D. Aussel, G. Bouza, S. Dempe, and S. Lepaul Multi-leader ... · D. Aussel, G. Bouza, S. Dempe, S....

ISSN 2512-3750

Fakultät für Mathematik und Informatik

PREPRINT 2018-10

D. Aussel, G. Bouza, S. Dempe, and S. Lepaul

Multi-leader disjoint-follower game: formulation as a bilevel optimization problem

D. Aussel, G. Bouza,

S. Dempe, and S. Lepaul

Multi-leader disjoint-follower game: formulation as a bilevel optimization problem

TU Bergakademie Freiberg

Fakultät für Mathematik und Informatik

Prüferstraße 9

09599 FREIBERG

http://tu-freiberg.de/fakult1

ISSN 1433 – 9307

Herausgeber: Dekan der Fakultät für Mathematik und Informatik

Herstellung: Medienzentrum der TU Bergakademie Freiberg

Multi-leader disjoint-follower game: formulation as abilevel optimization problem∗

D. Aussel, G. Bouza, S. Dempe, S. Lepaul

July 30, 2018

Abstract

Nowadays non-cooperative games have many applications. A particular case isthe so-called multi-leader disjoint-follower (MLDF) problem where several leadersact according to a Nash equilibrium and having each a number of exclusive fol-lowers. In this paper we consider the mathematical program with complementarityconstraints (MPCC) formulation of the bilevel problem corresponding to each leader.We focus in the case in which the sets of feasible solutions of the follower do notdepend on the variable of the corresponding leader. We show that, generically, goodproperties such as constraints qualification and non degeneracy of the solutions, aresatisfied at each bi-level problem. In particular, given a problem, we obtain thatexcept for a zero-Lebesgue measure set, with at most quadratic perturbations of theinvolved functions, these properties are satisfied. We also prove that these prop-erties will remain stable under small perturbations of the involved functions. Wediscuss the consequences of this result for the particular MLDF model that appearswhen agents has to design the contracts they will propose to their clients, knowingthat the clients will choose the best option.

Keywords. genericity, MPCC, muti-leader-disjoint-follower problems, regularity, sta-tionary points.

1 Introduction to Multi-Leader-Disjoint-Followers games

A Multi-Leader-Disjoint-Follower game (MLDF in short) is a particular Generalized Nashgame structure for which each of the players is facing a single-leader-multi-follower prob-lem with a private list of followers. So, his/her problem only depends on the decision

∗This research benefited from the support of the FMJH Program PGMO and from the support ofEDF.

1

1 INTRODUCTION TO MULTI-LEADER-DISJOINT-FOLLOWERS GAMES 2

variables of the other leaders and of the decision variables of a list of followers that onlyinteract with him. Here the single-leader-multi-followers problem of each leader is under-stood in the optimistic sense.

We will study the case in which the possible strategies of the followers does not dependon the strategies chosen by other agents, followers or leaders. The problem is

Solve simultaneously the N , i = 1, . . . , N , problems Pi:

(Pi) minxi,yi Fi(xi, x−i, yi)

s.t.

gik(xi, x−i, yi) ≤ 0, k = 1, . . . qi,

yli ∈ argminzli∈M lif li (xi, x−i, z

li),∀ l = 1, . . . , Li,

(1)

where yi =(yli)Lil=1

, M li =

zli ∈ Rmil : vlij(z

li) ≤ 0, j = 1, . . . , sli

.

In the case of one leader and one follower, the problem corresponds to the classicalbilevel programs. Therefore, it is clear its complexity. As in [5, 13], we substitute thelower level problem by the corresponding KKT system. So, we solve, simultaneously, theN , i = 1, . . . , N , MPCC problems


s.t.

gik(xi, x−i, yi) ≤ 0, k = 1, . . . qi,

∇ylif li (xi, x−i, y

li) +

∑slij=1∇yli

λlijvlij(y

li) = 0,∀ l = 1, . . . , Li,

0 ≥ vlij(yli)⊥λlij ≥ 0,∀ l = 1, . . . , Li, j = 1, . . . , sli

(2)


, M li =

zli ∈ Rmil : vlij(xi, z

li) ≤ 0, j = 1, . . . , sli

.

The goal of this paper is to present the genericity analysis of this model. This ap-proach is based on identifying each problem by the functions that define it, in this case(Fi, f

li , v

lij, gik), i = 1, . . . N, l = 1, . . . , Li, j = 1, . . . , sli, k = 1, . . . , qi, and considering the

corresponding topological functional space. Then, for a generic set of functions, the prop-erties that are fulfilled by the corresponding problems are obtained. Here a generic set isa countable intersection of open and dense sets with respect to the strong Whitney topol-ogy. We want to point out that this analysis had been made for studying the propertiesfulfilled at generic non-linear programming problem, MPCC and bi-level problems, see[16, 18, 4], respectively. Genericity for generalized Nash equilibrium problems has beenaddressed in [7].

In this paper we study if the non-degenerancy obtained in [18] for MPCC problemscan be generically fulfilled for each i = 1, . . . , N . As auxiliary results we obtain a locallystability and a perturbation result. That is, fixed a problem which satisfies the non-degeneracy condition, for each point of the domain there exists a neighborhood centered init and a neighborhood of the functions defining the problem in which the non-degeneracyholds. On the other hand, given the functions, we prove that for almost all quadratic

2 A MOTIVATION EXAMPLE IN ELECTRICITY CONTRACT PROBLEM 3

perturbation of (Fi, fli ), i = 1, . . . N, l = 1, . . . , Li and linear of vlij, gik, i = 1, . . . N, l =

1, . . . , Li, j = 1, . . . , sli, k = 1, . . . , qi, these conditions are satisfied.Compared with [17], the model we consider is simpler, because the set of the feasible

solutions of the lower level problems does not depend on the choice of the leaders. Despiteother multi-leaders-multi-follower games, see [1, 2, 12], there are not common constraints.This simplifies the analysis because there is no need of duplicating variables. Nevertheless,this model describe practical situations as described next. The rest paper is organizedas follows. Section 3 contains some definitions and basic results from the critical pointtheory and the genericity. The main results are presented in Section 4. Then, the resultis particularized to the example presented in Section 2. The papers ends with someconcluding remarks and future research lines.

2 A motivation example in electricity contract prob-

lem

Let us consider the following contract problem. We consider that N “independent actorsthat could be producers, smart grid manager or aggregators. Each one of them want tooptimize their contract offers to their client in order to maximize their own benefit. Weassume that :

• the considered period of time is splitted into time periods indexed by index h ∈1, . . . , H.

• for each time period h, each actor i has the possibility either to produce by its ownproduction plants, to buy electricity on the market or eventually to sell to the othersactors. This last option is particular important in case of a small producer or ofan aggregator. For any period of time h, the production of actor i is denoted byoplh while extih, int

i′

i,h and pii′,h stands respectively for the quantity of electricity thatactor i buy to the market, buy to actor i′ and unit price at which actor i proposesto sell electricity to actor i′.

• for actor i and at hour h, the cost to production is described by the function costih(·)

• the market price for each MWh at hour h is fixed and given by Mph.

• the different actors are competing in a non cooperative way.

• each actor i has its own clients C li , l = 1, . . . , Li that are fully dependent on him

(due essentially to network and geographical reasons).


• each actor i proposes NCi kinds of contracts each of them being described by a stepfunction

contci : 1, . . . N × R → R(h, q) 7→ contci(h, q) = phi,c.q

where phi,c is the unit price (euros/MWh) proposed by the contract c of actor i athour h . For any i, c and h, the unit price variable phi,c is assumed to be an elementof an interval [ph

i,c, phi,c]. The contract c of actor i is thus entirely characterized by

the vector (phi,c)h

• each client l of actor i can buy the quantity ql,h,ci at hour h through the contract c.An aditional cost Ch

i,c is associated to the unitary cost that client l of actor i has topay for the energy provided by contract c in the period h.

• each client l has a total consumption of Dl during the full period of time. Thisquantity can be splitted into the different time periods but for each time period, thenon shiftable demand is dj,h while the rest of the demand is shiftable between thetime periods.

• the aim of each client is of course to minimize its cost of energy.

Let us observe that the stepwise structure of the contracts offered by actors encourages theclients to use shift processes to reduce their costs. On the other hand, even if the clientsare linked to an actor, they can obtain (global) information on the offers of the otheractors and thus in order to secure the link with their clients, the following constraintsensure a certain equilibrium between the weighted price of the contracts of the actorswith respect to a a leader, actor 1.

∀ i, ∀ j,∑h

NCj∑c=1

whi,cpci,n =

∑h

NC1∑c=1

wh1,cpc1,n.

where the wi’s are coefficients/weights taking into account the ”relative size” of the actorsand the moments of the day they can provide energy. Of course we also assume that∑

h

∑NC1

c=1 wh1,cp

c1,n. ≤ ω

The proposed model to describe this contract management is the following:


Figure 1: General organization of the contract management model

The mathematical model for the client l of actor i is

(P li ) min(ql,h,ci )h,c

∑Hh=1

∑NCic=1 [phi,c + C l,h

i,c ]ql,h,ci

s.t.

∑NCi

c=1 ql,h,ci ≥ dl,hi , h = 1, ..H,∑NCi

c=1

∑Hh=1 q

l,h,ci = Dl

i,

ql,h,ci ≥ 0, h = 1, ..H, c = 1, .., NCi.

