementarity or vanishing constraints. We remind that for MPCC the C-index of topologically relevant C-stationary points depends on the number of negative pairs of bi-active Lagrange multipliers, see [9,17]. The same is true for MPVC, where just negative pairs of bi-active Lagrange multipliers of Tstationary points matter, see [1]. For switching constraints however all pairs of bi-active Lagrange multipliers contribute to the W-index. A global interpretation of the deformation and cell-attachment theorems is as follows. Suppose that the feasible set is compact and connected, that the linear independence constraint qualification (LICQ) holds, and that all W-stationary points are nondegenerate with pairwise different functional values. Then, passing a level corresponding to a local minimizer, a connected component of the lower level set is created. Different components can only be connected by attaching one-dimensional cells. This shows the existence of at least (r − 1) W-stationary saddle points with W-index equal to one, where r is the number of local minimizers, see also [2,7]. This mountain pass result for MPSC is shown in Theorem 5.
We point out that the crucial assumption in Morse theory is the nondegeneracy of Wstationary points. A W-stationary point is called nondegenerate, see Definition 5, if (ND1) LICQ is satisfied, (ND2) the Lagrange multipliers corresponding to active inequality constraints are positive, (ND3) the Lagrange multipliers corresponding to bi-active switching constraints do not vanish, and (ND4) the restricted Hessian of the Lagrangian is nonsingular. Are the requirements ND1-ND4 too restrictive to be assumed for W-stationary points of MPSC? First, we show that all W-stationary points of a generic MPSC are nondegenerate, see Theorem 3. Genericity means that the set of MPSC defining functions (f, g, h, F 1 , F 2 ), for which all W-stationary points are nondegenerate, is open and -dense with respect to the so-called strong (or Whitney) C 2 -topology, see e. g. [4]. Second, we examine the gap between nondegeneracy and strong stability of W-stationary points. Strongly stable Wstationary points in the sense of M. Kojima [14] do not only remain locally unique under sufficiently small C 2 -perturbations of the MPSC defining functions (f, h, g, F 1 , F 2 ), but can also be continuously tracked back. All nondegenerate W-stationary points are strongly stable, but not vice versa. In this paper, we present a full characterization of strong stability for W-stationary points by means of first and second order information of the MPSC defining functions under LICQ, see Theorem 6. In particular, ND3 is necessary for strong stability of W-stationary points, i. e. no bi-active Lagrange multipliers can vanish. This new issue is in strong contrast e. g. with the characterization of strong stability for Cstationary points in MPCC, see [10]. In MPCC, strong stability includes cases where one of the bi-active Lagrange multipliers may vanish. We conclude that in absence of active inequality constraints, nondegeneracy of W-stationary points is equivalent to their strong stability.
The article is organized as follows. In Section 2 we provide notation and auxiliary results which will be used later. Section 3 contains the exposition of Morse theory for MPSC including the proofs of the deformation and cell-attaching theorem. In Section 4 we completely characterize strongly stable W-stationary points of MPSC and draw comparisons to their nondegeneracy.
Our notation is standard. The n-dimensional Euclidean space is denoted by R n with norm · . We denote the set of nonnegative numbers by H. The solution set of the basic switching relation is DF denotes its Jacobian matrix. Given a differentiable function f : R n −→ R, Df denotes the row vector of partial derivatives and D T f stands for the transposed vector.
The index set J 0 (x) corresponds to the active inequality constraints and β(x) to the bi-active switching constraints atx. Without loss of generality, we assume throughout the whole article that at the particular point of interestx ∈ M it holds: Let us start by stating the MPSC-tailored linear independence constraint qualification, which turns out to be the crucial assumption for all results to follow. Definition 1 (LICQ, cf. [15]) The linear independence constraint qualification (LICQ) for MPSC is said to hold atx ∈ M if the vectors are linearly independent.
The assumption of LICQ is justified in the sense, that it generically holds on the MPSC feasible set. In order to formulate this assertion in mathematically precise terms, the space C 2 (R n , R) will be topologized by means of the strong (or Whitney-) C 2 -topology, denoted by C 2 s , cf. [4,7]. The C 2 s -topology is generated by allowing perturbations of the functions and their derivatives up to second order which are controlled by means of continuous positive functions. The product space C 2 (R n , R l ) ∼ = C 2 (R n , R) × · · · × C 2 (R n , R) will be topologized with the corresponding product topology.

