Notes on Some Constraint Qualifications for Mathematical Programs with Equilibrium Constraints

Abstract

We study the constraint qualifications for mathematical programs with equilibrium constraints (MPEC). Firstly, we investigate the weakest constraint qualifications for the Bouligand and Mordukhovich stationarities for MPEC. Then, we show that the MPEC relaxed constant positive linear dependence condition can ensure any locally optimal solution to be Mordukhovich stationary. Finally, we give the relations among the existing MPEC constraint qualifications.

Appendices

Appendix A: Existing Constraint Qualifications for MPEC

The constraint qualifications given for MPEC in the literature include the following ones: Let x ^∗∈X.

(a):

[7] We say that the MPEC linear independence constraint qualification (MPEC-LICQ) holds at x ^∗ iff the family of gradients

is linearly independent.

(b):

[9, 15] We say that the MPEC linear constraint qualification (MPEC Linear CQ) holds iff all functions {g,h,G,H} in the constraints are affine.

(c):

[6, 7] We say that the MPEC Mangasarian–Fromovitz constraint qualification (MPEC-MFCQ) holds at x ^∗ iff there is no nonzero multipliers {λ,μ,u,v} such that

(d):

[1, 6, 9] We say that the MPEC no nonzero abnormal multiplier constraint qualification (MPEC-NNAMCQ) holds at x ^∗ iff there is no nonzero multipliers {λ,μ,u,v} such that

(e):

[15, 24] We say that the MPEC constant rank constraint qualification (MPEC-CRCQ) holds at x ^∗ iff there exists δ>0 such that, for any $\mathcal{I}_{1}\subseteq \mathcal{I}_{g}^{*}, \mathcal{I}_{2}\subseteq\{1,\ldots, q\}$, $\mathcal{I}_{3}\subseteq\mathcal{I}^{*}\cup\mathcal{J}^{*}$, and $\mathcal{I}_{4} \subseteq \mathcal{K}^{*}\cup\mathcal{J}^{*}$, the family of gradients

has the same rank for each $x\in\mathcal{B}_{\delta}(x^{*})$.

(f):

[15] We say that the MPEC relaxed constant rank constraint qualification (MPEC-RCRCQ) holds at x ^∗ iff there exists δ>0 such that, for any $\mathcal{I}_{1}\subseteq \mathcal{I}_{g}^{*}$ and $\mathcal{I}_{2}, \mathcal{I}_{3}\subseteq \mathcal{J}^{*}$, the family of gradients

has the same rank for each $x\in\mathcal{B}_{\delta}(x^{*})$.

(g):

[24, 25] we say that the MPEC constant positive linear dependent condition (MPEC-CPLD) holds at x ^∗∈X iff, for any $\mathcal{I}_{1}\subseteq\mathcal{I}_{g}^{*}$, $\mathcal{I}_{2}\subseteq\{1,\ldots, q\}$, $\mathcal{I}_{3}\subseteq\mathcal{I}^{*}\cup\mathcal{J}^{*}$, and $\mathcal{I}_{4} \subseteq\mathcal{K}^{*}\cup\mathcal{J}^{*}$, whenever there exist multipliers {λ,μ,u,v}, not all zero, with λ _i ≥0 for each $i\in\mathcal{I}_{1}$, such that

there exists a neighborhood B(x ^∗) of x ^∗ such that, for any x∈B(x ^∗), the vectors

are linearly dependent.

(g′):

[15, 21] We say that the MPEC constant positive linear dependent condition (MPEC-CPLD) holds at x ^∗∈X iff, for any $\mathcal{I}_{1}\subseteq\mathcal{I}_{g}^{*}$, $\mathcal{I}_{2}\subseteq\{1,\ldots, q\}$, $\mathcal{I}_{3}\subseteq\mathcal{I}^{*}\cup\mathcal{J}^{*}$, and $\mathcal{I}_{4} \subseteq\mathcal{K}^{*}\cup\mathcal{J}^{*}$, whenever there exist multipliers {λ,μ,u,v}, not all zero, with λ _i ≥0 for each $i\in\mathcal{I}_{1}$, either u _l v _l =0 or u _l >0,v _l >0 for each $l \in\mathcal{J}^{*}$, such that

there exists a neighborhood B(x ^∗) of x ^∗ such that, for any x∈B(x ^∗), the vectors

are linearly dependent.