(3)

3 PRELIMINARIES AND NOTATIONS 6

while the problem of each producer i is described as

max(phi,c)h,c(exthi )h, (op

hi )h

(inti,hi′ )i′ 6=i,h

(pi′,hi )i

′ 6=i,h

max(ql,h,ci )h,c∑Li

l=1

∑Hh=1

∑NCic=1 p

hi,cq

l,h,ci +

∑i′ 6=i p

i′,hi .inti

′,hi

−∑H

h=1Mph.exthi −∑H

h=1 costhi (op

hi )−

∑i′ 6=i p

i,hi′ .int

i,hi′

ophi + exthi +∑

i′ 6=i inti,hi′ =

∑i′ 6=i int

i′,hi +

∑Lil=1

∑NCic=1 q

l,h,ci , h = 1, . . . , H,

phi,c ∈ [phi,c, phi,c], c = 1, . . . , NCi, h = 1, . . . , H,

pi′,ci ∈ [pi

′,ci, pi′,ci ], c = 1, . . . , NCi, h = 1, . . . , H,∑

h

∑NCjc=1 w

hi,cp

hi,c, =

∑h

∑NC1

c=1 wh1,cp

h1,c, i 6= 1,

or∑

h

∑NC1

c=1 wh1,cp

h1,c ≤ ω, i = 1,

exthi ≥ 0 h = 1, . . . , H,

ophi ≥ 0 h = 1, . . . , H,

inti,hi′ ≥ 0, h = 1, . . . , H,

(ql,h,ci )h,c solves (P li ), l = 1, . . . , Li.

(4)

Now for any i by setting xi = ((phi,c)h,c, (exthi )h, (op

hi )h, (int

i,hi′ )i′ 6=i,h, (p

i′,hi )i′ 6=i,h), and yi =

((ql,h,ci )h,c)l=1,Li we clearly have that this contract management problem fit with the struc-ture of ”first problem” and corresponds thus a multi-leader-disjoint-follower game.

costhi is usually considered as an increasing piecewise affine function. In the literature,it is common to approximate it by a quadratic function costhi (op

hi ) = ahi op

hi + bhi (op

hi )

2

3 Preliminaries and notations

We start with some basic concepts and results from MPCC theory and parametric opti-mization that will be needed in the convergence and genericity analysis.

Consider for the moment the problem

minx F (x)gi(x) ≤ 0, i = 1, . . . , q,hi(x) = 0, i = 1, . . . , p,0 ≤ ri(x)⊥si(x) ≥ 0, i = 1, . . . ,m,

(5)

where all functions are sufficiently smooth and map Rn into appropriate real spaces.When dealing with MPCC problems, the following active index set notation is required:


Ig(x) = i ∈ 1, . . . , q|gi(x) = 0, Is(x) = i ∈ 1, . . . ,m|ri(x) > 0, si(x) = 0,

Ir(x) = i ∈ 1, . . . ,m|ri(x) = 0, si(x) > 0, Irs(x) = i ∈ 1, . . . ,m|ri(x) = 0, si(x) = 0.

Definition 3.1 (Strict Complementarity (SC)) Let x ∈ F . We say that SC holdsat x if Irs(x) = ∅.

The regularity conditions for MPCCs constitute adaptations for their nonlinear pro-gramming counterpart, see [3]. Here we present the ones that will be used later.

Definition 3.2 Let x ∈ F . We say that the MPCC-LICQ holds at x if the system

∇gi(x)|i ∈ Ig(x) ∪ ∇hi(x)|i = 1, . . . , p ∪ ∇ri(x)|i ∈ Ir((x) ∪ Irs((x)∪∇si(x)|i ∈ Is(x)) ∪ Irs((x)

is linearly independent.

Definition 3.3 Let x ∈ F . We say that the MPCC-MFCQ holds at x if the system

∇hi(x)|i = 1, . . . , p ∪ ∇ri(x)|i ∈ Ir(x)) ∪ Irs((x) ∪ ∇si(x)|i ∈ Is(x)) ∪ Irs((x)

is linearly independent and there is a d ∈ Rn such that

∇gi(x)Td < 0, ∀ i ∈ Ig(x), ∇hi(x)Td = 0, ∀ i = 1, . . . , p,

∇ri(x)Td = 0, ∀ i ∈ Ir(x)) ∪ Irs((x), ∇si(x)Td = 0, ∀ i ∈ Is(x)) ∪ Irs((x).

Now we present the main stationarity concepts:

Definition 3.4 (Stationarity Concepts) Let x ∈ F . Then, x is called weakly station-ary (W-stationary) if there are multipliers (λ, µ, γ, ν) ∈ R|Ig(x)|+l+|Ir(x)|+|Is(x)| with λ ≥ 0such that

0 = ∇f(x)+∑i∈Ig(x)

λi∇gi(x)+l∑

i=1

µi∇hi(x)−∑

i∈Ir(x))∪Irs((x)

γi∇ri(x)−∑

i∈Is(x))∪Irs((x)

νi∇si(x).

In particular, a W- stationary point x with multipliers (λ, µ, γ, ν) is known to be:

(a) Clarke stationary (C-stationary) if γiνi ≥ 0 for all i ∈ Irs(x).

(b) Mordukhovich stationary (M- stationary) if either γiνi = 0 or γi, νi > 0 hold for alli ∈ Irs(x).


(c) Strongly stationary (S- stationary) if γi, νi ≥ 0 for all i ∈ Irs(x).

Clearly

S- stationarity ⇒ M- stationarity ⇒ C- stationarity ⇒ W- stationarity.

It is worth to point out that the S- stationarity condition is equivalent to the standardKKT condition applied directly to problem (5).

The following necessary condition holds:

Theorem 3.1 (First order necessary condition, cf. [8]) Let x be a local minimizerof problem (5) in which the MPCC-LICQ is satisfied. Then x is an S-stationary point.

This paper deals with the properties that are fulfilled at the solutions of the MLDFmodel. We base the analysis on the critical point theory, see [16]. Now, we will presentthe main results of this theory developed for non-linear programs:

min f(x) s.t. hi(x) = 0, i = 1, . . . , l, gj(x) ≤ 0, j = 1, . . . , q. (6)

We say that a point is critical if the gradient of the objective function is a linear com-bination of the gradients of the equality and the active inequality constraints. Moreover,if the coefficients associated to the gradients of the active inequality constraints are nonzero and the Hessian matrix of the Lagrangian function is regular, then the critical pointis non-degenerated.

Genericity analysis is done to know how strong it is to assume that critical points arenon-degenerated. To this aim, we define [Cr]kn as the space of Cr functions from Rn toRk, endowed with the strong topology, (see, e.g., [11]). We have the following definition

Definition 3.5 A set E ⊂ [Cr]kn is generic with respect to the strong topology if E =∩∞i=1Ei, where Ei are open and dense in this topological space.Suppose that the class of problems P is such that each instance is determined by an elementof [Cr]kn. We say that generically the class P fulfills the property P if there is a genericset of functions such that the problems they define, satisfy P.

Next theorem presents the properties satisfied by a generic non-linear program (6).

Theorem 3.2 (cf. [19, 9]) The set of functions(f, h1, . . . hl, g1, . . . , gq) ∈ [C2]1+l+q

n :LICQ holds at all feasible points,

and all critical points are non-degenerate

,

is open and dense with respect to the strong topology in [C2]1+l+qn .

4 GENERIC ANALYSIS FORMULTI-LEADER, DISJOINT FOLLOWER PROBLEM9

With respect to the proof we want to point out that the open part is a consequence ofthe continuity of the involved functions, their derivatives and an adequate locally finitecovering of Rn. The density is obtained using the following perturbation result

Theorem 3.3 (cf. [10]) Consider the problem P (b, d) defined analogously by the per-turbed functions

f(x) + bTx,

hi(x) + di, i = 1, . . . , l,

andgj(x, τ) + dj+l, j = 1, . . . , q,

where (b, d) ∈ Rn+l+q. Then, the set of perturbations (b, d) such that at P (b, d) LICQ failsat some feasible point or there is a degenerated critical points, has Lebesgue measure equalto zero.

Next lemma is the basis of the just presented perturbation result.

Lemma 3.1 (Parameterized Sard Lemma, cf. [10]) Let us assume that φ ∈ [Cκ]rn+p,κ > max 0, n− r, x ∈ Rn and z ∈ Rp. We suppose that for all (x, z) ∈ Rn × Rp suchthat φ(x, z) = 0, we have rank(∇x,zφ(x, z)) = r. Define φz : Rn → Rr, φz(x) = φ(x, z).Then for almost all z ∈ Rp, the matrix ∇xφz(x

∗) has rank r, for all zeros x∗ of the mapφz(x).

We end this section with some basic notation that will be used through the text. Thecanonical vectors will be denoted by ei. The open ball centered at x with radius ε will beBε(x) = x : ‖x− x‖ < ε, where ‖x‖ is the Euclidean norm.

4 Generic analysis for multi-leader, disjoint follower

problem

Following the notations of Section 1, we will consider the following version of the MLDFgame:

Solve simultaneously the N , i = 1, . . . , N , problems Pi:


s.t.

gik(xi, x−i, yi) ≤ 0, k = 1, . . . qi,

yli ∈ argminzli∈M lif li (xi, x−i, z

li),∀ l = 1, . . . , Li,

(7)


, M li =

zli ∈ Rmil : vlij(z

li) ≤ 0, j = 1, . . . , sli

.


The associated EPEC-formulation is thus:

Solve simultaneously the N , i = 1, . . . , N , problems MPECi:

(MPECi) minxi,yi,λi Fi(xi, x−i, yi)

s.t.

gik(xi, x−i, yi) ≤ 0, k = 1, . . . , qi,


li) +

∑slij=1 λ

lij∇yli

vlij(yli) = 0, l = 1, . . . , Li

vlij(yli) ≤ 0, l = 1, . . . , Li, j = 1, . . . , sli

λlij ≥ 0, l = 1, . . . , Li, j = 1, . . . , sli,

vlij(yli)λ

lij = 0, l = 1, . . . , Li, j = 1, . . . , sli.

(8)

We define the following multipliers associated to the respective constraints:

• µik, for gik.