Theorem 1 (LICQ is generic) Let
denote the subset of MPSC defining functions for which LICQ holds at all feasible points. Then, F is C 2 s -open and -dense.
Proof Let us consider the following disjunctive optimization problem: Obviously, it holds: In the case where LICQ holds at a W-stationary pointx ∈ M, the Lagrange multipliers in (2) are uniquely determined. It is well-known that under LICQ a local minimum of MPSC is W-stationary [15].
Let us now locally describe the MPSC feasible set under LICQ. This is done by an appropriate change of coordinates.
Definition 3 (Coordinate system) The feasible set M admits a local C r -coordinate system of R n (r ≥ 1) atx by means of a C r -diffeomorphism Φ : U −→ V with open R nneighborhoods U and V ofx and 0, respectively, if it holds: Lemma 1 (Local structure) Suppose that LICQ holds atx ∈ M. Then M admits a local C 2 -coordinate system of R n atx.
Proof Choose vectors ξ l ∈ R n , l = 1, . . . , p, which form -together with the vectors -a basis for R n . Next we put or, shortly, Thus, the property (ii) in Definition 3 follows from the definition of Φ.

Definition 4
We will refer to the C 2 -diffeomorphism Φ defined by (6) and (7) as standard diffeomorphism.
Let us interpret the first-and second-order derivatives of the objective function f •Φ −1 in new coordinates given by the standard diffeomorphism Φ. For that, we define the Lagrange function Atx ∈ M we further consider the local part of the feasible set:

Morse Theory
Our goal is to study topological changes of MPSC lower level sets as the level of the objective function varies. For the topological concepts we refer to [7,19].
In order to provide intuition on deformation and cell-attachment in presence of switching constraints, we first consider the following simple Example 1.

Example 1 Consider the MPSC:
We see that for all a < b < 0 the lower level sets M b and M a are homotopy-equivalent. More precisely, M b is a strong deformation retract of M a , see Fig. 1. This is due to the fact that M b a does not contain any W -stationary points. The situation changes dramatically if we pass the zero level corresponding to the W-stationary point (0, 0). For a < 0 < b the lower level sets M b and M a are not homotopy-equivalent anymore. It is possible to describe the topological difference between M b and M a by means of the cell-attachment procedure. Namely, M b is homotopy-equivalent to M a with a one-dimensional cell attached along its boundary, see Fig. 2. In order to adequately describe the topological changes of MPSC lower level sets, we introduce nondegenerate W-stationary points along with their W -index. The latter will provide us with the dimension of the cell to be attached while passing the corresponding W-stationary level.
is called nondegenerate if the following conditions are satisfied: Condition ND4 means that the matrix V T D 2 L(x)V is nonsingular, where V is some matrix whose columns form a basis for the tangent space TxM 0 (x) and D 2 xx L(x,x,λ,μ,σ ) is the Hessian of the Lagrange function with respect to x-variables.
Definition 6 (W-index) Letx ∈ M be a nondegenerate W-stationary point with unique Lagrange multipliers (x,λ,μ,σ ). The number of negative eigenvalues of Note that in absence of switching constraints, the W-index has only the QI -part and coincides with the well-known quadratic index of a nondegenerate Karush-Kuhn-Tucker-point in nonlinear programming or, equivalently, with the Morse index, cf. [7,14,16]. Confer that for MPCC and MPVC the BI -part of the topologically relevant C-stationary and Tstationary points, respectively, counts the pairs of negative bi-active Lagrange multipliers, see [9] and [1]. In MPSC, all pairs of bi-active Lagrange multipliers -independently of their sign -contribute to BI , which is an essentially new phenomenon here.
The following Theorem 2 describes the local structure of MPSC around a nondegenerate W-stationary point.
where (9) there are exactly QI negative squares.
Proof Without loss of generality, we may assume f (x) = 0. Let Φ : U −→ V be a standard diffeomorphism according to Definition (4).
From now on we may assume s = 0. In view of Remark 1 we have at the origin: . . , p and ∂ 2f ∂y k 1 +n−p ∂y k 2 +n−p 1≤k 1 ,k 2 ≤p is a nonsingular matrix with QI negative eigenvalues.
In view of (iii) we may apply the Morse Lemma on the C 2 -function f (0, Y p ), cf.
as new local C 1 -coordinates. Denoting the transformed function f again by f and, recalling the signs in (i)-(ii), we obtain (9) by a straightforward application of the inverse function theorem. Here, the coordinate transformation Ψ is understood as the composition of all previous ones.
As a by-product we elaborate on how the W-index can be used to characterize nondegenerate local minimizer of MPSC.

