Hierarchical Structures in Equilibrium Problems

and

Academy of Sciences of the Czech Republic

Institute of Information Theory and Automation

Doctoral Thesis

Mgr. Bc. Michal ƒervinka

Hierachical Structures in Equilibrium Problems

Supervisor:

Doc. Ing. Ji°í V. Outrata, DrSc.

Prague, May 2008

a

Akademie v¥d ƒeské Republiky

Ústav teorie informace a automatizace, v.v.i

Diserta£ní práce

Mgr. Bc. Michal ƒervinka

Hierarchické struktury v ekvilibriálních úlohách

’kolitel:

Doc. Ing. Ji°í V. Outrata, DrSc.

Praha, kv¥ten 2008

This doctoral thesis would not be written if not for my supervisor, Ji°íOutrata. When I

entered the Ph.D. program at the Charles University, I had very little if any knowledge

of modern variationalanalysis and nonlinear optimization. Overthe years, my supervisor

was patiently answering my questions, explainedmethe concepts of modern optimization

theory,eachtime pointed metorelevantbooksand papers. He carefullyread my working

papers and preprints uncountably many times, always improving the texts. His critical

reviews, insights and ideas lead to the creation of this thesis. I amdeeply indebt to him

for providingmewith this opportunity, for hisguidance and constant patience.

Also, ifnot for him,I would not havethe opportunity todiscussmyresults withother

researches in my eld. I am grateful to Boris S. Mordukhovich for interesting discussions

and topicswhichlead toajointworkwith himand Ji°íOutrata, resultinginSections 4.1,

4.3 and 5.3.

I would like to express my gratitude to Daniel Ralph, for allowing me to spend two

unforgettable months at the Judge Business School, University of Cambridge during the

summer2006. Ithank himfornearlyeverydaydiscussionsdespitehisbusyschedule,which

helped me to get deeper understanding of stationarity concepts discussed in Section 2.3.

Also,he proposed anidea to generalize the homotopy methodin such a way that itcould

be appliedtoaspecialtype ofEPEC. The results weachieved together duringmy stay in

Cambridgeform abasis of Section 5.2.

I alsothank to Jong-ShiPang andAlejandro Jofré forsuggesting the generalizationof

concept of solutions tomixed strategies, which lead metoresults in Section3.3.

TheInstituteofInformationTheoryandAutomationoftheAcademyofSciencesofthe

CzechRepublichas beenagreatworking environment. Iwould likethank forthenancial

support of The Ryoichi Sasakawa Young Leaders Fellowship Fund, which allowed me to

visit Daniel Ralph in Cambridge. The work on results presented in this thesis was also

supported by the Grant Agency of the Charles University under grant GAUK 7645/2007

and by the GrantAgency of the Academy of Sciences of the Czech Republic under grant

IAA 1030405.

A few other people deserve special mention: Michal Ko£vara, for help in creating

gures, Václav Kratochvíl, for his great help in creating computer codes in Matlab, my

students, forcheerful hours every week;and myformer and current colleagues,classmates

thesis is dedicated toyou.

Michal ƒervinka

Preface iv

Abbreviations ix

Notation x

1 Introduction 1

2 Mathematical Program with Equilibrium Constraints (MPEC) 5

2.1 Mathematicalformulation . . . 5

2.2 Necessary optimality conditions vianonlinear programming. . . 8

2.3 Mathematicalprogram with complementarity constraints . . . 9

2.3.1 Stationarityconditions for MPCCs . . . 10

2.3.2 Implicit programmingapproach and Clarkestationarity . . . 17

2.3.3 Equivalence of Clarkeand C-stationarity . . . 24

3 Equilibrium Problem with EquilibriumConstraints (EPEC) 33 3.1 Mathematicalformulation . . . 33

3.2 Sourceproblems . . . 36

3.2.1 Oligopolisticmarketproblem . . . 37

3.2.2 Forward-spot marketmodel . . . 38

3.2.3 Deregulated electricity market model . . . 39

3.2.4 Trac equilibriumproblemwith private toll roads . . . 42

3.3 Existence of solutions . . . 44

3.4 Stationarityconcepts and existenceof stationary points . . . 50

4 Multiobjective Problem with Equilibrium Constraints (MOPEC) 55 4.1 Mathematicalformulation . . . 55

4.2 Existence of weak Pareto solutions . . . 57

4.3 Necessary optimality conditions . . . 62

5 Solution Methods for EPECs and MOPECs 67 5.1 Overview. . . 67

5.1.2 Sequential nonlinear complementarity method . . . 68

5.1.3 Price-consistent NCPmethod . . . 69

5.2 Homotopy methodfor computationof C-stationary points toEPCCs . . . 71

5.2.1 Parameter-free problem. . . 72

5.2.2 A oneparametric problem . . . 76

5.2.3 Homotopy method . . . 82

5.2.4 Numericalresults . . . 89

5.3 Numericalmethod for MOPCCs . . . 91

Conclusion 97 A Variational Analysis 99 A.1 Multifunctions. . . 99

A.2 Generalizeddierentiation . . . 100

A.3 Variationalinequality and complementarity problem. . . 102

B LICQ and MFCQ of Standard Nonlinear Program 105

C Noncooperative Nash Games 107

Abbreviations

EPCC equilibriumproblemwith complementarityconstraints

EPEC equilibriumproblemwith equilibriumconstraints

GLICQ generalizedlinear independence constraintqualication

GMFCQ generalizedMangasarian-Fromowitz constraintqualication

ISO independent system operator

KKT Karush-Kuhn-Tucker

LICQ linear independence constraint qualication

MCP mixed complementarity problem

MFCQ Mangasarian-Fromowitzconstraint qualication

MOPCC multiobjective problemwith complementarity constraints

MOPEC multiobjective problemwith equilibriumconstraints

MPCC mathematicalprogramwith complementarity constraints

MPEC mathematicalprogramwith equilibriumconstraints

NCP nonlinear complementarity problem

NLP nonlinear program

OD origin-destination

SOPEC set-valued optimization problemwith equilibriumconstraints

SRC strongregularity condition

Notation

Spaces and Orthants

R

the real numbers

R

₋

the left halfline

R

₊

the righthalf line

R

n

the

n

-dimensional real vector space

R

n

−

the nonpositiveorthant in

R

n

R

n

+

the nonnegative orthantin

R

n

Sets

∅

empty set

{x}

the set consisting of the vector

x

{x}

⊥

the orthogonal complement of vector

x

(a, b)

anopen interval in

R

[a, b]

aclosed interval in

R

conv

S

convex hull of the set

S

cone

S

conic hullof the set

S

cl

S

closureof the set

S

int

S

interior of the set

S

rint

S

relativeinterior of the set

S

bdry

S

boundary of the set

S

S

1

⊂ S

2

S

1

isa subset of

S

2

|I|

cardinalityof a nite set

I

P(I)

the set of all subsetsof a nite set

I

S

1

× S

2

Carthesian product of sets

S

1

and

S

2

X

n

i=1

S

i

Carthesian product of sets

S

i

, i = 1, . . . , n

arg min

_x∈Ω

f (x)

the set of pointswhere the minimum of the real-valued function

f

onthe set

Ω

isattained

arg max

x∈Ω

f (x)

the set of pointswhere the maximum of the real-valued function

f

onthe set

Ω

isattained

B

the closed unit ball

B

_(x)

the closed unit ballaround

x

Cones

T (x; Ω)