• αli for ∇ylif li (xi, x−i, y

li) +

∑slij=1 λ

lij∇yli

vlij(yli) = 0.

• βlij for vlij(yli).

• γlij for λlij.

The active index sets are denoted as:

• J l0i(x, y, λ) := j : vlij(yli) = 0, λlij > 0.

• JΛl0i(x, y, λ) := j : vlij(y

li) = λlij = 0.

• Λl0i(x, y, λ) := j : vlij(y

li) < λlij = 0.

• J0ig(x, y) = k : gik(xi, x−i, yi) = 0.

Although some results can be obtained under simpler hypothesis, we will assume that(Fi, gik) ∈ C3, (f li , v

lij) ∈ C4, i = 1, . . . N, l = 1, . . . , Li, j = 1, . . . , sli.

Now, we present the critical point theory for the multi-leaders-disjoint-followers prob-lem (MLDF). First we introduce the concept of critical point, see [9].

Definition 4.1 We say that (x, y, λ) is a generalized critical point of problem (7) if forsome (µ0, α, β, µ, γ) 6= 0, µ0 ≥ 0

H0i (xi, x−i, y, λi, µ0i, αi, βi, µi, γi) = 0, i = 1, . . . , N, (9)

where


H0i (xi, x−i, y, λi, µ0i, αi, βi, µi, γi) =

µ0i∇xi,yiFi(xi, x−i, yi) +∑Li

l=1

∑slij=1 β

lij∇xi,yiv

lij(y

li)+

+∑

k∈J0igµik∇xi,yigik(xi, x−i, yi)+

+∑Li

l=1

(∇xi,yi [∇yli

f li (xi, x−i, yli) +

∑slij=1 λ

lij∇yli

vlij(yli)]α

li

)∇yli

vlij(yli)α

li, j ∈ J l0i, l = 1, . . . , Li,

∇ylivlij(y

li)α

li + γlij, j ∈ JΛl

0i ∪ Λl0i, l = 1, . . . , Li,

gik(xi, x−i, yi), k ∈ J0ig,


li) +

∑slij=1 λ

lij∇yli

vlij(yli), l = 1, . . . , Li,

vlij(yli), j ∈ J l0i ∪ JΛl

0i, l = 1, . . . , Li,λlij, j ∈ JΛl

0i ∪ Λl0i, l = 1, . . . , Li.

(10)

If µ0i = 1, for i = 1, . . . , N , then (x, y, λ) is a critical point.

For the necessary optimality conditions, certain multipliers must be non-negative.

Definition 4.2 A point (x, y, λ) is a FJ-weakly stationary point if it is a generalized crit-ical point with µ ≥ 0.If also µ0 = 1, then (x, y, λ) is a weakly-stationary point, for all i.If at a weakly stationary point, βlij, γ

lij ≥ 0, for all j ∈ JΛl

0i(xi, x−i, yi, λi), l = 1, . . . , Li, i =1, . . . , N , then (x, y, λ) is called a strongly stationary point.

Note that these definition in the framework of MLDF problems are clear generalizationsof the analogous concepts for MPCC. As necessary optimality condition, we have thefollowing evident result.

Proposition 4.1 If (x, y, λ) solves MLDF problem, then it is a FJ-weakly stationarypoint. Furthermore, if MPCC-LICQ holds at (xi, yi, λi) for all i = 1, . . . , N , then (x, y, λ)is a strong stationary point with unique multipliers (α, β, µ, γ).

Proof. If (x, y, λ) solves MLDF problem, then (xi, yi, λi) solves the MPCC Pi for alli = 1, . . . , N . The result is a consequence of the stationary condition for MPCC, see [8].

We start with the analysis of the constraints qualification in the generic case.As usual, in order to consider genericity properties of our EPEC problem (8), let us

consider the following perturbed EPEC problem defined, for i = 1, . . . , N , l = 1, . . . , Liand j = 1, . . . , sli by Fi(xi, x−i, yi)+b

Fi (xi, yi), f

li (xi, x−i, y

li)+b

fily

li+(xi, y

li)A

fi y

li, v

lij(y

li)+c

vijl

and gik(xi, x−i, yi) + cgik + bgik(xi, yli), that is:


Solve simultaneously the N , i = 1, . . . , N , problems MPECperti :

minxi,yi,λi Fi(xi, x−i, yi) + bFi (xi, yi)

s.t.

gik(xi, x−i, yi) + cgik + bgik(xi, yli) ≤ 0, k = 1, . . . , qi,

∇yli[f li (xi, x−i, y

li) + bfil.y

li + (xi, y

li)A

fi .y

li] +

∑slij=1 λ

lij∇yli

vlij(yli) = 0, l = 1, . . . , Li

vlij(yli) + cvijl ≤ 0, l = 1, . . . , Li, j = 1, . . . , sli

λlij ≥ 0, l = 1, . . . , Li, j = 1, . . . , sli,

[vlij(yli) + cvijl].λ

lij = 0, l = 1, . . . , Li, j = 1, . . . , sli.

(11)Let us denote respectively by bF , bf , Af , cv, cg and bg the vectors bF = (bFi )i ∈ ΠN

i=1R(mi+ni),bf = (bfil)i,l ∈ ΠL

i=1R(ni+mi), Af = (Afi )i ∈ ΠNi=1R(ni+mi)×(mi), cv = (cvijl)i,j,l ∈ RS,

cg = (cgik)i,k ∈ RQ and bg = (bgik)i,k ∈ R(ni+mi)Q

where L =∑N

i=1 Li, S =∑N

i=1

∑lil=1 s

li

and Q =∑N

i=1 qi.

Proposition 4.2 For almost all parameters bF , bf , Af , cv, cg and bg, if (x, y, λ) is ageneralized critical point of the perturbed EPEC problem (11) then, for any i = 1, . . . , N,the MPCC-LICQ for (MPECpert

i ) holds at (xi, yi, λi).

For shortness, we denote∑Li

l=1mil by mi. ⊗ represents a matrix of suitable dimension.

Proof. First assume that the point (xi, x−i, y, λ) is a generalized critical point of problem(8) where MPCC-LICQ fails. So, the system (9) has a solution and, by the failure of theMPCC-LICQ for some i, these two conditions hold:

(i) for some i∗, µ0i∗ = 0 and (α, β, µ, γ)i∗ 6= 0.

(ii) (µ0, α, β, µ, γ)i 6= 0 for all i 6= i∗.

For considering all the possibilities, take an arbitrary, but fixed set of active indicesand the system that describe FJ-weakly stationary points where MPCC-LICQ fails atsome problems Pi.

First fix (i, l) and J l0i ∪ JΛl0i. We get

vlij(yli) + cvijl = 0, j ∈ J l0i ∪ JΛl

0i.

For simplicity, we assume that J l0i ∪ JΛl0i = 1, . . . , r, r ≤ sli. Taking derivatives with

respect to yli and cvijl, we obtain the full rank matrix(∇yli

vlij(yli), j ∈ J l0i ∪ JΛl

0i Ir|0).

By Lemma 3.1, for almost all cvijl, the vectors∇ylivlij, l ∈ J l0i∪JΛl

0i are linearly independent.

Taking all the possible (finitely many) combinations of (i, l) and J l0i∪JΛl0i, we obtain that


LICQ holds for almost all cvijl, at the setyli ∈ Rmli : vlij(y

li) + cvijl ≤ 0, j = 1, . . . , si

, for

all i = 1, . . . , N, l = 1, . . . , Li∗ .If (α, µ)i∗ = 0, then, at the solutions of system (9), γ = 0. Considering the first

equation in (9), [∇xi,yivlij(y)]β = [0 × ∇yiv

lij(·)]β = 0. So, the (classical) LICQ fails at

the sety ∈ Rmli : vli∗j(y

li) + cvi∗jl ≤ 0, j = 1, . . . , si∗

, for all l = 1, . . . , Li∗ . But, as shown

before, the LICQ holds almost everywhere. So, for almost all cvi∗jl, there is no solution ofthe system (9) satisfying (i)-(ii) and with (α, µ)i∗ = 0.

We assume that cvijl is such that LICQ holds at the setyli ∈ Rmli : vlij(y

li) + cvijl ≤ 0, j = 1, . . . , si∗

, for all i = 1, . . . , N and l = 1, . . . , Li∗ .

Hence, for each i, we normalize the linear combination described in

H0i (xi, x−i, y, λi, µ0i, αi, βi, µi, γi) = 0

using one of the following (disjoint) possibilities:

(a) µ0i = 1.

(b) µ0i = 0 and µik = 1, for some k ∈ J0ig.

(c) µ0i = 0 and µik = 0, for all k ∈ J0ig, αlij = 1, for some l ∈ 1, . . . Li, j ∈

1, . . . ,mil.

Note that, by (i), at least for one index i the system H0i (xi, x−i, y, λi, µ0i, αi, βi, µi, γi) = 0

corresponds to cases (b) or (c).Take i ∈ 1, . . . , N. Depending on the cases (a), (b) or (c), the derivatives of the block

of equations H0i (xi, x−i, y, λi, µ0i, αi, βi, µi, γi) = 0, defined by the perturbed functions

correspond to exactly one of the following three possible cases.Case (a):

(xi, yi) λijl αli γijl bFi bfiil cgik⊗ ⊗ ⊗ 0 I ⊗ 0⊗ 0 ∇vlij, j ∈ J0i 0 0 0 0⊗ 0 ∇vlij, j ∈ JΛl

0i I|0 0 0 0⊗ 0 ∇vlij, j ∈ Λl

0i 0|I 0 0 0⊗ ⊗ 0 0 0 I 0⊗ 0 0 0 0 0 I

∇vlij, l ∈ J l0i ∪ JΛl0i 0 0 0 0 0 0

0 I|0 0 0 0 0 0

This matrix has as many rows as the dimension of the vector (xi, yi, λi, αi, µi, βi, γi).