Corollary 1 (Minimizers and W-index) Letx ∈ M be a nondegenerate W-stationary point. Then,x is a local minimizer of MPSC if and only if its W-index vanishes.
Proof Letx be a nondegenerate W-stationary point for MPSC. The application of Morse Lemma from Theorem 2 says that there exist neighborhoods U and V ofx and 0, respectively, and a local C 1 -coordinate system Ψ : U → V of R n aroundx such that (9) where y ∈ {0 s } × H |J 0 (x)| × R p . Thus, 0 is a local minimizer for (11). Vice versa, if 0 is a local minimizer for (9), then obviously β(x) = ∅ and QI = 0, hence, the W-index ofx vanishes.
The assumption of nondegeneracy is justified in the sense, that it generically holds at all W-stationary points of MPSC.

denote the subset MPSC defining functions for which each W-stationary point is nondegenerate. Then, F is C 2 s -open and -dense.
Proof Let us fix an index set J 0 ⊂ {1, . . . , |J |} of active inequality constraints, an index subset K 0 ⊂ J 0 of these active inequality constraints, pairwise disjoint index sets α, γ , β ⊂ {1, . . . , k} of switching constraints satisfying α ∪ γ ∪ β = {1, . . . , k}, an index subset δ ⊂ β of be-active switching constraints, and a number r ∈ N standing for the rank. For this choice we consider the set M J 0 ,K 0 ,α,γ ,β,δ,r of x ∈ R n such that the following conditions are satisfied: (m4) the matrix D 2 xx L(x, λ, μ, σ ) T x M(x) has rank r. Note that (m1) refers to equality and active inequality constraints, (m2) to switching constraints, while (m3) describes violation of ND2 and ND3. Furthermore, (m4) describes violation of ND4. Now, it suffices to show that M J 0 ,K 0 ,α,γ ,β,δ,r is generically empty whenever one of the sets K 0 or δ is nonempty or the rank r in (m4) is not full, i. e. r < dim (T x M(x)). In fact, the available degrees of freedom of the variables involved in each M J 0 ,K 0 ,α,γ ,β,δ,r are equal to n. The loss of freedom caused by (m1) is |I | + |J 0 |, and the loss of freedom caused by (m2) is |m α | + m γ + 2 m β . Due to Theorem 1, LICQ holds generically at any feasible x, i. e. (ND1) is fulfilled. Suppose that the sets K 0 and δ are empty, then (m3) causes a loss of freedom of n − |P | − |Q 0 | − |m α | − m γ − 2 m β . Hence, the total loss of freedom is n. We conclude that a further degeneracy, i. e. K 0 = ∅, δ = ∅ or r < dim (T x M(x)), would imply that the total available degrees of freedom n are exceeded. By virtue of the jet transversality theorem from [7], generically the sets M J 0 ,K 0 ,α,γ ,β,δ,r must be empty. For the openness result, we argue in a standard way. Locally, W-stationarity can be written via stable equations. Then, the implicit function theorem for Banach spaces can be applied to follow W-stationary points with respect to (local) C 2 -perturbations of defining functions. Finally, a standard globalization procedure exploiting the specific properties of the strong C 2 s -topology can be used to construct a (global) C 2 s -neighborhood of problem data for which the nondegeneracy property is stable, cf. [7]. Now, we are ready to state and prove deformation and cell-attachment theorems for MPSC, which constitute the core results of the paper.

