A Hybrid Newton Method for Solving Box Constrained Variational

A HYBRID NEWTON METHOD FOR SOLVING

BOX CONSTRAINED VARIATIONAL INEQUALITY PROBLEMS

VIA THE DGAP FUNCTION

JiMing Peng Christian Kanzow and Masao Fukushima

State Key Lab oratory of Scientic and Engineering Computing

Institute of Computational Mathematics and ScienticEngineering Computing

Academic Sinica

PO Box Beijing China

email p jmlsecccaccn

Institute of Applied Mathematics

University of Hamburg

Bundesstrasse

D Hamburg Germany

email kanzowmathunihamburgde

Department of Applied Mathematics and Physics

Graduate School of Engineering

Kyoto University

Kyoto Japan

email fukukuampkyotouacjp

December

Abstract A b ox constrained variational inequality problem can b e reformulated as an

unconstrained minimization problem through the Dgap function A hybrid Newtontype

metho d is prop osed for minimizing the Dgap function Under suitable conditions the

algorithm is shown to b e globally convergent and lo cally quadratically convergent Some

numerical results are also presented

Key words Variational inequality problem b ox constraints Dgap function Newtons

metho d unconstrained optimization global convergence quadratic convergence

The research of this author was supp orted by Pro ject of NSFC in China

Current address Octob er September Computer Sciences Department University

of Wisconsin Madison West Dayton Street Madison WI email kanzowcswiscedu The

research of this author was supp orted by the DFG Deutsche Forschungsgemeinschaft

The work of this author was supp orted in part by the Scientic Research GrantinAid from the Ministry

of Education Science and Culture Japan

Introduction

n n

Let F b e a mapping from into itself and X b e a nonempty closed convex subset of

The variational inequality problem VIP is to nd a vector x X such that

hF x y x i y X

where h i denotes the inner pro duct in In this pap er we study the case that X is a

b ox dened by

X fx j l x u i ng

i i i

where l fg and u fg with l u represent the lower and upp er

i i i i

b ounds on the variables resp ectively It is not diult to see Prop osition that the b ox

structure of the set X enables us to rewrite as

F x y x i n y X

i i

If the constraint set X is the nonnegative orthant then the VIP reduces to the com

plementarity problem CP This class of sp ecial VIPs has numerous imp ortant applications

in various elds such as mathematical programming economics and engineering see

and the references therein

A useful way to deal with VIP is to reformulate it rst as a system of equations or an

optimization problem via a merit function and then solve the resultant system of equations

or optimization problem Recently much attention has b een paid on the reformulation of

VIPs and CPs and various merit functions have b een prop osed and studied Well known

merit functions for VIPs include the gap function

g x sup hF x x y i

y 2X

rst presented by Auslender and then studied by Marcotte and Dussault the

regularized gap function

hF x x y i f x max ky xk

y 2X

introduced by Fukushima and Auchmuty and the Dgap function

g x f x f x

prop osed by Peng and Yamashita Taji and Fukushima where and are arbitrary

p ositive parameters such that Since a b ox constrained VIP is actually equivalent to

a system of KKT mixed complementarity conditions several merit functions based on the

KKT system of VIP have also b een prop osed and explored extensively see Qi and the

references therein

In this pap er we fo cus our attention to the Dgap function for VIP It is not dicult to

see that g x for all x and g x if and only if x is a solution of the VIP

Therefore the VIP can b e cast as the following unconstrained optimization problem

min g x

x2<

When the mapping F is dierentiable the Dgap function g is also dierentiable

However it is not twice dierentiable in general Therefore it is not straightforward to apply

conventional secondorder metho ds to problem As a remedy for this inconvenience Sun

Fukushima and Qi introduced the concept of a computable generalized Hessian of the D

gap function g and presented a Newtontype metho d for solving problem Restricting

themselves to the b ox constrained VIP Kanzow and Fukushima discussed a generalized