Case (b):

(xi, yi) λijl αli γijl Afil bgik bfiil cgik⊗ ⊗ ⊗ 0 ⊗ ⊗|I|⊗ 0 0⊗ 0 ∇vlij, j ∈ J0i 0 0 0 0 0⊗ 0 ∇vlij, j ∈ JΛl

0i I|0 0 0 0 0⊗ 0 ∇vlij, j ∈ Λl

0i 0|I 0 0 0 0⊗ ⊗ 0 0 ⊗ ⊗ I 0⊗ 0 0 0 0 ⊗ 0 I

∇vlij, l ∈ J l0i ∪ JΛl0i 0 0 0 0 0 0 0

0 I|0 0 0 0 0 0 0

This matrix has as many rows as the dimension of the vector (xi, yi, λi, αi, µi, βi, γi) + 1,because for some k ∈ J0ig, µik = 1 is constant.

Case (c):

(xi, yi) λijl αli γijl Afil bfiil cgik⊗ ⊗ ⊗ 0 ⊗ |Λ| ⊗ 0 0⊗ 0 ∇vlij, j ∈ J0i 0 0 0 0⊗ 0 ∇vlij, j ∈ JΛl

0i I|0 0 0 0⊗ 0 ∇vlij, j ∈ Λl

0i 0|I 0 0 0⊗ ⊗ 0 0 ⊗ I 0⊗ 0 0 0 0 0 I

∇vlij, l ∈ J l0i ∪ JΛl0i 0 0 0 0 0 0

0 I|0 0 0 0 0 0

This matrix has as many rows as the dimension of the vector (xi, yi, λi, αi, µi, βi, γi) + 1,because there exists l and j such that αlij = 1 is constant.

In cases (a) and (b), evidently, full row rank matrices were obtained. In (c), note thatAfil only appears at the first and fifth block of equations of

H0i (xi, x−i, y, λi, 0, αi, βi, 0, γi) = 0.

Let, without loss of generality, αli1 = 1. Then αli = (1, αli2, . . . , αlimil

), the deriva-tive with respect to the first row (and column, recall that Alif is symmetric) is Λ =(

1 αli2, . . . , αimil0 I

), which is a regular matrix. So, also in (c), the matrix of the deriva-

tive of H0i has full row rank, J0i, JΛ0i,Λ0i, J0ig for i = 1, . . . N .

Now, we consider the whole system and consider, for each i = 1, . . . , N , the case from(a) to (c) that corresponds. Note that, at least for one index i = i∗, the derivative corre-sponds to case (b) or (c). Moreover, the evident full row rank matrix for the i-th leaderinvolves λlij and parameters indexed by i. So, for each i1 6= i2 the corresponding matrices


of full row rank have no columns in common. Hence, the derivative of the whole perturbedsystem (9) has full row rank. Then, by Lemma 3.1, for almost all Q = (Af , bF , bv, bf , bg, cg)the derivative with respect to (xi, y, µ0, λ, α, β, µ, γ) of the corresponding system (9) hasfull row rank at their solutions. But, for at least one index i, the derivative corresponds tocase (b) or case (c). So, the total number of variables is strictly smaller than the numberof equations. Hence, for almost all Q, such a system cannot have a solution, given the,already, fixed set of active constraints. That is, cases (b) and (c) are not possible, foralmost all Q, given the set of active constraints.

Taking the intersection of the almost all sets for all the possible combinations of thethree cases, we obtain the failure of the MPCC-LICQ at a set QJ0i,JΛ0i,Λ0i,J0ig

of zeromeasure. Considering the set

R =

[cvijl]i,j,l :

for some i = 1, . . . , N, l = 1, . . . Li, LICQ fails atthe lower level problem of client j of producer i.

we get

R×⋃

i=1,...N

⋃J0i,JΛ0i,Λ0i,J0ig

QJ0i,JΛ0i,Λ0i,J0ig

is the product of two sets of zero measure. Since the MPCC-LICQ is satisfied at allgeneralized critical points for all Pi at the complement of this set, the result easily follows.

Remark 4.1 With this result it is clear that for almost all Q, generalized critical pointsare critical points. By Proposition 4.1, the solutions of problem (8) are FJ-weakly station-ary points. Since FJ-weakly stationary points are generalized critical points, for almostall Q, the solutions of (8) are weakly stationary.

Remark 4.2 We want to point out that we only proved that, for almost all perturbations,MPCC-LICQ is fulfilled at FJ-weakly stationary points for the associated perturbed prob-lem. In the case of one producer, i.e. the KKT-approach applied to bilevel problems, thisconstraint qualification holds generically at all feasible points. An analogous result doesnot hold for the MLDF. First, it is clear that at the proof of Theorem 4.2, we stronglyused the stationary condition at the problem of each producer. Feasibility is a very weakcondition, it is not enough for the rank condition. The next example shows that at nonFJ-weakly stationary points MPCC-LICQ may fail generically.

Example 4.1 Let (x1, x2, y1, y2) = (1, 1, 1, 0) Assume that ∇x2F2(x1, x2, y1, y2) 6= 0,∇y1f1(1, 1, 1, 0),∇y2f2(1, 1, 1, 0) < 0. Consider the problem

(P1) minx1,y1 F1(x1, x2, y1, y2)s.t. ∇y1f1 + λ1

1 + λ21 + λ3

1 = 0, y1 ≤ 1, y1 ≤ x2, , y1 ≤ x1, λ11, λ

21, λ

31 ≥ 0


(P2) minx2,y2

F2(x1, x2, y1, y2) s.t. ∇y2f2 + λ12 = 0, y2 ≤ 0, λ1

2 ≥ 0

Then (1, 1,−∇y1f1(1, 1, 1, 0), 0, 0) is a feasible point of the first problem and (1, 0,−∇y2f2(1, 1, 1, 0))of the second. MPCC-LICQ is violated at the first problem because

∇x1∇y1f1 ∇y1∇y1f1 1 1 10 1 0 0 00 1 0 0 01 1 0 0 00 0 0 0 1

has not full row rank while [

∇x2∇y2f2 ∇y2∇y2f2 10 1 0

]has full row rank and the MPCC-LICQ is satisfied at the second case. Since ∇x2F2 6= 0,(x2, y2) is not a critical point of problem P2. So, the point ((x1(ε), x2(ε), y1(ε), y2(ε)),where (x1(ε), y1(ε)) = ((1, 1,−∇y1f1(1, 1, 1, 0), 0, 0)) and (x2(ε), y2(ε) = ((1, 0,−∇y2f2(1, 1, 1, 0))),is not a FJ stationary point and MPCC-LICQ fails at one of the upper player problems.Moreover, take a small perturbation of the functions v1

1, v21, f1, f2. Then, by inverse func-

tion arguments, the perturbed system

y1 − 1 + ε1(x, y) = 0,y1 − x2 + ε2(x, y) = 0,y1 − x1 + ε3(x, y) = 0,

y2 + ε4(x, y) = 0,

(12)

has a solution. By the continuity of this solution with respect to εi, i = 1, . . . , 4 andthe continuity of the functions ∇y1f1 + ε5(x, y),∇y2f2 + ε6(x, y), they are strictly neg-ative at (x1(ε), x2(ε), y1(ε), y2(ε)), the solution of system (12). Moreover, as ∇ε7 issmall, ∇x2F2(x1(ε), x2(ε), y1(ε), y2(ε)) + ∇x2ε7(x1(ε), x2(ε), y1(ε), y2(ε)) 6= 0. So, thepoints (x1(ε), y1(ε),−∇y1f1 + ε5(x, y), 0, 0) and (x2(ε), y2(ε),−∇y2f2 + ε6(x, y)) are fea-sible solutions of problem P1 and P2 respectively. Writing the active constraints, for theperturbed case, we obtain that MPCC-LICQ fails at P1, holds at P2 and (x2, y2) is not acritical point of problem (P2). Then, for the perturbed problem the MPCC-LICQ, fails atfeasible points.

Now we introduce the concept of critical points.

Definition 4.3 A point (x, y, λ) is critical if (xi, y1i , . . . , y

Lii , λ

1i , . . . , λ

Lii ) is a critical point

of problem PKKTi for i = 1, . . . , N . i.e. if there exists (α, β, µ, γ) such that:

Hi(xi, x−i, y, λi, αi, βi, µi, γi) = 0, i = 1, . . . , N, (13)


where

Hi(xi, x−i, y, λi, αi, βi, µi, γi) =

∇xi,yiFi(xi, x−i, yi) +∑Li

l=1

∑slij=1 β

lij∇xi,yiv

lij(y

li)+

+∑

k∈J0igµik∇xi,yigik(xi, x−i, yi)+

+∑Li

l=1

(∇xi,yi [∇yli

f li (xi, x−i, yli) +

∑slij=1 λ

lij∇yli

vlij(yli)]α

li

)∇yiv

lij(y

li)α

li, j ∈ J l0i, l = 1, . . . , Li,

∇yivlij(y

li)α

li + γlij, j ∈ JΛl

0i ∪ Λl0i, l = 1, . . . , Li,

gik(xi, x−i, yi), k ∈ J0ig,

∇yifli (xi, x−i, y

li) +

∑slij=1 λ

lij∇yiv

lij(y

li), l = 1, . . . , Li,

vlij(yi), j ∈ J l0i ∪ JΛl0i, l = 1, . . . , Li,

λlij, j ∈ JΛl0i ∪ Λl

0i, l = 1, . . . , Li.

(x, y, λ) is a non-degnerate critical point if (xi, yi1, . . . , yiLi , λi1, . . . , λiLi) is a non-degeneratedcritical point of PKKT

i for i = 1, . . . , N , i.e. if MPCC-LICQ, MPCC-SC and MPCC-SOChold.