(h):

[6] We say that the MPEC pseudonormality holds at x ^∗ iff there is no multipliers {λ,μ,u,v} such that

• ∇g(x ^∗)λ+∇h(x ^∗)μ−∇G(x ^∗)u−∇H(x ^∗)v=0;

• $\lambda\geq0,g(x^{*})^{T}\lambda= 0, u_{i}=0\ \mathrm{for}\ i\in\mathcal{K}^{*}, v_{i}=0\ \mathrm{for}\ i\in\mathcal{I}^{*},\ \mathrm{either}\ u_{i}v_{i}=0 \mathrm{or} \ u_{i}> 0, v_{i}> 0\ \mathrm{for}\ i\in\mathcal{J}^{*}$;

• there exists a sequence {x ^k}→x ^∗ such that, for each k,

  $$\sum_{i=1}^p \lambda_i \nabla g_i\bigl(x^k\bigr) + \sum _{j=1}^q\mu_j \nabla h_j \bigl(x^k\bigr) - \sum_{i=1}^m u_\imath\nabla G_\imath\bigl(x^k\bigr) - \sum _{i=1}^m v_\jmath\nabla H_\jmath\bigl(x^k\bigr)>0. $$

(i):

[6] We say that the MPEC quasinormality holds at x ^∗ iff there is no nonzero multipliers {λ,μ,u,v} such that

• ∇g(x ^∗)λ+∇h(x ^∗)μ−∇G(x ^∗)u−∇H(x ^∗)v=0;

• $\lambda\geq0, g(x^{*})^{T}\lambda= 0, u_{i}=0\ \mathrm{for}\ i\in\mathcal{K}^{*}, v_{i}=0\ \mathrm{for} \ i\in\mathcal{I}^{*},\ \mathrm{either}\ u_{i}v_{i}=0 \mathrm{or} \ u_{i}> 0, v_{i}> 0 \ \mathrm{for}\ i\in\mathcal{J}^{*}$;

• there exists a sequence {x ^k}→x ^∗ such that, for each k,

  

(j):

[9, 18] We say that the MPEC Abadie constraint qualification (MPEC Abadie CQ) holds at x ^∗ iff $\mathcal{T}_{X}(x^{*})=\mathcal{L}_{\mathit{MPEC}}(x^{*})$.

(k):

[5] We say that the MPEC Guignard constraint qualification (MPEC Guignard CQ) holds at x ^∗ iff $\mathcal{T}_{X}(x^{*})^{o}=\mathcal{L}_{\mathit{MPEC}}(x^{*})^{o}$.

It is easy to see that the MPEC-CPLD given in [15, 21] is weaker than the one given in [24, 25]. Note that the MPEC-CPLD in Fig. 1 is understood in the sense of [24, 25].

Appendix B: Proof of “MPEC-RCRCQ ⟹ MPEC Abadie CQ”

Suppose that the MPEC-RCRCQ holds at x ^∗∈X. Let

$$\mathcal{P}\bigl(\mathcal{J}^*\bigr):=\bigl\{\bigl(\mathcal{J}^*_1, \mathcal{J}^*_2\bigr) \mid \mathcal{J}^*_1\cup \mathcal{J}^*_2=\mathcal{J}^*, \mathcal{J}^*_1\cap \mathcal{J}^*_2=\emptyset\bigr\}. $$

For each partition $(\mathcal{J}^{*}_{1},\mathcal{J}^{*}_{2})\in\mathcal{P}(\mathcal{J}^{*})$, we consider the following restricted problem associated with (1):

(13)

Denote by $X(\mathcal{J}^{*}_{1},\mathcal{J}^{*}_{2})$ the feasible region of (13). It is not difficult to see that

(14)

where $\mathcal{L}_{X(\mathcal{J}^{*}_{1},\mathcal{J}^{*}_{2})}(x^{*})$ is the linearized cone of $X(\mathcal{J}^{*}_{1},\mathcal{J}^{*}_{2})$ at x ^∗.

Since x ^∗ satisfies the MPEC-RCRCQ, the RCRCQ holds at x ^∗ for each partition $(\mathcal{J}^{*}_{1},\mathcal{J}^{*}_{2})\in\mathcal{P}(\mathcal{J}^{*})$. It follows from Lemma 6 in [26] that

This, together with (14), indicates that $\mathcal{T}_{X}(x^{*})=\mathcal{L}_{\mathit{MPEC}}(x^{*})$ and hence the MPEC Abadie CQ holds at x ^∗.