Hessian of the Dgap function and prop osed a GaussNewtontype metho d to minimize it

Further prop erties of the Dgap function have b een investigated in

This work is motivated by the recent pap er in which the Dgap function is used to

globalize the classical JosephyNewton metho d for general VIPs The main purp ose of

this work is to further study the algorithm prop osed in by restricting ourselves to b ox

constrained VIPs and test the eectiveness of the algorithm The pap er is organized as

follows In Section we review some basic results that will b e used in the pap er In Section

we present the algorithm and study its convergence prop erties Some numerical results

are presented in Section Finally we conclude the pap er with some remarks in Section

Preliminaries

We rst review some concepts related to the VIP and state some prop erties of the Dgap

function dened by

n n

The mapping F is said to b e a P function if

max x y F x F y x y x y

i i i i

1in

x 6=y

i i

a P function if

max x y F x F y x y x y

i i i i

and a uniform P function with modulus if

max x y F x F y jjx y jj x y

i i i i

An n n matrix M is a P matrix if

max z M z z z

i i

1in

z 6=0

and a P matrix if

max z M z z z

i i

It is easy to see that if M is a P matrix then there exists a constant such that

max z M z kz k z

i i

It is known Theorem that if F is a dierentiable P function then rF x is a

P matrix for each x Moreover if F is a dierentiable uniform P function with mo dulus

then rF x is a uniform P matrix with mo dulus in the sense that

T n n

max z rF x z kz k z x

i i

Since we are not aware of any explicit reference for the formula we include a short pro of

for it

n n

Lemma Let F be a dierentiable uniform P function with modulus

Then holds

n n

Pro of Let x and z b e arbitrary but xed Let ft g b e a sequence

of p ositive numbers converging to Since F is a uniform P function and the index set

f ng is nite there exists an index i f ng indep endent of k and a subsequence

ft g such that z and

k K i

t z F x t z F x t kz k k K

k i i k i

0 0 0

Dividing this expression by t taking the limit k k K and using the assumed

dierentiability of F we obtain

z rF x z kz k

i i

0 0

This implies

max z rF x z kz k

i i

and completes the pro of 2

The following lemma will play an imp ortant role in the analyses of this pap er Let X b e a

n n

b ox M b e a P matrix and b For any v let xv X denote the unique solution

of the following ane VIP

hb v M x y xi y X

or equivalently

b v M x y x i n y X

i i i i i

Lemma Let xv be the unique solution of the ane VIP where X is a box and M

is a P matrix Let be a constant satisfying Then we have

0 0 0 n

kxv xv k kv v k v v

0 n 0

Pro of Let v v b e arbitrary Since xv X xv X and X is a b ox it follows

from that

b v M xv x v x v i n

i i i i i

and

0 0 0

b v M xv x v x v i n

i i i i

Adding the ab ove two inequalities we obtain

0 0 0 0

v v x v x v x v x v M xv xv i n

i i i i i i

Since M is a P matrix there exists a constant and some index i such that

0 0 0

x v x v M xv xv kxv xv k

i i i

0 0 0

This inequality together with implies that

0 0 0

v v x v x v kxv xv k

i i i

i 0 0 0

Since

0 0 0 0

v v x v x v kv v k kxv xv k

i i i

0 0 0

the inequality yields

0 0

kxv xv k kv v k

This completes the pro of of the lemma 2

Next we give another result on the continuity of a solution of the b ox constrained ane VIP