the Bouligand-Severicontingent cone to

Ω

at

x

T

C

(x; Ω)

the Clarke tangentcone to

Ω

at

x

N(x; Ω)

the limitingnormal cone to

Ω

at

x

N

C

(x; Ω)

the Clarke normalcone to

Ω

at

x

ˆ

N(x; Ω)

the Fréchet normal cone to

Ω

at

x

K(x, y; Ω)

the critical coneof

Ω

with respect to

x

and

x − y

K

∗

the polarcone to

K

K

−

x ∈ R

n

columnvector in

R

n

x

>

transpose of vector

x

(x, y)

columnvector

(x

>

_{, y}

>

₎

>

x

i

i

thcomponent of vector

x

x

−i

the vector in

R

n−1

consisting of components

x

j

, j 6= i

x

I

the vector in

R

|I|

consisting of components

x

i

, i ∈ I

x

−i

the vector

(x

1

_{, . . . , x}

i−1

_{, x}

i+1

_{, . . . , x}

m

₎

with

x

j

_{∈ R}

n

_{, j = 1, . . . , m}

x ≥ y

componentwise comparison

x

i

≥ y

i

, i = 1, . . . , n

x > y

componentwise strict comparison

x

i

> y

i

, i = 1, . . . , n

hx, yi := x

>

_y

the standard inner product of vectors in

R

n

||x||

the Euclidean norm of a vector

x ∈ R

n

min

{x, y}

the vector whose

i

th componentis min

{x

i

, y

i

}

x⊥y

orthogonality of vectors

x

and

y

in

R

n

Functions and Multifunction

f : R

n

_{→ R}

m

afunction that maps

R

n

to

R

m

f

i

: R

n

→ R

the

i

thcomponent function of

f

F : R

n

_{⇒ R}

m

amultifunctionthat maps

R

n

tosubsets of

R

m

epi

f

the epigraph of function

f

epi

F

the generalized epigraph of multifunction

F

E

F

the epigraphicalmultifunctionof multifunction

F

Dom

F

the domainof multifunction

F

Gph

F

the graph of multifunction

F

Ker

F

the kernel of operator

F

∇f (x)

the Jacobianof

f : R

n

_{→ R}

m

(the gradient of

f : R

n

_{→ R}

)

∇

x

f (x)

the partial Jacobianof

f : R

n

_{→ R}

m

(the partial

gradient of

f : R

n

_{→ R}

) with respect to

x

∇f

I

(x)

the submatrixof the

m × n

matrix

∇f (x)

with rows indexed by

i ∈ I ⊂ {1, . . . , m}

∇f

I,J

(x)

the submatrixof the

m × n

matrix

∇f (x)

with rows

indexed by

i ∈ I ⊂ {1, . . . , m}

and columns by

j ∈ J ⊂ {1, . . . , n}

∂f (x)

limitingsubdierential of

f

at

x

¯

∂f (x)

generalizedJacobian (Clarkesubdierential)of

f

at

x

∂

K

F (x, y)

limitingsubdierential of multifunction

F

at

(x, y) ∈

epi

F

with respect toa cone

K

∂

∞

K

F (x, y)

singularsubdierentialof multifunction

F

at

(x, y) ∈

epi

F

with respect toa cone

K

D

∗

_{F (x, y)}

coderivativeof a multifunction

F

at

(x, y) ∈

Gph

F

Π(x; Ω)

the Euclidean projector of

x

ontothe closure of

Ω

dist

(x; Ω)

Euclidean distance between

x

and

Ω

E

the identity matrix of appropriate order

A

>

transpose of a matrix

A

A

j

the

j

th rowof a matrix

A

A

I

the submatrixof amatrix

A

with rows

A

j

, j ∈ J

A

>

I

transpose of the submatrixof a matrix

A

with rows

A

j

, j ∈ J

A

x

i

the submatrixof amatrix

A

with rows of

A

which correspond tocomponents of vector

x

i

in the product

Ax, x = (x

1

_{, . . . , x}

n

₎

Q

x

i

_,x

i

the square submatrix of asquare matrix

Q

with rows

and columnsof

Q

whichcorrespond to components of vector

x

i

inthe product

Qx, x = (x

1

_{, . . . , x}

n

₎

det

A

determinant of amatrix

A

Adj

A

adjunct matrix of amatrix

A

A

−1

inverse matrix of a matrix

A

diag

(A

1

_{, . . . , A}

n

₎

block diagonalmatrix with the

i

th block equaltomatrix

A

i

Sequences

{x

(k)

_}

asequence in

R

n

x → ¯

x

x

converges to

x

¯

x

−

→ ¯

Ω

x

x

converges to

x

¯

with

x ∈ Ω

x & ¯

x

x

converges to

x

¯

with

x > ¯

x

liminf lowerlimitfor real numbers

limsup upper limitfor real numbers

Liminf lower/inner limitfor multifunctions

Limsup upper/outer limitfor multifunctions

Oligopolistic market problem

x

i

_{∈ R}

productionof the

i

th leader

y

j

_{∈ R}

productionof the

j

thfollower

ω ⊂ R

n

the set of geometric constraints of leaders

T

overall productionquantity on the market

p

inverse demand function/marketprice

ϕ

i

objectivefunction of the

i

th leader

f

j

objectivefunction of the

j

the follower

c

i

cost function of the

i

thproducer Forward-spot market model

x

productionvector

s

spotsales vector

f

forward position vector

p

inverse demand function/spotprice

L

set of links

q

i

injection/withdrawalatnode

i

C

ij

transmissionlimiton the link

ij

φ

ij,k

contributionof injection/withdrawalatnode

k

to the link

ij

p

i

price atnode

i

Trac equilibriumproblem

G

transportationnetwork

N

set of nodes

A

set of arcs

W

set of OD pairs in

G

R

w

set of all pathsconnecting OD pair

w ∈ W

R

set of all routes

F

r

owon route

r ∈ R

v

a

owon arc

a ∈ A

∆

incidence matrix with elements

δ

ar

C

r

costs of using route

r ∈ R

D

w

tracdemand between OD pair

w ∈ W

µ

w

minimumtravel costs between OD pair

w ∈ W

y

a

capacity onarc

a ∈ A

α

the value of time

t

a

travel time onarc

a ∈ A

Introduction

Inpast century,the study of conictingsituation,acollisionofinterest,received a

consid-erablescienticinterest. Althoughsomegame-theoreticalresultscanbetracedtothe18th

century,the rst rigorousresultswere developed inthe 1920sbyBorel andvonNeumann.

The establishment of game theory asa scientic eld isusually related tothe publication

of [50] in1944. Since then, agreat variety ofscientic disciplines, likeeconomics, biology,

sociology and politics,becomeinterested instudy of conictingsituations.

An individual facing a decisiontakes intoaccount dierent outcomes. However, he or

she may not be the onlydecision-makingperson and the resultingoutcome oftendepends

onmulti-persondecision. Inthiscase,optimalityisnot awelldenedconcept andinstead,

we speak of equilibria.

There is a greatvariety of dierent equilibriumconcepts. Among the two widely used

belongs a solution to a noncooperative game, where, roughly speaking, each player can

not improvehis or her outcome by altering his orher decision unilaterally. This concept,

named Nash equilibrium concept, was introduced in the early 1950s in [34]. A dierent

situation arises when cooperation is present. We then speak of a Pareto optimal solution

when there isno other joint decisionsuchthat the performance of at least one playercan

be improved without degrading the performance of the others.