Theorem 4 Let M b
a be compact and suppose that LICQ is satisfied at all points x ∈ M b a .

(a) (Deformation) If M b a does not contain any W-stationary point for MPSC, then M a is a strong deformation retract of M b . (b) (Cell-attachment) If M b a contains exactly one W-stationary point for MPSC, sayx, and if a < f (x) < b and the W-index ofx is equal to w, then M b is homotopyequivalent to M a with a w-cell attached.
Proof (a) Due to LICQ at all x ∈ M b a there exist real numbers λ i (x), i ∈ I , σ 1,m α (x), l = 1, . . . , p such that: where vectors ξ l , l = 1, . . . , p are chosen as in Lemma 1. We set: Since eachx ∈ M b a is not W-stationary for MPSC, we getx ∈ A ∪ B. The proof consists of a local argument and its globalization which are more or less standard here, cf. [9, Theorem 3.2]. Let us briefly recall the details for the sake of completeness.
First, we show the local argument: for eachx ∈ M b a there exist an (R n )-neighborhood Ux ofx, tx > 0 and a mapping Ψx : Obviously, the level sets of f are locally mapped onto the level sets of f • Φ −1 , where Φ is a C 1 -diffeomorphism according to Definition 3. Applying the standard diffeomorphism Φ from Definition 4, we consider f • Φ −1 (denoted by f again). Thus, we havex = 0 and f is given on the feasible set   + f (x 1 , . . . ,x j , . . . , x n ).
Define a local C 1 -vector field Fx as follows: Fx (x 1 , . . . , x j , . . . , x n ) = (0, . . . , 1, . . . , 0) Thus, we obtain for x ∈ M b a a unique t a (x) > 0 with Ψ (t a (x), x) ∈ M a . It is not hard (but technical) to realize that t a : x −→ t a (x) is Lipschitz continuous. Finally, we define r : [0, 1] × M b −→ M b as follows: a , τ ∈ [0, 1]. The mapping r provides that M a is a strong deformation retract of M b .
(b) Part (a) allows deformations up to an arbitrarily small neighborhood of the Wstationary pointx. In such a neighborhood, we may assume without loss of generality that x = 0 and f has the following form as from Theorem 2 (for simplicity we omit the trivial part by assuming s = 0): where x ∈ H |J 0 (x)| × S |β(x)| × R p , and the number of negative squares in (12) equals QI . In terms of [3] the set H |J 0 (x)| × S |β(x)| × R p can be interpreted as the product of the tangential part R p and the normal part H |J 0 (x)| ×S |β(x)| . The cell-attachment along the tangential part is standard. Analogously to the unconstrained case, a QI -dimensional cell has to be attached on R p . The cell-attachment along the normal part is more involved. First, we emphasize that the linear terms x j , j ∈ J 0 (x), in (12) do not contribute to the dimension of the cell to be attached. In fact, with respect to lower level sets, the 1-dimensional constrained singularity x on H plays the same role as the unconstrained singularity x 2 . In this sense the linear terms on H |J 0 (x)| can be neglected for cell-attachment. Second, the linear terms x 2m+q−1 + x 2m+q , m ∈ β(x), in (12) contribute each with the one-dimensional cell due to Example 1. Hence, the linear terms on S |β(x)| are responsible for the attachment of a BI -dimensional cellrecall that BI = |β(x)|. Finally, we apply Theorem 3.7 from Part I in [3], which says that the local Morse data is the product of tangential and normal Morse data. Hence, the dimensions of the attached cells add together. Here, a (QI + BI )-dimensional cell has to be attached. The latter corresponds to the W-index of the W-stationary pointx.