with a P matrix For any P matrix M and vectors b p let xM X denote the

unique solution of the following ane VIP

hb M p M x y xi y X

or equivalently

b M p M x y x i n y X

i i i i i

Lemma Suppose that M and N are P matrices and X is a box Let be a constant

satisfying Then we have

kM N k kxN pk kxM xN k

Pro of The lemma trivially holds if xM xN Hence we only need to consider the

case where xM xN Since X is a b ox it follows from and the denitions of xM

and xN that

b M p M xM x N x M i n

i i i i i

and

b N p N xN x M x N i n

i i i i i

Adding the ab ove two inequalities we get

M p N p M xM N xN x N x M i n

i i i i i i

which implies that

M N xN p x N x M M xM xN x M x N i n

i i i i i i

Since M is a P matrix it follows from and xM xN that there exists an index

i such that

N kxM xN k M x x M xM xN

i i i

0 0 0

Then and imply

kxM xN k M N xN p x N x M

i i i

0 0 0

kM N k kxN pk kxN xM k

This completes the pro of 2

Now let x b e a given p oint in and consider the linearized variational inequality problem

of nding a p oint z X such that

hF x rF x z x y z i y X

or equivalently

F x rF x z x y z i n y X

i i i i

If rF x is a P matrix then problem has a unique solution which we denote z x

The following lemma is a renement of Prop osition in

Lemma Suppose that F is a continuously dierentiable uniform P function and X is

a box Then the solution z x of the ane VIP is continuous as a function of x

Moreover x is a solution of the VIP if and only if x z x

Pro of First note that since F is a uniform P function is satised with some constant

0 n 0

indep endent of x For two arbitrary p oints x x let z x and z x b e the unique

solutions of the linearized VIPs at x and x resp ectively Also let z denote the unique

solution of the ane VIP

0 T

hF x rF x z x y z i y X

0 T 0 0 0 T 0

It then follows from Lemma with v F x rF x x v F x rF x x b

0 T

and M rF x that

0 0 0 T 0

kz z x k kF x F x rF x x xk

On the other hand by Lemma with b F x p x N rF x and M

0 T

rF x we have

0 T T

krF x rF x k kz x xk kz z xk

It then follows that

kz x z x k

kz z x k kz z xk

0 0 T 0 0 T T

kF x F x rF x x xk krF x rF x k kz x xk

Consequently for any xed x we obtain

lim kz x z x k

x !x

This proves the rst half of the lemma

To prove the second half supp ose rst that z x x Then it follows immediately from

that x solves Conversely supp ose that x is a solution of Since X is a b ox and

z x X yields

F xz x x i n

i i i

Similarly from x X and we have

F xx z x rF x z x x x z x i n

i i i i i i

The inequalities and give

z x x rF x z x x i n

i i i

Since rF x is a P matrix it follows from that there exists an index i such that

z x x rF x z x x kz x xk

i i i

0 0 0

Combining with we get z x x 2

A Hybrid Newton Metho d

In this section we consider the hybrid Newton metho d prop osed in for solving the VIP

with general convex constraints Our aim is to rene the convergence results obtained

in by restricting ourselves to the sp ecial case where the VIP is b ox constrained

The algorithm is stated as follows

Algorithm

Step Cho ose x and suciently small

Let k

k k

Step If g x or krg x k stop

Step Find z X such that

k k T k k k

hF x rF x z x x z i x X

k k k

and let d z x If

k k k

g x d g x

then let and go to Step If the linearized VIP is not solvable or if d do es not

satisfy the condition

k k k k

hrg x d i maxfkrg x k kd k g

k k

then set d rg x

Step Find the smallest nonnegative integer m satisfying

k m k k m k k

k k

g x d g x hrg x d i

and let

k k k

Step Set x x d and k k Go to Step

Global convergence of the algorithm has b een established in For completeness we quote

the global convergence theorem without pro of

Theorem Suppose that the mapping F is continuously dierentiable Let and

suppose that the algorithm generates an innite sequence fx g Then any accumulation

point x of the sequence fx g is a stationary point of the Dgap function g

If F is a uniform P function then by Theorem the level sets of Dgap function

g are b ounded Since fg x g is nonincreasing the b oundedness of level sets guarantees

the b oundedness of the generated sequence fx g and hence the existence of at least one

accumulation p oint of fx g On the other hand by Theorem any stationary p oint

x of g such that rF x is a P matrix is a solution of the VIP Therefore if F is

a uniform P function then it follows from Theorem that any accumulation p oint of

the generated sequence fx g solves the VIP Because the b ox constrained VIP with a

uniform P function has a unique solution we obtain the next corollary to Theorem

Corollary Suppose that F is a continuously dierentiable uniform P function and X

n k

is a box Then for any starting point x the sequence fx g generated by the algorithm

converges to the unique solution of the VIP

Now we turn our attention to the convergence rate of the algorithm A solution x of the