Probably the rst study of a hierarchical model of conicting situations is due to

Stackelberg[51]. Nowadays,aStackelberg (orsometimestermedalsosingle-leader-follower)

game is used to model an economic situation when on the market the dominant rm

(e.g., due to some temporaladvantage), called the market leader (or upper-level player),

maximizesitsprotsundertheassumptionthatallotherrmspresentonthemarket,called

followers (or lower-level players), play a noncooperative strategy. Mathematically, this

situation is modeled via bilevel optimization problems (namely when only one follower is

presentonthemarket) andmathematicalprogramswithequilibrium constraints (MPECs).

The MPEC class of optimization problems was introduced in 1970s motivated by other

applicationstomechanicsandnetworkdesign. InpastdecadeMPECsreceivedanextensive

interest of mathematicians. Followingthe progress in computationalpowerof computers,

Ourmaininterestinthisthesis,however, isfocusedontheconictingsituationsleading

to problems which in a sense liein between Nashand Stackelberg games, to the so-called

multi-leader-followergames. This situationoccurs,asthe namesuggests, whenmorethan

oneplayerisinadominantpositionandhencehastotakeintoaccountnotjustthereaction

of players on the lowerlevel but alsoof the remainingleaders.

Concerning the behavior of the leaders, one can again distinguish two situations: the

decisionmaking of the leaders formsa Nashequilibrium onthe upper level, orall leaders

cooperate in order to achieve an upper-level Pareto optimal strategy. To express

mathe-maticallytheformersituationone canusethenovelparadigmofequilibrium problems with

equilibrium constraints (EPECs). This class of hierarchical decision making models was

probablydirectly addressedfor the rsttime in[47]. Thelatter situationleads toa

dier-entclassofhierarchicalproblems,nowadayscalledmultiobjectiveproblemswithequilibrium

constraints (MOPECs).

The aim is, of course, to nd (local) solutions to the mentioned problems. For this

purpose, various stationarity concepts have been introduced. To verify that a given point

is stationary is in general easier then to check that it is a local solution. However, for a

local solution to be stationary, certain constraint qualication must hold true. One can

observe two approaches to the study of MPECs: to restrict the attention to problems

constrained by a nonlinear complementarity problemand to study the Lagrange function

andbehaviorofthecorrespondingmultipliers;ortoimposearatherstrictassumptionthat

the lower problemattains (locally)a unique solution. The latter restriction enables us to

apply successfully the so-called implicit programming approach.

In this thesis, weinvestigatestationarity concepts tailoredtoMPECs and EPECs and

theconnectionbetweenthevariousstationarityconcepts. Duetothestructuraldependence

of EPECs on MPECs, we naturally build upon known results about MPECs. We pay

the main attention to a subclass of MPECs constrained by a nonlinear complementarity

problem since this isthe case of currently known applicationsof EPECs.

Oneofthemainaimswastoconstructabridgebetweenstationarityconditionsresulting

from the above mentioned approaches. To this end we use many results from [45], [39]

and [36]. However, the structure of our considered problem is slightly dierent, hence

we decided to present most of the results with full proofs. This is done in Chapter 2.

The main attention is paid to the so-called Clarke stationarity and C-stationarity, both

based on application of Clarke generalized calculus. These two stationarity concepts are

of particular importanceto EPECs.

InChapter3wegivemathematicalformulationofEPEC.Interestingly,thestudyofthis

class ofproblemswasboostedbymodelingofconicting behaviorofagentsinderegulated

electricity markets; we devote a separate section to several source problems which are

currently of high scientic interest. Weaimto addressthe questionof existence of Clarke

and C-stationary points and alsoof solutions toEPECs in mixed strategies.

Chapter4isdevotedtoMOPECs. Wederivenecessaryoptimalityconditionsandusing

the novel subdierential calculus for set-valued mappings by Mordukhovich we establish

algorithms to nd solution to EPEC depend directly on techniques to solve MPECs

nu-merically, in some cases due to very strong assumptions imposed on the data of EPEC.

For this reason we attempt to derive an alternative algorithm based on the homotopy

methodtailoredspecicallytoaspecialsubclass ofEPECs. Finally,aneectivenumerical

technique tosolveMOPECs is developed.

Parts of the original work which could be found in this thesis have already appeared

in separate publications [8], [9] and [31] and working papers [10] and [11], some previous

results by the author have been completely reworked and generalized to t the structure

of this thesis or complemented with additional results. Other sources have been also

used throughout the thesis when appropriate or necessary. In each case, this is carefully

Mathematical Program with

Equilibrium Constraints (MPEC)

In thischapterweinvestigateMPECs and associatedrst ordernecessary optimality

con-ditions. In the center of focus of this chapter are stationarity concepts for MPECs with

equilibrium constraints in the form of a nonlinear complementarity problem. We discuss

the relationsbetween stationarity concepts, inparticular, of those based onClarke

gener-alizedcalculus. Also,wediscussthequalicationconditionswhichareessentialinderiving

necessary optimality conditions forMPECs of the considered structure.

2.1 Mathematical formulation

An MPEC is an optimizationproblemwith two sets of players; one leader trying tosolve

anupper-levelminimizationproblemandone ormorelower-levelplayers, followers, trying

toreachaparameterized(bytheupper-leveldecisionvariable)Nashequilibriumbysolving

a lower-levelequilibrium problemamongthemselves.

More precisely, this problemis dened as follows. Let

(x, y)

denote the multistrategy composed from the strategies

x ∈ R

l

1

of the leader and multistrategy

y ∈ R

ml

2

of

m

followers. Suppose that

ϕ : R

l

1

+ml

2

→ R

is the objective function of the leader and

κ ⊂ R

l

1

+ml

2

is a nonempty and closed set of constraints. For the feasible strategy

x

, let the set of solutionstothe lower-level equilibriumproblem,denoted by

S(x),

be closed. Denition 2.1. (solution to abstract MPEC)

An admissible multistrategy vector

(¯

x, ¯

y) ∈ R

l

1

+ml

2

is a solution to an abstract MPEC if

(¯

x, ¯

y)

is a solution to the following optimization problem

minimize

x,y

ϕ(x, y)

subject to

y ∈ S(x),

(x, y) ∈ κ.

Thesolutiontothelowerproblemrepresentsanequilibriumconditionand

S(x)

species the set of such equilibria. This is the reason for the term equilibrium constraints in

MPEC.

Note that the minimization inmathematical program(2.1) is considered inboth

vari-ables,

x

and

y

, and hence we implicitly assume the so-called optimistic (orweak) formu-lation of MPEC. By the term optimistic we mean that whenever the lower problem has

multiplesolutionsforagiven

x

,thelower-levelplayerschoose oneofthebest inthesense that it minimizesthe upper-level objective fora xed

x

. Wecan explicitlyexpress this in the reformulation of(2.1) to minimize

x

ϕ

o

_(x),

(2.2) where

ϕ

o

_{(x) :=}

inf

{ϕ(x, y) | y ∈ S(x), (x, y) ∈ κ}.

(2.3) In a similar way we can obtain a pessimistic (or strong) formulation, assuming that the

lower-levelplayerschoose oneofthe worst multistrategieswithrespecttotheupper-level

objectivewhenmultipleoptionsarepossible. Replacinginf bysup in(2.3)henceresults

in amin-max formulation of MPEC.