Let us present a global interpretation of Theorem 4 by showing a mountain pass theorem for MPSC.
Theorem 5 (Mountain pass) Let the MPSC feasible set M be compact and connected, and all W-stationary points of MPSC be nondegenerate. Then, it holds: where r is the number of local minimizers of MPSC and r s is the number of W-stationary saddle points with W-index equal to one.
Proof We assume without loss of generality that the objective function f has pairwise different values at all W-stationarity points of MPSC. If it is not the case, we may enforce this property by sufficiently small perturbations of the objective function. Due to the openness part in Theorem 3, all W-stationarity points of such a perturbed MPSC remain nondegenerate. Moreover, (13) is still valid since it does not depend on the functional values of f . Further, let q a denote the number of connected components of the lower level set M a . We focus on how q a changes as a ∈ R increases. Due to Theorem 4(a), q a can change only if passing through a value corresponding to a W-stationary pointx, i. e. a = f (x). In fact, Theorem 4(a) allows homotopic deformations of lower level sets up to an arbitrarily small neighborhood of the W-stationary pointx. Then, we have to estimate the difference between q a and q a−ε , where ε > 0 is arbitrarily, but sufficiently small, and a = f (x). This is done by a local argument. For that, let the W-index ofx be QI + BI . We use Theorem 4(b) which says that M a is homotopy-equivalent to M a−ε with a (QI + BI )-dimensional cell attached. Let us distinguish the following cases: 1)x is a local minimizer with vanishing W-index, cf. Corollary 1, i. e. QI = BI = 0.
Then, we attach to M a−ε the cell of dimension zero. Consequently, a new connected component is created, and it holds: 2)x is a saddle point with W-index equal to one, i. e. either QI = 1, BI = 0 or QI = 0, BI = 1 holds. Then, we attach to M a−ε the cell of dimension one. Consequently, at most one connected component disappears, and it holds: The case QI = 1, BI = 0 is well known from nonlinear programming, see e. g. [7]. The case QI = 0, BI = 1 is new and characteristic for MPSC, cf. Example 2. 4)x is W-stationary with W-index greater than one, i. e. QI + BI > 1. The boundary of the to be attached (QI + BI )-dimensional cell is thus connected. Consequently, the number of connected components of M a remains unchanged, and it holds: Now, we proceed with the global argument. Compactness of M implies that there exists c ∈ R such that M c is empty, thus, q c = 0. Additionally, there exists d ∈ R such that M d W-stationary points. The concept of strong stability is defined by means of an appropriate semi-norm. To this aim let bex ∈ R n , r > 0. For defining functions (f, h, g, F 1 , F 2 ) from (1) the seminorm (f, h, g, F 1 , F 2 ) C 2 B(x,r) is defined to be the maximum modulus of the function values and partial derivatives up to order two of f, h, g, F 1 , F 2 .
Example 4 (Instability II) Consider the MPSC: Obviously, (0, 0) is the unique W-stationary point for (17) with exactly one vanishing biactive Lagrange multiplier. Consider the following perturbation of (19) with respect to parameter t > 0: min (20) It is easy to see that (0, 0) and (0, t) are W-stationary points for (20). It means that (0, 0) is not a strongly stable W-stationary point for (19).
For characterizing strong stability, we shall use some auxiliary objects associated with a W-stationary pointx ∈ M and its multipliers (λ,μ,γ ). Let the index set of positive multipliers corresponding to the inequality constraints be given by Obviously, M 0 (x) ⊂ M * (x) and, in the case where LICQ holds atx, Proof In virtue of LICQ atx, Lemma 2 allows us to deal equivalently with the strong stability of the W-stationary pair (x,λ,μ,σ ).
This new issue is in strong contrast e. g. with the characterization of strong stability of C-stationary points in MPCC, see [10]. In MPCC, strong stability includes cases where one of the bi-active Lagrange multipliers may vanish. In other words, there is an essential discrepancy between nondegeneracy and strong stability of C-stationary points in MPCC. In MPSC, the situation is rather different: in absence of active inequality constraints, nondegeneracy of W-stationary points is equivalent to their strong stability.

Corollary 3
Let inequality constraints be inactive at a W-stationary pointx ∈ M, i. e. J 0 (x) = ∅. Then,x is nondegenerate if and only if it is strongly stable and satisfies LICQ.
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