Note that this concept is an extension of the definition of non-degenerated generalizedcritical point for non-linear programs, see [9, 10] for more details.

In the forthcoming result let us consider the following perturbed problem:

Solve simultaneously the N , i = 1, . . . , N , problems MPECperti :

minxi,yi,λi Fi(xi, x−i, yi) + bFi (xi, yi) + (xi, yi).AFi .yi

s.t.

gik(xi, x−i, yi) + cgik + bgik(xi, yli) ≤ 0, k = 1, . . . , qi,

∇yli[f li (xi, x−i, y

li) + bfil.y

li + (xi, y

li)A

fi .y

li] +

∑slij=1 λ

lij∇yli

vlij(yli) = 0, l = 1, . . . , Li

vlij(yli) + cvijl ≤ 0, l = 1, . . . , Li, j = 1, . . . , sli

λlij ≥ 0, l = 1, . . . , Li, j = 1, . . . , sli,

[vlij(yli) + cvijl].λ

lij = 0, l = 1, . . . , Li, j = 1, . . . , sli.

(14)Let us denote by AF the matrix AF = (AFi )i of R(ni+mi)×(n1+mi)×N).

Theorem 4.1 Given (F, f, v1, . . . , vl, g1, . . . , gq). Then, for almost all AF , bF , Af , bf , cv,bg and cg, the generalized critical points of the perturbed problem (14) are non-degeneratedcritical points.

Proof. The idea of the proof is similar to the proof of the Theorem 6.18, [9]. Let (x, y, λ)be a generalized critical point of (14). Then, there exist (α, β, µ, γ) such that the systemof equations

H0i (xi, x−i, yi, λi, 1, αi, βi, µi, γi) = 0, i = 1, . . . , N,


is satisfied. By Proposition 4.2, for almost all bF , Af , bf , cv, bg, cg, such that (x, y, λ) is acritical point, (LICQ) is satisfied at the lower level problem of all clients of all producers.Hence, we consider the critical points where MPCC-LICQ holds and that LICQ holds atall feasible points of

yli ∈ Rmli : vlij(yli) + cvijl ≤ 0, j = 1, . . . , si

,

for all i = 1, . . . , N, l = 1, . . . , Li. To verify the theorem we have to show that MPCC-SOCand MPCC-SC are also satisfied for almost all bFi ∈ Rmil , Afi ∈ R(ni+mi)(mi) , bfii ∈ Rni+mi ,cvijl ∈ R, bgik ∈ Rni+mi , cgik ∈ R. To start with MPCC-SOC, we consider the matrix

∇Hi =

Ai ⊗ ⊗ ⊗ ⊗ 0⊗ 0 0 ∇vlij, j ∈ J0i 0 0⊗ 0 0 ∇vlij, j ∈ JΛl

0i 0 I|0⊗ 0 0 ∇vlij, j ∈ Λl

0i 0 0|I∇xi,yigik(xi, yi), k ∈ J0ig 0 0 0 0 0

∇(xi,yi)[∇yifi(xi, x−i, yi) +∑sli

j=1 λi∇yivij(yi)] Ui 0 0 0 00|∇yiv

lij(yi), j ∈ J0i ∪ JΛ0i 0 0 0 0 0

0 0|IJΛ0i∪Λ0i0 0 0 0

.

(15)where the columns correspond to the derivatives with respect to ((xi, yi), λi, µi, αi, βi, γi)and Ui = [∇vlij]T . Note that

Ai = ∇2xi,yi

Fi(xi, x−i, yi) +∇2xi,yi

[∇ylifi(xi, x−i, yi) +

∑slij=1 λij∇yli

vij(yi)]αi+∑slij=1 βij∇2

xi,yivij(yi) +

∑qik=1 µik∇2

xi,yigik(xi, x−i, yi).

As ∇yivij(yli), j ∈ J0i ∪ JΛ0i are linearly independent, the matrix

B0i =

∇vlij, j ∈ J0i 0∇vlij, j ∈ JΛl

0i I|0∇vlij, j ∈ Λl

0i 0|I

has full row rank sli. Let Bi be a square regular sub-matrix of dimension sli of B0

i . As theMPCC-LICQ holds, the sub-matrix

B1i =

⊗ ⊗ ⊗ 00 ∇vlij, j ∈ J0i 0 00 ∇vlij, j ∈ JΛl

0i 0 I|00 ∇vlij, j ∈ Λl

0i 0 0|I

of (15) has full (column) rank.


Combining both rank conditions, we can assume that there exists a regular matrix

Ci, which has Bi as a sub-matrix, such that rank(Ci) = rank(B1i ). Hence,

(⊗ CiCTi 0

)is

regular. If the assertion MPCC-SOC fails, w.l.o.g., the matrix ∇Hi can be partitioned asfollows:

∇Hi =

(A1i Bi

BTi Di

)and

A1i = BiD−1

i BTi ,

whereA1i is a sub-matrix ofAi andDi is a square non-singular matrix such that rank(Di) =

rank(∇Hi) and Ci is a sub-matrix of Di. If MPCC-SOC holds Di = ∇Hi.For describing the violation of MPCC-SC, we consider the set of active indices corre-

sponding to the vanishing multipliers. That is βlij = 0, j ∈ JΛl,10i , γ

lij = 0, j ∈ JΛl,2

0i , µik ∈J1

0ig, where i = 1, . . . , N , j = 1, . . . Li.Now, we fix the set of active indices, the set of vanishing multipliers and the columns

of A1i and consider the solutions of the following system

H0i (xi, x−i, yi, λi, 1, αi, βi, µi, γi) = 0, i = 1, . . . , N,

A1i −BiD−1

i BTi = 0, i = 1, . . . , N.

βlij = 0, j ∈ JΛl,10i ⊂ JΛl

0i,

γlij = 0, j ∈ JΛl,20i ⊂ JΛl

0i,µik = 0, i ∈ J1

0ig ⊂ J0ig

(16)

They corresponds with the critical point with active indices set J0i, JΛ0i,Λ0i, J0ig, i =

1, . . . , N, vanishing multiplier JΛl,10i , JΛl,2

0i , J10ig and the rank condition of ∇Hi given by

the fixed columns.For applying the Sard Lemma we consider the derivatives with respect to parameters

and variables. We obtain

∂(x,y) ∂λijl ∂αli ∂βijl ∂µik ∂γijl ∂bFi ∂AFi⊗ ⊗ ⊗ ⊗ ⊗ 0 I ⊗⊗ 0 ∇vlij, j ∈ J0i 0 0 0 0 0⊗ 0 ∇vlij, j ∈ JΛl

0i 0 0 I|0 0 0H0i = 0 ⊗ 0 ∇vlij, j ∈ Λl

0i 0 0 0|I 0 0⊗ ⊗ 0 0 0 0 0 0⊗ 0 0 0 0 0 0 0

∇vlij, l ∈ J l0i ∪ JΛl0i 0 0 0 0 0 0 0

0 I|0 0 0 0 0 0 0A1i −BiD−1

i BTi = 0 ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ 0 I|⊗

0 0 0 I1|0 0 0 0 00 multipliers 0 0 0 0 I2|0 0 0 0

0 0 0 0 0 I3|0 0 0


As the MPCC-LICQ hold and ∇vlij, j ∈ J0i,∇vlij, j ∈ JΛl0i are linearly independent,

the previous matrix has full row rank. By the Sard Lemma, for almost all bF , AF thematrix

∇x,y,α,β,µ,γ

Hi(xi, x−i, yi, λi, 1, αi, βi, µi, γi), i = 1, . . . , N,

A1i −BiD−1

i BTi , i = 1, . . . , N.

βlij, j ∈ JΛl,10i ⊂ JΛl

0i,

γlij, j ∈ JΛl,20i ⊂ JΛl

0i,µik, j ∈ J1

0ig ⊂ J0ig

has full row rank. In particular, the number of variables of the system can not be strictlysmaller than the number of equations. So, only if the sets JΛl,1

0i , JΛl,20i , JΛ1

0ig are emptyand A1 has 0 dimension, system (16) may have a solution. Therefore, MPCC-SC andMPCC-SOC holds. Moreover the matrix

[∇x,y,α,β,µ,γHi(xi, x−i, yi, λi, αi, βi, µi, γi), i = 1, . . . , N ] (17)

has full row rank.So, for almost all AF , bF , at the critical points of problem (14) in which MPCC-LICQ

holds, MPCC-SOC and MPCC-SC are satisfied. By the Theorem of Fubbini, [14], thesame conclusion holds true for almost all AF , bF , Af , bf , cv, bg, cg.

Now, we consider the generalized critical points where MPCC-LICQ fails. By Proposi-tion 4.2, only for a zero measure sets of parameters AF , bF , Af , bf , cv, bg, cg at the general-ized critical points the MPCC-LICQ fails. Intersecting both sets, for almost all parameter,all the generalized critical points of problem (14) are non-degenerated.

Remark 4.3 Similar perturbations results are known for non linear programs and mathe-matical programs with complementarity constraints, see [9, 18]. Only linear perturbationson the objective functions and constants for the constraints functions were needed. Analo-gously, at bilevel programs, only linear perturbations at the objective function of the upperlevel problem are needed for characterizing the second order condition at the critical pointsalmost everywhere, see [4]. The non-cooperative character of the problem, i.e. the para-metric character of the problem of each producer, implies that the MPCC-SOC has to beexplicitly expressed and quadratic perturbations of Fi, i = 1, . . . N , are need.

As already stated in [4] even for almost all perturbations, degenerate critical pointsmay appear. Next corollary shows a special case where only non degenerated criticalpoints may appear a.e.

Corollary 4.1 Given a bilevel problem in which the set of feasible points in the lowerlevel problem does not depend on the upper level problem, i.e.

minF (x, y)s.t. gi(x, y) ≤ 0, i = 1, . . . , q,

y ∈ arg min f(x, y), s.t. vi(y) ≤ 0, i = 1, . . . s.