Observe that we can equivalently rewrite the constraints in(2.1) ina compact form

(x, y) ∈ κ ∩

Gph

S.

The set

κ ∩

Gph

S

is hence called the feasible region of MPEC (2.1).

Since the mathematical program (2.1) is generally nonconvex due to its hierarchical

structure, inorder to guarantee the existence of itssolutionwe need toimpose additional

restrictions onthe data.

Theorem 2.2. Let

ϕ

be lowersemicontinuous, Gph

S

be closed andthere exista constant

c ∈ R

such thatthe set

Ξ

c

= {(x, y) ∈ κ ∩

Gph

S | ϕ(x, y) ≤ c}

is nonempty and bounded. ThenMPEC (2.1) possesses a solution.

Proof. The existence of solution is due to the classical Bolzano-Weierstrass theorem. For

details,see [39, Proposition 1.1].

Let

κ = U × R

ml

2

_,

where

U

is a closed set of feasible strategies of the leader and let

V

1

_{, . . . , V}

m

_{⊂ R}

l

2

denote closed convex sets of admissible strategies of followers. Let

f

j

_{: R}

l

1

+ml

2

→ R, j = 1, . . . , m,

denote the individual objective of the

j

th follower and assume that for each

j = 1, . . . , m,

the objectives

f

j

are continuously dierentiable on an

open set containing

U × Ω

, where

Ω :=

X

m

j=1

V

j

. Finally,dene

F (x, y) :=







∇

y

1

f

1

(x, y)

. . .

∇

y

m

f

m

(x, y)







.

Then wecan replacethe equilibriumconstraint

y ∈ S(x)

in(2.1) by the equivalent gener-alized equation

0 ∈ F (x, y) + N(y; Ω).

(2.4)

Thusanadmissiblemultistrategyvector

(¯

x, ¯

y) ∈ R

l

1

+ml

2

isasolutiontoMPEC if

(¯

x, ¯

y)

is a solutiontothe following optimizationproblem

minimize

x,y

ϕ(x, y),

subjectto

0 ∈ F (x, y) + N(y; Ω),

x ∈ U.

(2.5)

This particularproblembelongs toabroadsubclass ofproblems of MPECs (2.1) withthe

solution map in the form

S(x) = {y ∈ R

ml

2

_{|0 ∈ f (x, y) + Q(x, y)}}

with function

f : R

l

1

+ml

2

_{→ R}

ml

2

and multifunction

Q : R

l

1

+ml

2

_{⇒ R}

ml

2

_.

Mathematical program minimize

x,y

ϕ(x, y)

subjectto

0 ∈ f (x, y) + Q(x, y),

(x, y) ∈ κ

(2.6)

covers optimizationproblems constrainedby classical variationalinequalitiesand

comple-mentarity problems. In this thesis we are particularly interested in the latter, i.e., the

subclass of MPECs given by the mathematicalprograms

minimize

x,y

ϕ(x, y)

subject to

0 ≤ F

1

_{(x, y) ⊥ F}

2

_{(x, y) ≥ 0,}

x ∈ U,

(2.7) with functions

F

1

_{, F}

2

_{: R}

l

1

+ml

2

_{→ R}

ml

2

continuously dierentiable onanopen set

contain-ing

U × R

ml

2

. Toemphasize thepresence of complementarityconstraints,wereferto(2.7)

as tothe mathematical program with complementarity constraints (MPCC).

Foradeeperinsighttothe analysisof MPECsand MPCCs,wereferthereaderstothe

monographs [25], [39] and [30].

Anotherclass ofhierarchicalproblems withoneupper-levelplayerare bilevelprograms.

Theseproblemsarecharacterizedbythelowerproblemintheformofoptimizationproblem

minimize

y

f (x, y)

subject to

y ∈ V (x)

(2.8)

with the solution map

S(x) = arg min

y∈V (x)

Note that bilevel programs constitute a subclass of MPECs in the sense of Denition

2.1. Thus just like in the case of an abstract MPEC, if the solution to the lower-level

problemisnotunique, theupper-levelobjective functionisnotwelldeterminedand hence

the problem is ill-possed. The optimisticreformulation is the usual way how to overcome

this ill-possedness.

On the other hand, MPEC (2.6) can be understood as the generalization of a bilevel

programonly when the lower problemis replaced by its necessary and sucient

optimal-ity conditions, either represented by the generalized equation, variational inequality or

Karush-Kuhn-Tucker(KKT)conditions inthe formofcomplementarityproblem,entering

the upperproblemasconstraints. Note, thatthis ispossibleif the problem(2.8) isconvex

andalsosomeconstraintqualication,e.g,Slaterconstraintqualication,issatised.

Oth-erwise, one can detect stationarypointswhich are not even feasible in the original bilevel

program.

A bilevel program is in turn a special case of a hierarchical mathematical program

whichpossessesmultiplelevelsofoptimization. Suchmultilevelmathematicalprogramsare

usefulinmodelingofhierarchicaldecisionmakingprocessesandoptimizationofengineering

designs, see [25, Chapter 1.2] and references therein.

Though on the rst glance it might look appealing, the equivalent reformulation of

(2.8) tothe form

z ∈ V (x),

f (x, z) ≤

inf

{f (x, y) | y ∈ V (x)}

(2.9)

does not ease the investigation of the bilevel problems. This is due to the fact that the

second constraint in (2.9) does not satisfy any constraint qualication. For more on this

subject, see early work [35], a recent paper [14] and the references therein. For other

relationsbetweenbilevelprogramsorMPECsandotherwell-knownoptimizationproblems,

solutionalgorithmsand applications, see [13] and the references therein.

2.2 Necessary optimality conditions via nonlinear

pro-gramming

Some MPECs can beconverted tothe following form

minimize

x

ϕ(x, y)

subjectto

y = S(x),

x ∈ U.

(2.10) Assumethat

ϕ : R

l

1

+ml

2

_{→ R}

and

S : R

l

1

_{→ R}

ml

2

arelocallyLipschitzcontinuousfunctions

and that

U ⊂ R

l

1

isa closed set. Then, if weset

h(x) := ϕ ◦ Φ(x)

with

Φ(x) :=



x

S(x)



,

the MPEC (2.10) turnsout tobe a nonlinear program(NLP) minimize

x

h(x)

subject to

x ∈ U,

(2.11) where

h : R

l

1

→ R

is locally Lipschitz continuous function. If

x

¯

is a local minimizer of (2.11), then one has

0 ∈ ∂h(¯

x) + N(¯

x; U).

(2.12)

Using the formula for upper approximation of limiting subdierential of composite

function, the necessary optimalityconditions for MPEC (2.10) are asfollows.

Theorem2.3. Let

(¯

x, ¯

y)

bealocalminimizerof (2.10). Thenthereexistvectors

(u

∗

_{, v}

∗

_{) ∈}

∂ϕ(x, y)

such that

0 ∈ u

∗

+ D

∗

S(¯

x)(v

∗

) + N(¯

x; U).

(2.13) Proof. Forproof see [38, Theorem 1.6].

Fromnowon,assume

ϕ

tobecontinuouslydierentiable. Thusthegeneralizedequation (2.13) attainsthe form

0 ∈ ∇

x

ϕ(¯

x, ¯

y) + D

∗

S(¯

x)(∇

y

ϕ(¯

x, ¯

y)) + N(¯

x; U).