VIP is said to b e regular in the sense of Robinson see also if there exist a

neighborho o d of x and a neighborho o d V of such that for every v V the

p erturb ed VIP of nding a vector x X such that

hF x v y xi y X

where F x v F x v rF x x x has a unique solution xv that is

Lipschitz continuous as a function of v ie

0 0 0

kxv xv k kv v k v v V

for some

Regularity conditions have b een widely used in the study of variational inequality prob

lems particularly in the analysis of lo cal convergence prop erties of iterative metho ds for

VIPs In a strong but simple sucient condition for a solution x of the VIP to b e

regular was given Here we give a dierent condition p ertaining to the b ox constrained VIP

Lemma Assume that x is a solution of the VIP with X being a box If the matrix

rF x is a P matrix then x is a regular solution

Pro of The theorem follows from Lemma Alternatively the statement can b e directly

derived from a characterization of strong regularity given in Theorem 2

To study the convergence rate of the algorithm we need the following results concerning

an error b ound prop erty of the Dgap function We denote by y x the unique maximizer

on the righthand side in the dening equation of the regularized gap function f Note

that y x x F x where denotes the pro jection op erator on X We dene

X X

R x x y x Moreover y x and R x are dened similarly Let B denote the

closed sphere centered at x with radius ie

B fx j kx x k g

Following the pro of of Lemma in we have the next lemma

Lemma Let x be a solution of the VIP with X being a box Suppose that F is

a uniform P function with modulus Suppose also that F is Lipschitz continuous with

constant on B for some Then there exists a constant such that

kx x k kR xk x B

where

The next lemma shows that the Dgap function provides a lo cal error b ound for the VIP

under suitable assumptions This result will b e useful in establishing the quadratic

convergence of the prop osed algorithm

Lemma Let x be a solution of the VIP with X being a box Suppose that F is

a uniform P function with modulus Suppose also that F is Lipschitz continuous with

constant on B for some Then there exist constants c c such that

c kx x k g x c kx x k x B

Pro of By Prop osition we have

kR xk g x kR xk x

It follows from Lemma and the left part of the ab ove inequality that

kx x k x g x

where which shows that the left inequality in is true

Next observe that

kR xk kx y x x y x k

kx x k k x F x x F x k

X X

kx x k kx F x x F x k

kx x k

for all x B where the equality follows from the denition of R and the fact that

R x the rst inequality follows from the triangle inequality and the denition of y

the second inequality follows from the nonexpansiveness of the pro jection op erator and

the last inequality follows from the Lipschitz continuity of F The right inequality in

then follows from the right inequality of The pro of is complete 2

We are ready to prove quadratic convergence of the prop osed algorithm

Theorem Suppose that F is continuously dierentiable and X is a box Let x be an

accumulation point of the sequence fx g generated by the algorithm If F is a uniform P

function and rF is locally Lipschitzian then x is a solution of the VIP and the sequence

fx g converges quadratically to x

Pro of By Theorem x is a stationary p oint of the Dgap function g Since rF x

is a P matrix by Lemma it follows from Theorem in that x is already a solution

of the VIP Moreover Lemma shows that x is a regular solution Since F is

dierentiable it is lo cally Lipschitzian hence there exists a such that F is is Lipschitz