Then for almost all bF , bf ∈ Rn+m, cg ∈ Rq and cv ∈ Rs, at the problem (8) correspondingto the bi-level problem

minF (x, y) + bF (x, y)s.t. gi(x, y) + cgi ≤ 0, i = 1, . . . , q,

y ∈ arg min f(x, y) + bf (x, y), s.t. vi(y) + cvi ≤ 0, i = 1, . . . s,

MPCC-LICQ holds at all feasible points and all critical points are non degenerated.

Proof. For the MPCC-LICQ, we refer to Theorem 3.1 in [4]. Now we follow the same linesof the proof of Theorem 4.1 taking the combinations of set of indices J0g, J0, JΛ0,Λ0, JΛ1

0,JΛ2

0, J10g and assuming that D = ∇H at system (16), i.e. the block of equations A1 −

BD−1BT = 0 is not included. So, for almost all (bF , bf , cg, cv), the MPCC-SC holds and∇H is regular. As for N = 1, this matrix is equal to

∇2

F (x, y) +s∑j=1

βjvj(y) +∑k∈J0g

µkgk(x, y) + [∇yf(x, y) +s∑j=1

λj∇yvj(y)]α

,MPCC-SOC also holds.

Now we will characterize the generic problem.First, we introduce the regularity definition

Definition 4.4 We say that a problem (8) is regular if

(a) LICQ holds atyli ∈ Rmli : vlij(y

li) ≤ 0, j = 1, . . . , si

, for all i = 1, . . . , N, l =

1, . . . , Li.

(b) MPCC-LICQ holds at all generalized critical points.

(c) All generalized critical points are non-degenerated.

For the open part, we first show the following auxiliary result.

Lemma 4.1 Suppose that the functions vlij, i = 1, . . . N, l = 1, . . . , Li, j = 1, . . . , sli

fulfill that the LICQ holds at M li =

yli ∈ Rmli : vlij(y

li) ≤ 0, j = 1, . . . , si

, for all i =

1, . . . , N, l = 1, . . . , Li. Then, for all y there exists ε0y, ε

1y > 0 such that for all y satisfying

‖y− y‖ < ε0y and for all vlij fulfilling ‖vlij(y)− vlij(y)‖ < ε1

y, ∀y : ‖y− y‖ < ε0y, the following

holds:

(i) If for some (i, l), yli /∈M li , then yli /∈


li) ≤ 0, j = 1, . . . , si

.


(ii) If yli ∈M li , then LICQ is fulfilled yli ∈


li) ≤ 0, j = 1, . . . , si

.

Proof. If (i) or (ii) are not satisfied, there exist yn → y, (εy)n → 0 and a sequence offunctions (vlij)n, ‖(vlij)n − vlij‖ < εy such that (i) or (ii) fails. It is clear that, by theconvergence of yn and (εy)n,

∇r(vlij∗)n(yn)→ ∇rvlij∗(yli). (18)

for r = 0, 1.

If yli /∈yli ∈ Rmli : vlij(y

li) ≤ 0, j = 1, . . . , si

, there exists j∗ such that vlij∗(y

li) > 0.

Since vlij ∈ C1 by (18), (vlij∗)n(yn) > 0. So, if (i) fails, then vlij(yn) ≤ 0 and y /∈M li . Taking

limits we obtain that 0 < vlij(y) ≤ 0, recall (18) for r = 0, which is a contradiction.If (ii) fails, the LICQ fails. Considering (18) for r = 1, we obtain the failure of the

LICQ at yli, a contradiction.

Theorem 4.2 (Local stability theorem) Suppose that the functions Fi, gik, fli , v

lij, i =

1, . . . N, l = 1, . . . , Li, j = 1, . . . , sli are such that problem (8) is regular. Then, for all (x, y)there exists ε0

x,y, ε1x,y > 0 such that for all Fi, gik, f

li , v

lij fulfilling

‖∇r(Fi, gik)−∇r(Fi, gik)‖ < ε1x,y, r = 0, . . . , 2, ‖∇r(f li , v

lij)−∇r(f li , v

lij)‖ < ε1

x,y, r = 0, . . . , 3,

at (x, y) : ‖(x, y) − (x, y)‖ < ε0x,y, the following holds for all (x, y) such that ‖(x, y) −

(x, y)‖ < ε0x,y:

(i) If (x, y, λ) is infeasible for all λ then, for all λ, (x, y, λ) is not a feasible point of theproblem defined by the functions Fi, gik, f

li , v

lij.

(ii) If (x, y, λ) is a generalized critical point of the problem defined by the functionsFi, gik, f

li , v

lij then (a)-(b), at Definition 4.4 hold.

(iii) If (x, y, λ) is a critical point of the problem defined by the function Fi, gik, fli , v

lij,

then MPCC-LICQ, MPCC-SC and MPCC-SOC are satisfied.

Proof. Following the ideas of Lemma 6.3 in [9], we assume that there exists a point (x, y),a sequence (xn, yn)→ (x, y) and functions (Fi, gik, f

li , v

lij)n such that

‖∇r(Fi, gik)−∇r(Fi, gik)‖ → 0, r = 0, . . . , 3, ‖∇r(f li , vlij)−∇r(f li , v

lij))‖ → 0, r = 1, . . . , 4,

and that (i), (ii) or (iii) fail. As stated in Lemma 4.1, for i = 1, . . . N, l = 1, . . . , Li, j =1, . . . , sli:

(Fi, gik, fli , v

lij)n(xn, yn)→ (Fi, gik, f

li , v

lij)n(x, y),


as well as their first and second order derivatives.If (i) fails, there exists (x, y, λ) which is infeasible for all λ and λn such that (xn, yn, λn)

is feasible. Moreover, by Lemma 4.1(i), yli ∈M li for all (i, l). So, by Lemma 4.1(ii), LICQ

holds at [yli]n, for all (i, l), recall it is a feasible point of the lower level problem (i, l).Hence, for n large enough λn is well-determined.

Furthermore, since (xn, yn) is a sequence of feasible points [gik]n(xn, yn) ≤ 0, for all(i, k) gik(x, y) ≤ 0.Finally, as yn converges by (18) for r = 1, ∇yli

(vlij)n([ylj]n) → ∇yli(vlij)([y

lj]) and, analo-

gously, ∇yli(f li )n(xn, yn)→ ∇yli

(f li )(xn, yn). Moreover, as the LICQ holds at (yli)n and yli,

λn has a limit λ, λ ≥ 0 and complements with vlij(yli). So, (x, y, λ) is feasible, contradicting

the assumption that (i) fails.Suppose that (ii) fails. Let (xn, yn)→ (x, y) be such that (xn, yn, λn) is a generalized

critical point. By Lemma (4.1), the LICQ holds at [yli]n, for all (i, l) and λn → λ. Then(a) is fulfilled. Since (xn, yn, λn) is a generalized point, there exists (µ0i, αi, βi, µi, γi)n 6= 0,for all i = 1, . . . , N , such that the following system has a solution

H0i ((xi, x−i, y, λi, µ0i, αi, βi, µi, γi)n) = 0, i = 1, . . . , N, (19)

and for some in, (µin0)n = 0.W.l.o.g. we assume that in = 1, ‖(µ0i, αi, βi, µi, γi)n‖ = 1. For the limit

(µ0i, αi, βi, µi, γi)n → (µ0i, αi, βi, µi, γi),

it holds ‖µ0i, αi, βi, µi, γi‖ = 1 and µin0 = 0. Using the same arguments of (i), λn convergesto λ and (x, y, λ) is feasible. Taking limits in (19) and using the continuity of the H0

i , asin (18), we obtain:

H0i ((xi, x−i, y, λi, µ0i, αi, βi, µi, γi)n)→ H0

i (xi, x−i, y, λi, µ0i, αi, βi, µi, γi)) = 0, i = 1, . . . , N.

So, (x, y, λ) is a generalized critical point and, since µ0i = 0, the MPCC-LICQ fails.Therefore, (b) holds.

The failure of (iii) is possible if MPCC-LICQ, MPCC-SC or MPCC-SOC fails. By (ii)only MPCC-SC and MPCC-SOC shall be analyzed. As MPCC-LICQ holds the multi-pliers for all i =, 1, . . . , N, and n large, (αi, βi, µi, γi)n are unique. Continuity argumentscontradict the existence of zero multipliers (βlij, γ

lij, )n j ∈ [JΛl

0i]n or (µik)n, k ∈ [Jg0k]n.Similarly by the continuity of the second derivatives, MPCC-SOC is satisfied at (x, y, λ)n.

Remark 4.4 In Theorem 4.1 we needed to include the second order condition. So, com-pared with the previous result, a higher degree of differentiability was needed, namely(Fi, gik) ∈ C3 and (f li , v

lij) ∈ C4. The same conclusion also holds for this higher degree of

differentiability. We only need to restrict the neighborhood to those functions whose thirdor fourth order derivative satisfies the corresponding inequality.


Now we can present the genericity result

Theorem 4.3 The set of functions Fi, gik, fli , v

lij, i = 1, . . . N, l = 1, . . . , Li, j = 1, . . . , sli

which define a regular MLDF model , is generic.

Proof. For the open part, fix Fi, gik, fli , v

lij, i = 1, . . . N, l = 1, . . . , Li, j = 1, . . . , sli such

that the MLDF problem they define, is regular. For each (x, y) take the set B((x, y), ε0x,y),

that is the open ball centered at (x, y) and radius ε0(x,y) obtained at Theorem 4.2 and

Remark 4.4. Take a locally finite sub-covering S. Consider the function

ε(x, y) = min(x,y):(x,y)∈B((x,y),ε0x,y),B((x,y),ε0x,y)∈S

ε1x,y

As there are finitely many sets satisfying (x, y) ∈ B((x, y), ε0x,y), and B((x, y), ε0

x,y) ∈ S thefunction ε(x, y) is continuous and positive. For all elements at the neighborhood definedby ε(x, y), at the corresponding MLDF model conditions (i)-(iii) hold, recall Theorem4.2. This neighborhood is an open set and all its elements fulfill the non-degeneracy ofthe generalized critical points and LICQ holds at all feasible points of the lower levelproblems.