(2.14)

In MPECs, the set

U

has frequently implicit structure and hence to obtain necessary conditions in terms of the original data of the problemone needs touse the chain rule to

compute upper approximation of

N(¯

x; U)

undersuitable constraint qualication.

In accordance with nonlinear programming, the generalized equation (2.14) denes a

natural stationary concept. However, in most cases we may not be able to compute the

coderivative

D

∗

_S(¯

_x)(∇

y

ϕ(¯

x, ¯

y))

exactly. Then we have to conne ourself with its upper approximation and thus weaker stationarity conditions.

For

S

locally single-valued around

x

¯

and locally Lipschitz, one such possible upper approximation can be

( ¯

∂S(¯

x)

>

_∇

y

ϕ(¯

x, ¯

y)

or even its upper approximation. Clearly, this leads to stillweaker stationarityconditions.

2.3 Mathematical program with complementarity

con-straints

Let ustake acloser lookat the mathematical program(2.7). Note that for a special case

when

F

2

_{(x, y) := y}

, MPCC attains the formof (2.5) with

Ω = R

ml

2

+

since

S(x) = {y ∈ R

ml

2

_{|0 ≤ F}

1

_{(x, y) ⊥ F}

2

_{(x, y) ≥ 0}}

(2.15)

= {y ∈ R

ml

2

_{|0 ∈ F}

1

_{(x, y) + N(F}

2

_{(x, y); R}

ml

2

+

)}.

(2.16)

There are also other ways how to express the solution map

S

which assigns

x ∈ R

the solutionset of the nonlinear complementarity problem (NCP)

nd

y

suchthat

0 ≤ F

1

_{(x, y) ⊥ F}

2

_{(x, y) ≥ 0,}

(2.17) e.g., via the so-calledPang NCP function

S(x) = {y ∈ R

ml

2

_{|0 = min{F}

1

i

(x, y), F

i

2

(x, y)}, i = 1, . . . , ml

2

},

(2.18) orusing the graphof normal cone mapping

S(x) =



y ∈ R

m

|0 ∈



F

2

_{(x, y)}

−F

1

_{(x, y)}



∈

Gph

N(·; R

m

+

)



.

(2.19)

Another possibilityisto workwith an enhancedversion of the solutionmap,

S

e

, inwhich

we introduce extra variable

ν = F

1

_{(x, y)}

and obtain

S

e

(x) =



(y, ν) ∈ R

m

× R

m

_{|0 ∈}

 F

1

(x, y) − ν

F

2

_{(x, y)}



+ N(y, ν; R

m

× R

m

+

)



.

(2.20) The multifunction

S

e

isrelated tothe the solutionmap

S

by the following relationship

S

e

(x) =



S(x)

F

1

_{(x, S(x))}



.

2.3.1 Stationarity conditions for MPCCs

We can look at the MPCC (2.7) as a special constrained mathematical program having

additionally to a general constraint set

U

also nitely many functional constraints of in-equality and equality types. From this perspective we can work with a whole class of

stationaryconcepts forMPCCswhicharecenteredaroundLagrangefunction. Forobvious

reasons these are sometimes called KKT-type stationarity concepts.

First, let usintroduce the sets of indices relatedto activities of constraintsin

comple-mentarityproblem (2.17)at

(¯

x, ¯

y)

I

+

(¯

x, ¯

y) = {i ∈ {1, . . . , ml

2

}|F

i

1

(¯

x, ¯

y) > 0, F

i

2

(¯

x, ¯

y) = 0},

L(¯

x, ¯

y) = {i ∈ {1, . . . , ml

2

}|F

i

1

(¯

x, ¯

y) = 0, F

i

2

(¯

x, ¯

y) > 0},

I

0

(¯

x, ¯

y) = {i ∈ {1, . . . , ml

2

}|F

i

1

(¯

x, ¯

y) = 0, F

i

2

(¯

x, ¯

y) = 0}.

If there is nodoubt about the reference point,we write only

I

+

_{, L}

and

I

0

. The index set

I

0

is usually called the index set of biactive inequality constraints. Forbrevity, we denote

a

+

_{= |I}

+

_(¯

_{x, ¯}

_y)|

and

a

0

_{= |I}

0

_(¯

_{x, ¯}

_y)|

.

Consider the following auxiliarynonlinear program

minimize

x,y

ϕ(x, y)

subject to

F

1

_{(x, y) ≥ 0, F}

2

_{(x, y) ≥ 0,}

x ∈ U,

(2.21)

which results from the MPCC (2.7) by ignoring the complementarity structure of

con-straints. The rst order optimality conditions of the NLP (2.21) are asfollows:

There existmultipliers

(λ

1

_{, λ}

2

₎

and a vector

ξ ∈ N(x; Ω)

such that

0 = ∇

x

ϕ(x, y) −

ml

2

X

i=1

λ

1

i

∇

x

F

i

1

(x, y) −

ml

2

X

i=1

λ

2

i

∇

x

F

i

2

(x, y) + ξ,

0 = ∇

y

ϕ(x, y) −

ml

2

X

i=1

λ

1

i

∇

y

F

i

1

(x, y) −

ml

2

X

i=1

λ

2

i

∇

y

F

i

2

(x, y),

0 ≤ F

1

(x, y)⊥λ

1

≥ 0,

0 ≤ F

2

(x, y)⊥λ

2

≥ 0,

ξ ∈ N(x; U).

(2.22) Set

G(x, y) = (F

1

_{(x, y))}

>

_F

2

_{(x, y)}

. Then similarlyto the conditions above, the rst order

optimality conditions of the MPCC (2.7) are given by:

There existmultipliers

(λ

1

_{, λ}

2

_{, λ}

G

₎

and avector

ξ ∈ N(x; Ω)

such that

0 = ∇

x

ϕ(x, y) −

ml

2

X

i=1

λ

1

i

∇

x

F

i

1

(x, y) −

ml

2

X

i=1

λ

2

i

∇

x

F

i

2

(x, y) − λ

G

_∇

x

G(x, y) + ξ,

0 = ∇

y

ϕ(x, y) −

ml

2

X

i=1

λ

1

i

∇

y

F

i

1

(x, y) −

ml

2

X

i=1

λ

2

i

∇

y

F

i

2

(x, y) − λ

G

_∇

y

G(x, y),

G(x, y) = 0

F

_L∪I

1

0

(x, y) = 0, F

_I

1

+

(x, y) > 0,

F

I

2

+

_∪I

0

(x, y) = 0,

F

_L

2

(x, y) > 0,

λ

1

I

+

= 0,

λ

1

_I

0

≥ 0,

λ

2

L

= 0, λ

2

I

0

≥ 0,

ξ ∈ N(x; U).

(2.23) Now, since

(∇G(x, y))

>

= F

1

(x, y)

>

∇F

2

(x, y) + F

2

(x, y)

>

∇F

1

(x, y),

letus rearrange conditions (2.23), setting

λ

F

L

1

= λ

1

L

+ λ

G

F

L

2

(x, y),

λ

F

2

I

+

= λ

2

_I

+

+ λ

G

F

_I

1

+

(x, y),

(2.24)

λ

F

I

+

1

_∪I

0

= λ

1

_I

+

_∪I

0

,

λ

F

2

L∪I

0

= λ

2

_L∪I

0

.