continuous on B Hence by Lemma we have

c kx x k g x c kx x k x B

for some c c Moreover by choosing a smaller if necessary we may assume that rF

is Lipschitz continuous on B Then by the basic result on JosephyNewtons metho d

for the VIP there exists a such that for any initial p oint chosen from B

the Newton iteration is welldened and

kz x x k c kx x k x B

holds for some constant c Let min Then it follows from and

that

g z x c c kx x k x B

Let

min

c c

Then it follows from that for any x B

g z x c kx x k g x

where the rst inequality follows from the choice of and the second inequality follows

k k k k

from the left inequality in This implies that when x B we have d z x x

k k

and the step size is accepted ie x z x Consequently it follows from

that the sequence fx g converges to x quadratically 2

Similarly we may prove sup erlinear convergence of the algorithm under slightly weaker as

sumptions The pro of is omitted

Theorem Suppose that F is continuously dierentiable and X is a box Let x be an

accumulation point of the sequence fx g generated by the algorithm If F is a uniform P

function then x is a solution of the VIP and the sequence fx g converges superlinearly

to x

Numerical Results

We implemented the hybrid Newtontype metho d suggested in this pap er in MATLAB and

run it on a SUN SPARC station We rst give a brief description of the implementation

Let

r x x Pro j x F x

l u

denote the natural residual of the b ox constrained variational inequality problem We ter

minate our metho d if

k k

kr x k or g x

for some iterate x where

and

In addition the iteration was stopp ed if

k k

max

with

max

For the Dgap function g we used the parameters

and

In the line search rule we used

and

However we replaced the standard monotone Armijorule by a nonmonotone variant see

Gripp o Lampariello and Lucidi for details

As a solver for the linearized variational inequality problems we used the semismo oth

Newtontype metho d from In contrast to what is said in the description of our algorithm

however we always accept the corresp onding search direction d whenever it satises the

descent test

k T k

rg x d

note that this guarantees that the Armijo line search is welldened In particular we

accept this search direction d even if we were not able to solve the corresp onding linearized

variational inequality problem In this way we try to overcome the problem that we have

to take to o many gradient steps in a row which is obviously not very desirable

In order to improve the eciency of our algorithm however we also used a prepro cessor

more precisely we rst try to solve our test examples by using the recently prop osed metho d

from Kanzow and Fukushima This is a nonsmo oth Newtontype metho d applied to the

residual equation

r x

and globalized by the Dgap function g see for details The motivation for doing this

is quite simple The metho d from works extremely well whenever it solves a problem

successfully Unfortunately it do es not seem to b e very robust unless relatively strong

assumptions are satised

So we rst apply the nonsmo oth Newtontype metho d from in order to solve a test

example but we stop this prepro cessing iteration if either the termination criterion

is satised or if a certain test indicates that the prepro cessor runs into diculties In the

latter case we switch to the hybrid Newton metho d introduced in this pap er which is not

as ecient as the metho d from but which seems to b e considerably more reliable

Basically our criterion for switching from the prepro cessor to the hybrid Newton metho d

is as follows If

k k

t t or krg x k cg x

k min

then terminate the prepro cessing iteration and go to the hybrid Newton metho d using the

previous iterate x as a starting p oint The actual parameters used in are

t and c

min

If the prepro cessor is successful and converges to a solution of the b ox constrained variational

inequality problem which satises the standard regularity conditions used in for the lo cal

k k

convergence theory then t for all k suciently large and g x O krg x k

so none of the tests in will b e satised

We applied the metho d just describ ed to all test problems from the MCPLIB and GAM

SLIB libraries see using all the dierent starting p oints which are available within the