For the density we use the same ideas of the proof of Theorem 6.25 in [9]. Given acontinuous and positive function ε(x, y) and (Fi, gik, f

li , v

lij), i = 1, . . . N, l = 1, . . . , Li, j =

1, . . . , sli , we will prove the existence of functions (Fi, gik, fli , v

lij) at the closure of the cor-

responding neighborhood of (Fi, gik, fli , v

lij) satisfying the LICQ and the non-degeneracy

condition.To this aim we take the C∞-functions ξn(x, y) such that

- ξ1(x, y) = 1, if ‖(x, y)‖ ≤ 2 and ξ1(x, y) = 0, if ‖(x, y)‖ > 3.

- ξn(x, y) = 1, if n ≤ ‖(x, y)‖ ≤ n + 1 and ξn(x, y) = 0, if ‖(x, y)‖ < n − 1 or‖(x, y)‖ > n+ 2.

Their existence is guaranteed via partitions of the unity, see [14]. Now, we define (F, g, f, v)nas

(F, g, f, v)n−1+ξn(x, y)[(bFi (xi, yi) + (xi, yi)A

Fi yi, c

gik + bgik(xi, y

li), b

fily

li + (xi, y

li)A

fi y

li, c

vijl

)],

i = 1, . . . N, l = 1, . . . , Li, j = 1, . . . , sli.Here AF , bF , Af , bf , cv is taken in such a way that

‖[(bFi (xi, yi) + (xi, yi)A

Fi yi, c

gik + bgik(xi, y

li), b

fily

li + (xi, y

li)A

fi y

li, c

vijl

)]‖ < inf

(x,y):‖x,y‖<n+1ε(x, y).

Moreover, by Theorem 4.1, we can assume that AF , bF , Af , bf , cv are such that the corre-sponding parametric problem is regular.

So, this auxiliary sequence of problems satisfies

5 REGULARITY FOR OUR ELECTRICITY MARKET MODEL 25

• (F, g, f, v)n is an element of the neighborhood of (F , g, f , v) given by ε(x, y).

• (F, g, f, v)n = (F, g, f, v)n−1 at (x, y) : ‖x, y‖ ≤ n− 1.

• The non-degeneracy of the generalized critical points and the LICQ conditions aresatisfies for all feasible element of (x, y) : ‖x, y‖ ≤ n+ 1.

Defining (F, g, f, v)(x, y) = (F, g, f, v)n(x,y)(x, y), where n(x, y) = minn : ‖(x, y‖ < n,the problem is evidently regular and belongs to the closure of the neighborhood of(Fi, gik, f

li , v

lij) given by ε(x, y).

As in the case of Theorem 4.1, the following results holds for bilevel problems

Corollary 4.2 Given a bilevel problem in which the set of feasible points in the lowerlevel problem does not depend on the upper level problem, i.e.

minF (x, y)s.t. gi(x, y) ≤ 0, i = 1, . . . , q,

y ∈ arg min f(x, y), s.t. vi(y) ≤ 0, i = 1, . . . s.

Then for a generic set of functions F, f, gi, i = 1, . . . q, vi, i = 1, . . . s at the correspond-ing problem (8), MPCC-LICQ holds at all feasible points and all critical points are nondegenerated.

Proof. As before, the open part is a consequence of Theorem 4.2 for N = 1 and thedensity follows the same lines of the proof of the previous theorem using Corollary 4.1 asthe perturbation result.

If the upper level variables has dimension 1, the generic structure of the set of feasiblesolutions of the bilevel problem has been studied in [15]. In our case, we consider arbitrarydimensions. We use the MPCC approach, where the lower level multipliers are includedas variables. We have shown that, generically, the LICQ holds and, hence, the multipliersare uniquely determined by the lower and the upper level variables. This implies that, asstated in [6], local minimizers (x, y, λ) such that (x, y) is a feasible point of the originalproblem, then (x, y) is also a local minimizer of the bilevel model.

5 Regularity for our electricity market model

In this part, we consider the model (4),(3), discussed in Section 2 and provide conditionsfor the regularity.


Proposition 5.1 Consider problem (4). For all i = 1, . . . , N , l = 1, . . . , Li, such thatdl,hi > 0 and

∑Hh=1 d

l,hi < Dl

i the LICQ holds at problem (3) corresponding to client l ofclient actor i.

Proof: Consider the active indices set of problem (3). It is clear that LICQ holds at∑NCic=1 q

l,h,ci ≥ dl,hi , ql,h,ci ≥ 0, h = 1, ..H, c = 1, .., NCi, because for each (l, h), such that∑NCi

c=1 ql,h,ci ≥ dl,hi > 0 there exists c satisfying that ql,h,ci > 0. Moreover, summing up

Dli =

H∑h=1

NCi∑c=1

ql,h,ci ≥H∑h=1

dl,hi ,

as∑NCi

c=1 ql,h,ci > dl,hi , there exists indexes (l, h) fulfilling that

∑NCic=1 q

l,h,ci > dl,hi because

otherwise the demand Dli will not be satisfied. W.l.o.g. we assume that

∑NCic=1 q

1,1,ci > d1,1

i

and q1,h,ci > 0.

So, the gradients of the maximal active constraints are

constraints q1,1,11 qLi,H,NCiN

0, . . . , 0 −1, . . . ,−1 0, . . . , 0 0, . . . , 0

dl,hi −∑NCi

c=1 ql,h,ci = 0 0, . . . , 0 0, . . . , 0

. . . 0, . . . , 00, . . . , 0 0, . . . , 0 0, . . . , 0 −1, . . . ,−1∑H

h=1

∑NCic=1 q

l,h,ci = Dl

i 1, . . . , 1 1, . . . , 1 1, . . . , 1 1, . . . , 10| − I|0 0 0 0

−ql,h,ci = 0 0 0| − I|0 0 00 0 0| − I|0 00 0 0 0| − I|0

(20)

As the previous matrix has full row rank, LICQ holds.Now, we apply the KKT approach to the models described in (4) and analyze the

fulfillment of the MPCC-LICQ at the resulting set of feasible solutions of leader i, i =1, . . . , N . Writing the constraints we have


ph,ci + C l,h,ci − λl,hi − λ

l,h,ci + δli = 0 h = 1, . . . , H,

c = 1, . . . , NCi,l = 1, . . . Li

0 ≤ −dl,hi +∑NCi

c=1 ql,h,ci ⊥ λl,hi ≥ 0 h = 1, . . . , H,

l = 1, . . . Li∑NCic=1

∑Hh=1 q

l,h,ci = Dl

i h = 1, . . . , H,l = 1, . . . Li

0 ≤ ql,h,ci ⊥ λl,h,ci ≥ 0, h = 1, . . . , H,c = 1, . . . , NCi,l = 1, . . . Li

ophi + exthi +∑

i′ 6=i inti,hi′ −

∑Lil=1

∑NCic=1 q

l,h,ci =

∑i′ 6=i int

i′,hi , h = 1, . . . H

exthi ≥ 0 h = 1, . . . , H

ophi ≥ 0 h = 1, . . . , H

inti,hi′ ≥ 0, h = 1, . . . , H

phi,c ∈ [phi,c, phi,c], c = 1, . . . , NCi,

h = 1, . . . , H

pi′,ci ∈ [pi,c

i, pi′,ci ], c = 1, . . . , NCi,

i 6= i′∑h

∑NCjc=1 w

hi,cp

hi,c −

∑h

∑NC1

c=1 wh1,cp

h1,c = 0, i > 1

or∑

h

∑NC1

c=1 wh1,cp

h1,c ≤ ω, i = 1

(21)

Considering the gradients of a possible set of possible active constraints, we obtain fori = 1, . . . , N, l = 1, . . . , Li:

constraints ∂ph,ci

∂ql,h,ci

∂δli

∂λl,hi

∂λl,h,ci

∂ophi,exth

i,int

i,hi′

∂pi′,hi

ph,ci − λl,hi − λl,h,ci + δli = 0 I 0 eNCi ETLi,H,NCi

−I 0 0

dl,hi −

∑NCic=1 q

l,h,ci = 0∑H

h=1

∑NCic=1 q

l,h,ci = Dli 0 Ωi,l 0 0 0 0 0

−ql,h,ci = 0

−λl,h,ci = 0 0 0 0 0 0|I|0 0 0

−λl,hi = 0 0 0 0 0|I|0 0 0 0

phi,c = phi,c

or phi,c0 I 00 0 −I 0 0 0 0 0 0

(22)

constraints ∂ph,ci

∂ql,h,ci

∂δli

∂λl,hi

∂λl,h,ci

∂ophi,exth

i,int

i,hi′

∂pi′,hi

ophi , exthi , int

i,h

i′ , pi′,hi ≥ 0 0 0 0 0 0 0|I 0

0 ⊗ 0 0 0 EH 0

pi′,ci = pi,c

ior p

i′,ci 0 0 0 0 0 0

0 I 00 0 −I

(23)


Here eNCi = (1, . . . , 1)T ∈ RNCi , ELi,H,NCi =

0 −1, . . . ,−1 0 00 0 0 00 0 0 −1, . . . , 1

and

Ωi,l is given in (20).