(2.25) Duetothenatureofindex sets

I

+

_{, L}

and

I

0

There existmultipliers

(λ

, λ

)

and a vector

ξ ∈ N(x; Ω)

suchthat

0 = ∇

x

ϕ(x, y) −

X

i∈L∪I

0

λ

F

1

i

∇

x

F

i

1

(x, y) −

X

i∈I

+

_∪I

0

λ

F

2

i

∇

x

F

i

2

(x, y) + ξ,

0 = ∇

y

ϕ(x, y) −

X

i∈L∪I

0

λ

F

i

1

∇

y

F

i

1

(x, y) −

X

i∈I

+

_∪I

0

λ

F

i

2

∇

y

F

i

2

(x, y),

λ

F

1

I

0

≥ 0, λ

F

2

I

0

≥ 0,

ξ ∈ N(x; U).

(2.26)

Following the terminology coined in [45], the conditions (2.26) are called strong

sta-tionarity conditions. The investigationof MPCCs gave rise toa wholeseries of stationary

concepts tailored to MPCCs. Their respective conditions dier only in requirements

im-posed on vectors

λ

F

1

I

0

and

λ

F

2

I

0

. In this respect, the weakest stationarity concept involves norestrictions on biactivemultipliers.

Denition 2.4. (weakly, C-, M- and strongly stationary point)

Let

(¯

x, ¯

y)

be feasible for the MPCC (2.7). Then we callthe point

(¯

x, ¯

y)

i) weakly stationary (or critical) if there existmultipliers

(λ

F

1

, λ

F

2

)

and a normal

ξ ∈

N(x; U)

such that the conditions

0 = ∇

x

ϕ(¯

x, ¯

y) −

X

i∈L∪I

0

λ

F

1

i

∇

x

F

i

1

(¯

x, ¯

y) −

X

i∈I

+

_∪I

0

λ

F

2

i

∇

x

F

i

2

(¯

x, ¯

y) + ξ,

0 = ∇

y

ϕ(¯

x, ¯

y) −

X

i∈L∪I

0

λ

F

1

i

∇

y

F

i

1

(¯

x, ¯

y) −

X

i∈I

+

_∪I

0

λ

F

2

i

∇

y

F

i

2

(¯

x, ¯

y),

(2.27) are satised.

ii) C-stationary if it is a weakly stationary point and, additionally,

λ

F

1

i

λ

F

2

i

≥ 0

for all

i ∈ I

0

.

iii) M-stationary if it is a weakly stationary point and, additionally, either

λ

F

1

i

> 0

and

λ

F

2

i

> 0

, or

λ

F

1

i

λ

F

2

i

= 0

for all

i ∈ I

0

.

iv) strongly stationary if it is a weakly stationary point and, additionally,

λ

F

1

I

0

≥ 0

,

λ

F

2

I

0

≥ 0

.

In theabovedenition,M and C standsforMordukhovichandClarke,respectively.

Note that if

I

0

_{= ∅}

, i.e., in the (lower-level) strict complementarity case, strong, M-,

C-and weak stationarity concepts coincide. Also, the restrictions imposed upon biactive

multipliersdirectly result in the following chain of implications

strongstationarity

⇒

M-stationarity

⇒

C-stationarity

⇒

weak stationarity

.

Clearly, Slater constraint qualication can never hold at any feasible point of (2.7).

quali-This phenomenon is closely related to the geometry of the complementarity structure of

constraints and results in the unbounded set of Lagrangian multipliers. This leaves the

conventional numerical optimization methods with a possibility of failure of convergence

toa solution.

In [45] one can nd suitable variants of both LICQ and MFCQ for MPCCs with

ge-ometric constraints given by nitely many functional constraints of the inequality and

equalitytypes. Then wesaythat the MPCC(2.7) satisestheMPEC linear independence

constraintqualication (MPEC-LICQ)andthe MPEC Mangasarian-Fromowitz constraint

qualication (MPEC-MFCQ) ata feasible point

(¯

x, ¯

y)

if the auxiliarynonlinear program minimize

x,y

ϕ(x, y)

subject to

F

1

L∪I

0

(x, y) = 0, F

_I

1

+

(x, y) ≥ 0,

F

I

2

+

_∪I

0

(x, y) = 0, F

_L

2

(x, y) ≥ 0,

x ∈ U

(2.28)

satisesLICQandMFCQ at

(¯

x, ¯

y),

respectively. ThefeasibleregionoftheNLP(2.28)isa subset of the feasible regionof the MPCC (2.7) locallyaround

(¯

x, ¯

y).

So, every minimizer of the MPCC is also a local minimizer of the corresponding NLP (2.28). This is the

reasonwhy thisprogramiscalledtightened nonlinearprogram (TNLP).Notethatthere is

awhole listof constraint qualicationstailoredspecically toMPCCs, withMPEC-LICQ

and MPEC-MFCQ amongthe strongest ones, cf. [18].

However, unlikein [45] or [18], we do not impose at this point any structural

require-mentsontheset

U

ofgeometricconstraints,thusweneedtoworkwithgeneralized versions of the respective constraintqualications.

Denition 2.5. (MPEC generalizedLICQ and MFCQ)

The MPCC (2.7) is saidto satisfy

i) the MPECgeneralizedLICQ(MPEC-GLICQ)atafeasiblepoint

(¯

x, ¯

y)

iftherelation

 (∇

x

F

_I

2

+

_∪I

0

(¯

x, ¯

y))

>

(∇

x

F

_L∪I

1

0

(¯

x, ¯

y))

>

(∇

y

F

I

2

+

_∪I

0

(¯

x, ¯

y))

>

(∇

y

F

_L∪I

1

0

(¯

x, ¯

y))

>

 

˜

u

˜

v



∈

 −N(¯

x; U)

0



(2.29) with

(˜

u, ˜

v) ∈ R

a

+

_+a

0

× R

ml

2

−a

+

implies

(˜

u, ˜

v) = 0.

ii) the MPEC generalized MFCQ (MPEC-GMFCQ) at a feasible point

(¯

x, ¯

y)

if the re-lation(2.29) with

(˜

u, ˜

v) ∈ R

a

+

_+a

0

× R

ml

2

−a

+

such thatfor each

i ∈ I

0

either

u

˜

i

˜

v

i

= 0

or

u

˜

i

< 0

and

v

˜

i

< 0,

implies

(˜

u, ˜

v) = 0.

Note that

0 ∈ N(¯

x; U),

hence(2.29) impliesin particular



∇F

_I

2

+

_∪I

0

(¯

x, ¯

y))

>

u + ∇F

˜

_L∪I

1

0

(¯

x, ¯

y))

>

v = 0, (˜

˜

u, ˜

v) ∈ R

a

+

_+a

0

× R

ml

2

−a

+



_{⇒ (˜}

_{u, ˜}

_{v) = 0.}

This is, however, true only if all the gradient vectors

∇F

1

i

(¯

x, ¯

y), ∇F

j

2

(¯

x, ¯

y), i ∈ I

+

∪

linear independence constraint qualication for MPCCs. Clearly, MPEC-GLICQ implies

MPEC-GMFCQ, since the latterrestricts the values of

(˜

u, ˜

v)

.

It turns out that MPEC-GMFCQ is just strong enough for M-stationarity conditions

tobenecessary optimality conditions. The following theorem isa modied version of [36,

Theorem 3.1] where the statement isproved forthe MPEC (2.5) with

Ω = R

ml

2

+

.