MATLAB environment

We rep ort the numerical results in Table for the MCPLIB test problems and in Table

for the GAMSLIB test problems The columns in these tables have the following meanings

problem name of the test problem in MCPLIB

n number of variables

m number of nite b ounds on the variables x

SP starting p oint

Psteps number of iterations used in the prepro cessing phase

Nsteps number of Newton steps used in the hybrid Newton phase

Gsteps number of gradient steps used in the hybrid Newton phase

F eval number of function evaluations

f f

g x value of g x at the nal iterate x x

f f

kr x k value of kr xk at the nal iterate x x

Lo oking at Tables and we see that we have just a few failures on some dicult test

problems whereas the overall b ehaviour of our metho d is quite go o d Although many of

the simple problems were solved by the prepro cessor ie there are no N and no Gsteps

the hybrid Newtontype metho d introduced in this pap er was necessary in order to solve a

number of other test examples

In fact we made the following observation during the testing phase for our algorithm

Both the prepro cessor from and the hybrid Newtontype metho d discussed in this pap er

try to minimize the Dgap function g Now the Dgap function might have a lo cal mini

mum which do es not corresp ond to a solution of the b ox constrained variational inequality

problem In that case we would exp ect b oth algorithms to run into diculties by converging

to one of these lo cal minima in particular since the search directions computed by b oth

metho ds are based on some lo cal information of the variational inequality problem In fact

this diculty arises eg for the billups example In general however our observation is

that the metho d from tends to converge to a lo cal minimum of g much more often

than the metho d discussed here This seems to indicate that from a global p oint of view

the search direction computed by our hybrid Newtontype metho d is a much b etter search

direction than the one computed by the nonsmo oth Newtontype metho d in

It is therefore our feeling that the robustness of many existing solvers can b e improved

by using the search direction from our hybrid Newtontype metho d whenever the underlying

solver do es not seem to converge

Concluding Remarks

The variational inequality problem is reformulated as an unconstrained minimization prob

lem by using the Dgap function g A hybrid Newtontype metho d is then prop osed to

minimize the function g Under mild conditions the prop osed metho d is shown to b e glob

ally convergent If some additional assumptions are satised then the sequence converges

quadratically or sup erlinearly to a solution of the original variational inequality problem A

sucient condition is given for a solution x of the VIP to b e regular This condition is only

concerned with the mapping F unlike the conditions in

Table Numerical results for MCPLIB test problems

f f

problem n m SP Psteps Nsteps Gsteps F eval g x kr x k

b ertsekas e e

b ert o c e e

billups

bratu e e

choi e e

colvdual

colvnlp e e

cycle e e

ehl k e e

ehl kost e e

explcp

freeb ert e e

gafni e e

hanskoop e e

hydroc e e

jel e e

josephy e e

Table continued Numerical results for MCPLIB test problems

f f

problem n m SP Psteps Nsteps Gsteps F eval g x kr x k

ko jshin e e

ko jshin

ko jshin e e

mathinum e e

mathisum e e

methan e e

nash e e

obstacle e e

opt cont e e

pgvon e e

pgvon

pies e e

p owell e e

p owell mcp e e

scarfanum e e

scarfasum e e

scarfbnum

Table continued Numerical results for MCPLIB test problems

f f

problem n m SP Psteps Nsteps Gsteps F eval g x kr x k

scarfbsum e e

spp e e e

tobin e e

Table Numerical results for GAMSLIB test problems

f f

problem n m SP Psteps Nsteps Gsteps F eval g x kr x k

cafemge e e

cammcp e e

cirimge e e

comge e e

dmcmge e e

ersmcp e e

etamge e e

hansmcp e e

hansmge e e

harkmcp e e

harmge e e

kehomge e e

kormcp e e

mrmcp e e

nsmge e e

oligomcp e e

scarfmcp e e

scarfmge e e

transmcp e e

twomcp e e

unstmge e e

vonthmcp

vonthmge e e

wallmcp e e

Acknowledgments The rst author would like to thank Prof Y Yuan for his constant

help and encouragement

References

G Auchmuty Variational principles for variational inequalities Numerical Functional