Proposition 5.2 Consider problem (4) in which the client l, l = 1, . . . , Li, has an addi-tional known costs of C l,h,c

i per unit of contracted electricity in period h. We assume thatwi is not the null vector for all i, dl,hi > 0 for all l, h and

∑Hh=1 dl,h < Dl

i.Then for almost all C the MPCC- LICQ holds at the bilevel problem of leader i , for

all i = 1, . . . , N ,.

Now we take i > 1, fix the set of active index set and consider the constraints which doesnot depend on (ophi , ext

hi , int

i,hi′ ), (oph1 , ext

h1 , int

1,hi′ ) variables corresponding to problem i

and 1 respectively. That is

ph,ci + C l,h,ci − λl,hi − λ

l,h,ci + δli = 0 h = 1, . . . , H,

c = 1, . . . , NCi,l = 1, . . . Li

0 ≤ −dl,hi +∑NCi

c=1 ql,h,ci ⊥ λl,hi ≥ 0 h = 1, . . . , H,

l = 1, . . . Li∑NCic=1

∑Hh=1 q

l,h,ci = Dl

i

0 ≤ ql,h,ci ⊥ λl,h,ci ≥ 0, h = 1, . . . , H,c = 1, . . . , NCi,l = 1, . . . Li

phi,c ∈ [phi,c, phi,c], c = 1, . . . , NCi,

h = 1, . . . , H

∑h

∑NCjc=1 w

h,ci phi,c −

∑NC1

c=1 wh,c1 ph1,c = 0, i > 1

or∑

h

∑NC1

c=1 wh,c1 ph1,c ≤ ω, i = 1

(24)


Taking derivatives at the constraints of agent i and agent 1 we obtain:

∂ph,c1 ,p

h,ci

∂ql,h,c1 ,q

l,h,ci

∂δl1,δ

li

∂λl,h1 ,λ

l,h1

∂λl,h,c1 ,λ

l,h,c1

∂C1,Ciw

I 0

.

.

.I 0

0 diagl(eNCi )|0 −diagl(ETLi,H,NCi )|0 −I|0 I|0 0

0 diagl(Ωi,l)|0 0 0 0 0 0

0 0 0 0 0|diagl(I)|0|0 0 00 0 0 0|diagl(I)|0|0 0 0 0

0 I 00 0 −I|0 0 0 0 0 0 0

w1|0 0 0 0 0 0 10 I

.

.

.0 I

0 0|diagl(eNCi ) 0| − diagl(ETLi,H,NCi ) 0| − I 0|I 0

0 0| − diagl(Ωi,l) 0 0 0 0 0

0 0 0 0 0|0|diagl(I)|0 0 00 0 0 00|diagl(I)|0 0 0 0

0|0 I 00 0 −I 0 0 0 0 0 0

−w1|wi 0 0 0 0 0 0

By Parameterized Sard Lemma, for almost all C1, Ci, ω the previous matrix has full rowrank. This means that the following matrix has full row rank

∂ph,c1 ,p

h,ci

∂ql,h,c1 ,q

l,h,ci

∂δl1,δ

li

∂λl,h1 ,λ

l,h1

∂λl,h,c1 ,λ

l,h,c1

I 0

.

.

.I 0

0 diagl(eNCi )|0 −diagl(ETLi,H,NCi )|0 −I|0

0 −diagl(Ωi,l)|0 0 0 0

0 0 0 0 0|diagl(I)|0|00 0 0 0|diagl(I)|0|0 0

0 I 00 0 −I|0 0 0 0 0

w1|0 0 0 0 00 I

.

.

.0 I

0 0|diagl(eNCi ) 0| − diagl(ETLi,H,NCi )

0 0| − diagl(Ωi,l) 0 0 0

0 0 0 0 0|0|diagl(I)|00 0 0 0|0|diagl(I)|0 0

0|0 I 00 0 −I 0 0 0 0

−w1|wi 0 0 0 0

Summing the rows wi| − w1 0 and 0|w1 0 corresponding toe the derivatives of the

constraints∑

h

∑NCjc=1 w

h,ci phi,c−

∑NC1

c=1 wh,c1 ph1,c = 0 and

∑h

∑NC1

c=1 wh,c1 ph1,c ≤ ω, respectively


we get that the following matrix has also full row rank :

∂ph,c1 ,p

h,ci

∂ql,h,c1 ,q

l,h,ci

∂δl1,δ

li

∂λl,h1 ,λ

l,h1

∂λl,h,c1 ,λ

l,h,c1

I 0

.

.

.I 0

0 diagl(eNCi )|0 −diagl(ETLi,H,NCi )|0 −I|0

0 −diagl(Ωi,l)|0 0 0 0

0 0 0 0 0|diagl(I)|0|00 0 0 0|diagl(I)|0|0 0

0 I 00 0 −I|0 0 0 0 0

w1|0 0 0 0 00 I

.

.

.0 I

0 0|diagl(eNCi ) 0| − diagl(ETLi,H,NCi )

0 0| − diagl(Ωi,l) 0 0 0

0 0 0 0 0|0|diagl(I)|00 0 0 0|0|diagl(I)|0 0

0|0 I 00 0 −I 0 0 0 0

0|wi 0 0 0 0

Using the block structure of this matrix, we can separate the part corresponding tothe i agent from the part corresponding to the first agent and obtain a matrix with thefollowing structure:

∂ph,c1 ,ql,h,c1 ,δl1,λl,h1 ,λl,h,c1 , ∂ph,ci ,ql,h,ci ,δli,λ

l,h1 ,λl,h,c1

⊗1 00 ⊗2

So, the matrices ⊗1 and ⊗2 have full row rank. In particular, they correspond to thederivatives of the constraints, see (22), for agent 1 and i respectively, adding 0 to thederivatives that correspond to (oph1 , ext

h1 , int

1,hi′ ) and (ophi , ext

hi , int

i,hi′ ), also respectively.

Now we consider all the active constraints for agent 1. Adding the matrix in (23) corre-sponding to agent i, we obtain

∂ph,c1 ,ql,h,c1 ,δl1,λl,h1 ,λl,h,c1 , ∂oph1 ,exth1 ,int

1,h

i′

⊗1 0matrix (23)

As it has full row rank and it coincides with the gradients of the active constraints ofthe problem 1, the MPCC-LICQ holds. An analogous result is obtained for agent i using⊗2|0. Taking all possible combinations of active index sets, we obtain that, for almost all(Ci, C1, ω), the MPCC-LICQ holds at the problems of agent i and 1. The result is true ifwe consider the almost all in the RC1,w ×Ni=1 RCi space corresponding to the parameters(C1, w)[×i−1

j=2Cj] × Ci[×Nj=i+1Cj] in which the almost all original set is embedded in thelarger dimension space. Intersecting these sets for all i, we get the fulfillment of theMPCC-LICQ for almost all perturbation.

6 CONCLUDING REMARKS 31

Remark 5.1 Using the linearity of the constraints we obtained that MPCC-LICQ holdsnot only at generalized critical points, but at all feasible points. The special constraintseven allow to use less parameters for the almost all result than in the general case.

Now we study the properties of non-degenerate critical points. At first glance wecan think that as in the case of MPCC-LICQ only the description of the KKT-system isneeded for the almost all structure. For the MPCC-LICQ, we could isolate the constraintsdepending on the parameter, obtain the full rank condition for the others and completethe gradients keeping full rank independently of the value of the other agents variables.However, writing the second order condition we can see that the non degeneracy stronglyrelays on the choice of the parameter, made by the other players.

Theorem 5.1 Denoting z =(phi,c)h,c, (ext

hi )h, (op

hi )h, (int

i,hi′ )i

′ 6=i,h, (pi′,hi )i

′ 6=i,h)

, consider

the perturbed model, for i = 1, . . . , N ,

maxz

max(ql,h,ci )h,c

Fi(z, q) s.t. (24)

where Fi(z, q) =∑Li

l=1

∑Hh=1

∑NCic=1 p

hi,cq

l,h,ci +

∑i′ 6=i p

i′,hi .inti

′,hi −

∑Hh=1Mph.exthi

−∑H

h=1 costhi (op

hi )−

∑i′ 6=i p

i,hi′ .int

i,hi′ + (z,q)TAi(z,q)

2+ bi(z, q).

Then, for almost all A1, . . . , AN , b1, . . . bN , Cl,h,ci , if dl,h > 0 and

∑Hh=1 dl,h < Dl the

MPCC-LICQ holds at all feasible point and the generalized critical points are non de-generated

Proof: By Proposition 5.2 the MPCC-LICQ holds, so all generalized critical points arecritical. We consider the system of critical points and the condition that some multipliersassociated to inequality constraints are zero. For each i = 1, . . . , N

Then we apply Theorem 4.1, and the result follows.

6 Concluding remarks

In this paper we studied the Multi-Leader Disjoint Follower game problem. This bilevelstructure is substituted by the KKT-approach, leading to a Generalized Nash equilibriummodel in which each leader solves an MPCC problem depending on the decision of theother leaders. It is clear that in the general case the failure of constraint qualificationsremains stable under perturbation.However, if the constraints of the lower level problemsdo not depend on the upper variables, we proved that the LICQ is generically fulfilledat all the lower level problems. Hence, based in [6], we can easily see that generically alocal optimal solution of the KKT transformation is also a local optimal solution of theoriginal problem. On the other hand, we obtained the non degeneracy of the solutions of

REFERENCES 32

each MPCC. From a numerical viewpoint this result guarantees that local solutions of themodel are isolated. Indeed, in the proof of Theorem 4.1, we have already pointed out thatmatrix (17) has full row rank. Moreover, it is non-singular. As this matrix coincides withthe Jacobian of the system corresponding to the critical point condition, good conditionsfor the convergence of Newton type methods can be ensured. An appealing solutionalgorithm for this kind of models was proposed in [20]. The non-degeneracy is a desiredcondition for the iterated solution of the MPCC corresponding to each leader.

References

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