Theorem 2.6. Let

(¯

x, ¯

y)

be a local minimizer of the MPCC (2.7). If MPEC-GMFCQ holds at

(¯

x, ¯

y)

then there exist multipliers

λ

F

1

, λ

F

2

and

ξ ∈ N(¯

x; U)

such that (2.27) hold and either

λ

F

1

i

> 0

and

λ

F

2

i

> 0

, or

λ

F

1

i

λ

F

2

i

= 0

for all

i ∈ I

0

. In particular,

(¯

x, ¯

y)

is M-stationary.

Proof. WhenMPEC-GMFCQholdswecancomputeanupperapproximationofthenormal

cone tothe feasible region



(x, y) ∈ U × R

ml

2

_|



F

2

_{(x, y)}

−F

1

_{(x, y)}



∈

Gph

N(·; R

ml

2

+

)



.

Recallingtherstordernecessaryoptimalityconditionsfornonlinearprograms,

(¯

x, ¯

y)

thus satises conditions

0 ∈∇ϕ(¯

x, ¯

y)+

+

 (∇

x

F

2

_(¯

_{x, ¯}

_y)

>

_−(∇

x

F

1

(¯

x, ¯

y)

>

(∇

y

F

2

(¯

x, ¯

y)

>

−(∇

y

F

1

(¯

x, ¯

y)

>



N(F

2

(¯

x, ¯

y), −F

1

(¯

x, ¯

y);

Gph

N(·, R

ml

2

+

))

+ N(¯

x; U) × {0}.

Takeintoaccount that

N(F

2

(¯

x, ¯

y), −F

1

(¯

x, ¯

y);

Gph

N(·, R

ml

2

+

)) =

ml

2

X

i=1

N(F

2

i

(¯

x, ¯

y), −F

i

1

(¯

x, ¯

y);

Gph

N(·, R

+

))

and that

N(F

_i

2

(¯

x, ¯

y), −F

_i

1

(¯

x, ¯

y);

Gph

N(·, R

+

)) =











{0} × R,

i ∈ L,

R

_{× {0},}

_{i ∈ I}

+

_,

({0} × R) ∪ (R × {0}) ∪ (R

−

× R

+

), i ∈ I

0

.

Now, consider arbitrary

(u, v) ∈ N(F

2

_(¯

_{x, ¯}

_{y), −F}

1

_(¯

_{x, ¯}

_y);

Gph

N(·, R

ml

2

+

))

and set

λ

F

1

:= v

and

λ

F

2

:= −u

. Then we arriveexactly at M-stationarity conditions. This completes the proof.

The M-stationarityconditionsare clearlythe propercounterpartofMordukhovich

sta-tionarity known from nonlinear programming, hence the choice for the name of the

sta-tionarity concept.

To provedirectly thatunderMPEC-GLICQlocalminimizersof (2.7)are C-stationary,

one just needs to properly modify [45, Lemma 1], although this statement follows from

Theorem2.7. Let

(¯

x, ¯

y)

bea localminimizerof theMPCC(2.7). If MPEC-GLICQholds at

(¯

x, ¯

y)

then there exist multipliers

λ

F

1

, λ

F

2

and

ξ ∈ N(¯

x; U)

such that conditions (2.27) hold and

λ

F

1

i

λ

F

2

i

≥ 0

for all

i ∈ I

0

. In particular,

(¯

x, ¯

y)

is C-stationary. Proof. Letus rewritethe MPCC (2.7) as

minimize

ϕ(x, y)

subject to

0 = min{F

1

i

(x, y), F

2

i

(x, y)}, i = 1, . . . , ml

2

,

x ∈ U.

From [30, Theorem 5.19 (ii)]and [29, Theorem 3.36] we get the followingversion of Fritz

John conditions. There exist multipliers

r ≥ 0, λ

min

i

, i = 1, . . . , ml

2

,

not all zero, and

ξ ∈ N(¯

x; U)

such that

0 = r∇

x

ϕ(¯

x, ¯

y) +

ml

2

X

i=1

λ

min

i

c

i

+ ξ,

0 = r∇

y

ϕ(¯

x, ¯

y) +

ml

2

X

i=1

λ

min

i

d

i

,

(2.30) with

(c

i

, d

i

) ∈ ¯

∂ min{F

i

1

(x, y), F

i

2

(x, y)} =











∇F

1

i

(¯

x, ¯

y),

i ∈ L,

conv

{∇F

1

i

(¯

x, ¯

y), ∇F

i

2

(¯

x, ¯

y)}, i ∈ I

0

,

∇F

2

i

(¯

x, ¯

y),

i ∈ I

+

.

For every

i ∈ I

0

there is

α

i

∈ [0, 1]

such that

c

i

= α

i

∇

x

F

1

(¯

x, ¯

y) + (1 − α

i

)∇

x

F

2

(¯

x, ¯

y),

d

i

= α

i

∇

y

F

1

(¯

x, ¯

y) + (1 − α

i

)∇

y

F

2

(¯

x, ¯

y).

Set

λ

F

i

1

=











−λ

min

i

,

i ∈ L,

−α

i

λ

min

i

,

i ∈ I

0

,

0,

i ∈ I

+

_,

λ

F

2

i

=











0,

i ∈ L,

(1 − α

i

)λ

min

i

, i ∈ I

0

,

−λ

min

i

,

i ∈ I

+

.

Then, since

α

i

∈ [0, 1],

we have

λ

F

1

i

λ

F

2

i

= α

i

(1 − α

i

)(λ

min

i

)

2

≥ 0

for each

i ∈ I

0

This resultsinthefollowingconditionswhichdierfromC-stationarityconditionsonly

in the presence of a nonnegativemultiplier

r

.

0 = r∇

x

ϕ(¯

x, ¯

y) −

X

i∈L∪I

0

λ

F

i

1

∇

x

F

i

1

(¯

x, ¯

y) −

X

i∈I

+

_∪I

0

λ

F

i

2

∇

x

F

i

2

(¯

x, ¯

y) + ξ,

0 = r∇

y

ϕ(¯

x, ¯

y) −

X

i∈L∪I

0

λ

F

1

i

∇

y

F

i

1

(¯

x, ¯

y) −

X

i∈I

+

_∪I

0

λ

F

2

i

∇

y

F

i

2

(¯

x, ¯

y),

λ

F

1

i

λ

F

2

i

≥ 0, i ∈ I

0

_,

ξ ∈ N(¯

x; U).

(2.31)

Assume now, that

r = 0

. Then the rst two lines of (2.31) may be writtenas

−ξ = −

X

i∈L∪I

0

λ

F

i

1

∇

x

F

i

1

(¯

x, ¯

y) −

X

i∈I

+

_∪I

0

λ

F

i

2

∇

x

F

i

2

(¯

x, ¯

y),

0 = −

X

i∈L∪I

0

λ

F

1

i

∇

y

F

i

1

(¯

x, ¯

y) −

X

i∈I

+

_∪I

0

λ

F

2

i

∇

y

F

i

2

(¯

x, ¯

y).

Setting

u = −λ

˜

F

2

I

+

_∪I

0

and

v = −λ

˜

F

1

L∪I

0

, from MPEC-GLICQ we get

λ

F

2

I

+

_∪I

0

= λ

F

1

L∪I

0

= 0.

This impliesalso

λ

min

i

= 0

forall

i = 1, . . . , ml

2

.