Analysis and Application

A Auslender Optimisation Methodes Numeriques Masson Paris France

DP Bertsekas and JN Tsitsiklis Parallel and Distributed Computation Numerical

Methods PrenticeHall Englewoo d Clis NJ

SP Dirkse and MC Ferris MCPLIB A collection of nonlinear mixed complementar

ity problems Optimization Methods and Software

F Facchinei A Fischer and C Kanzow Regularity prop erties of a semismo oth refor

mulation of variational inequalities SIAM Journal on Optimization to app ear

F Facchinei A Fischer and C Kanzow A semismo oth Newton metho d for variational

inequalities The case of b ox constraints Complementarity and Variational Problems

State of the Art MC Ferris and JS Pang eds SIAM Philadelphia PA

MC Ferris and JS Pang Engineering and economic applications of complementarity

problems SIAM Review to app ear

MC Ferris and TF Rutherford Accessing realistic mixed complementarity prob

lems within MATLAB Nonlinear Optimization and Applications G Di Pillo and F

Giannessi eds Plenum Press New York NY

M Fukushima Equivalent dierentiable optimization problems and descent metho ds

for asymmetric variational inequality problems Mathematical Programming

M Fukushima Merit functions for variational inequality and complementarity prob

lems Nonlinear Optimization and Applications G Di Pillo and F Giannessi eds

Plenum Press New York NY

M Fukushima and JS Pang Minimizing and stationary sequences of merit functions

for complementarity problems and variational inequalities Complementarity and Vari

ational Problems State of the Art MC Ferris and JS Pang eds SIAM Philadel

phia PA

L Gripp o F Lampariello and S Lucidi A nonmonotone linesearch technique for

Newtons metho d SIAM Journal on Numerical Analysis

PT Harker and JS Pang Finite dimensional variational inequality and nonlinear

complementarity problems A survey of theory algorithms and applications Mathe

matical Programming

NH Josephy Newtons metho d for generalized equations Technical Rep ort No

Mathematics Research Center University of Wisconsin Madison WI

C Kanzow and M Fukushima Theoretical and numerical investigation of the Dgap

function for b ox constrained variational inequalities Mathematical Programming to

app ear

C Kanzow and M Fukushima Solving b ox constrained variational inequality prob

lems by using the natural residual with Dgap function globalization Mathematical

Programming Technical Rep ort Computer Sciences Department University of

Wisconsin Madison Madison WI November

JM Liu Strong stability in variational inequalities SIAM Journal on Control and

Optimization

P Marcotte A new algorithm for solving variational inequalities with application to

the trac assignment problem Mathematical Programming

P Marcotte and JP Dussault A note on a globally convergent Newton metho d for

solving monotone variational inequalities Operations Research Letters

JJ More and WC Rheinboldt On P and S functions and related classes of n

dimensional nonlinear mappings Linear Algebra and Its Applications

JM Peng Equivalence of variational inequality problems to unconstrained minimiza

tion Mathematical Programming

JM Peng and M Fukushima A hybrid Newton metho d for solving the variational

inequality problem via the Dgap function Preprint Institute of Computational Math

ematics and ScienticEngineering Computing Chinese Academy of Science Beijing

China May

L Qi Regular pseudosmo oth NCP and BVIP functions and globally and quadratically

convergent generalized Newton metho ds for complementarity and variational inequality

problems Applied Mathematics Rep ort School of Mathematics The University

of New South Wales Sydney Australia July revised September

SM Robinson Strongly regular generalized equations Mathematics of Operations

Research

D Sun M Fukushima and L Qi A computable generalized Hessian of the Dgap func

tion and Newtontype metho ds for variational inequality problems Complementarity

and Variational Problems State of the Art MC Ferris and JS Pang eds SIAM

Philadelphia PA

K Taji M Fukushima and T Ibaraki A globally convergent Newton metho d for solv

ing strongly monotone variational inequalities Mathematical Programming

N Yamashita K Taji and M Fukushima Unconstrained optimization reformulations

of variational inequality problems Journal of Optimization Theory and Applications