Thelatter is,ofcourse, acontradiction to the statement that multipliers

r ≥ 0, λ

min

i

, i = 1, . . . , ml

2

,

are not allsimultaneously zero. Hence,

r 6= 0

and scaling yields

r = 1

. This completes the proof.

It turns out that toprovethe above statement directly,MPEC-GMFCQ isinsucient

to prevent the case of vanishing multiplier

r

. Nevertheless, recall that M-stationarity implies C-stationarity, hence MPEC-GMFCQ implies C-stationarity of local minimizers.

This is the statement of the following corollary.

Corollary 2.8. Let

(¯

x, ¯

y)

be local minimizer of MPEC (2.7). If MPEC-GMFCQ holds at

(¯

x, ¯

y)

then there exist multipliers

λ

F

1

, λ

F

2

and

ξ ∈ N(¯

x; U)

such that (2.27) hold and

λ

F

1

i

λ

F

2

i

≥ 0

for all

i ∈ I

0

. In particular,

(¯

x, ¯

y)

is C-stationary.

NotealsothatMPEC-GLICQdoesnotprovideuniqueness ofmultipliersif

U 6= ∅

. The followingexampleshows thatMPEC-GLICQcanbesatisedandyettheremaybeatleast

two dierentsets of multiplierssatisfyingC-stationarity conditions.

Example 2.9. Consider an MPCC minimize

x

1

,x

2

,y

2x

1

+ 2x

2

+ y

subject to

0 ≤ x

1

− x

2

− y ⊥ y ≥ 0,

x

1

, x

2

≥ 0.

atthe feasible point

(¯

x

1

, ¯

x

2

, ¯

y) = (0, 0, 0).

Then conditions

u ≥ 0,

−u ≥ 0,

−u + v = 0,

imply

u = v = 0

andhence MPEC-GLICQholds. On the other handone can easilycheck that there are multiple sets of vectors

(λ

F

1

, λ

F

2

, ξ

1

, ξ

2

)

with

(ξ

1

, ξ

2

) ∈ R

2

−

satisfying the conditions (2.27), e.g., (1,2,-1,-3)or (2,3,0, -4).

Clearly, our reference point is even strongly stationary. In fact, itis the unique global

minimizerof our MPCC.

4

2.3.2 Implicit programming approach and Clarke stationarity

In this section we consider an alternative approach to MPECs. We are particularly

in-terested in various criteria under which the lower-level complementarity problem locally

denes an implicit function. Most of the results in this section follow directly from [39],

although, for slightly dierent structure of an MPCC. Using the combination of the

cal-culus of Mordukhovich and of Clarke, however, we derive stronger optimality conditions

then in[39]. Only when webelieve itis appropriate,we present the the fullproof.

Consider the generalized equation(2.4) with the solution map

S(x) = {y ∈ R

ml

2

_{|0 ∈ F (x, y) + N(y; Ω)}.}

In what follows we work with the following condition of Robinson [43] concerning the

multivalued map

Σ : R

ml

2

_{⇒ R}

ml

2

generated by partial linearizationof

F (¯

x, ¯

y)

in(2.4). Denition 2.10. (Strong regularity condition)

Let

y ∈ S(¯

¯

x)

. Suppose that there exist neighborhoods

V

of

y

¯

and

O

of

0 ∈ R

ml

2

such that

the map

ξ → Σ(ξ) ∩ V

is single-valued and Lipschitz continuous on

O

, where

Σ(ξ) = {y ∈ R

ml

2

_{|ξ ∈ F (¯}

_{x, ¯}

_{y) + ∇}

y

F (¯

x, ¯

y)(y − ¯

y) + N(y; Ω)}.

Thenwe say thatthe generalizedequation(2.4)is strongly regularat

(¯

x, ¯

y)

or thatat this point the generalized equation(2.4) satises the strongregularitycondition (SRC).

Thestrongregularityconditionplaysanimportantroleinimplicitprogrammingmainly

due to the following result.

Theorem 2.11. Let the generalized equation (2.4) be strongly regular

(¯

x, ¯

y)

. Then there is a neighborhood

U

of

x

¯

and

V

of

y

¯

such that the map

σ(x) = S(x) ∩ V

is single-valued and locally Lipschitz continuous on

U.

Proof. Forproof see [43].

For

Ω

being a convex polyhedral set we get a useful characterization of the strong regularity condition.

Theorem 2.12. Let

Ω

be a convex polyhedron. Thenthe followingstatements are equiva-lent.

ii) The generalized equation

ξ ∈ ∇

y

F (¯

x, ¯

y)η + N(η; K(¯

y − F (¯

x, ¯

y), ¯

y))

(2.32) is single-valued on

R

ml

2

.

Proof. See, e.g., [39, Theorem 5.3].

We can apply Theorem 2.12 also to to the underlying generalized equation in (2.20).

This enables us to derive rather simple linear algebraic criteria for single-valuedness and

Lipschitz behaviorofthe map

σ

around

x

¯

. Notethat the thirdargument

ν

¯

ofthe general-ized equation in(2.20) isuniquely determined by

x

¯

and

y

¯

via relation

ν = F

¯

1

_(¯

_{x, ¯}

_y)

. Thus

we can referjust to the point

(¯

x, ¯

y)

.

IfSRCholdsat

(¯

x, ¯

y)

,thenthereexistneighborhoods

U

of

x

¯

and

V

of

y

¯

andaLipschitz continuous map

σ : U → R

m

_{× R}

m

such that

σ(¯

x) = (¯

y, F

1

(¯

x, ¯

y))

and

σ(x) = S

e

(x) ∩ (V × F

1

(x, V))

for all

x ∈ U.

The map

σ

can be split into two Lipschitz operators

σ

y

and

σ

ν

which correspond, locallyaround

x

¯

,tothe

y−

and

ν−

componentofthesolutiontotheunderlyinggeneralized equation in(2.20). Moreover, itsuces toanalyze just the operator

σ

y

since

σ

ν

(x) = F

1

(x, σ

y

(x))

for all

x ∈ U.

The criterion of SRC for the generalized equation in (2.20) is stated in the following

theorem. Theorem 2.13. Denote by

Z(x, y)

an

(ml

2

+ a

+

_{+ a}

0

_{) × (ml}

2

+ a

+

+ a

0

)

matrix given by

Z(x, y) =





∇

y

F

1

(x, y)

−E

_I

>

+

−E

_I

>

0

∇

y

F

_I

2

+

(x, y)

0

0

∇

y

F

_I

2

0

(x, y)

0

0





.

Then the generalized equation in (2.20) is strongly regular at

(¯

x, ¯

y)

if and only if the generalized equation

ξ ∈ Z(¯

x, ¯

y)η + N(η; R

ml

2

+a

+

_{× R}

a

0

+

)

possesses a unique solution

η

for all

ξ ∈ R

ml

2

+a

+

+a

0

.

Proof. In this case, the generalized equation (2.32) attainsthe form

ξ ∈

 ∇

y

F

1

_(¯

_{x, ¯}

_{y) −E}

∇

y

F

2

(¯

x, ¯

y)

0



η + N(η; K)

with

K = {(u, v) ∈ R

ml

2

_{× R}

ml

2

_|v

L∪I

0

≥ 0} ∩ {(u, v) ∈ R

ml

2

× R

ml

2

|v

_L

= 0}

= {(u, v) ∈ R

ml

2

_{× R}

ml

2

_|v

I

0

≥ 0, v

_L

= 0}.