a positive algorithm for the nonlinear complementarity...

SIAM J. OPTIMIZATIONVol. 5, No. 1, pp. 129-148, February 1995

1995 Society for Industrial and Applied Mathematics007

A POSITIVE ALGORITHM FOR THE NONLINEARCOMPLEMENTARITY PROBLEM*

RENATO D. C. MONTEIROt, JONG-SHI PANGS:, AND TAO WANG$

Abstract. In this paper, the authors describe and establish the convergence of a new iterativemethod for solving the (nonmonotone) nonlinear complementarity problem (NCP). The method uti-lizes ideas from two distinct approaches for solving this problem and combines them into one unifiedframework. One of these is the infeasible-interior-point approach that computes an approximatesolution to the NCP by staying in the interior of the nonnegative orthant; the other approach istypified by the NE/SQP method which is based on a generalized Gauss-Newton scheme applied toa constrained nonsmooth-equations formulation of the complementarity problem. The new method,called a positive algorithm for the NCP, generates a sequence of positive vectors by solving a sequenceof linear equations (as in a typical interior-point method) whose solutions (if nonzero) provide descentdirections for a certain merit function that is derived from the NE/SQP iteration function modifiedfor use in an interior-point context.

Key words, complementarity problems, interior-point methods, nonsmooth equations

AMS subject classifications. 90C30, 90C33, 49M37

1. Introduction. The idea of solving complementarity problems by staying inthe interior of the feasible region can be traced to a paper published in 1980 byMcLinden [18]. Although no explicit algorithm was formulated, the idea of tracing aninterior path as a possible solution procedure was quite evident in this paper and theexistence of the "central path" was demonstrated in the case of a complementarityproblem with a maximal monotone multifunction. Unfortunately, this paper was notwidely known. Of course, McLinden’s idea is central to many of today’s interior-pointmethods for solving a wide variety of mathematical programming problems.

In recent years, interior-point methods for solving complementarity problems havebeen the subject of many studies [4], [6]-[14], [19], [20], [26], [28], [31]-[35]. Amongthese, the monograph [9] presents a unified treatment of the original family of (feasible)interior-point methods for the linear complementarity problem (LCP) that requires alliterates to be strictly feasible; this volume also contains an extensive list of referencesfor the interior-point methods up to the year 1990.

A proposal by Lustig [16] and the subsequent computational study [17] have ledresearchers to investigate the family of infeasible interior-point methods. The mainfeature of these methods for solving a complementarity problem is that the iteratesare positive vectors, albeit not necessarily feasible to the problem, and have somedesirable limiting properties. There are many papers dealing with these methods forsolving linear programs; for the linear complementarity problem, we mention [31] and[34]. Most recently, the paper by Kojima, Noma, and Yoshise [15] presents a wide class

Received by the editors December 31, 1992; accepted for publication (in revised form) November15, 1993.

School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, Georgia30332 (monteiro(C)+/-sye.gatech.edu). The work of this author was based on research supported byNational Science Foundation grant DDM-9109404 and Office of Naval Research grant N00014-93-1-0234. This work w- done while this author was an assistant professor in the Systems and IndustrialEngineering Department of the University of Arizona.

Department of Mathematical Sciences, The Johns Hopkins University, Baltimore, Maryland21218 ([email protected]) and (wang_tbrutus.mts.jhu.edu). The work of these authors wasbased on research supported by National Science Foundation grants DDM-9104078 and CCR-9213739and Office of Naval Research grant N00014-93-1-0228.

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130 .R.D. C. MONTEIRO, J.-S. PANG, AND T. WANG

of infeasible interior point methods for solving a monotone nonlinear complementarityproblem (NCP).

Inclusive of the early work of McLinden, all interior-point methods for solvingcomplementarity problems to date have invariably relied on a certain monotonicityassumption (or more generally, a P0-property) see [8], [9], [15] and [i, 5.9]. This facthelps to explain why the interior-point methods proposed so far for solving nonlinearprograms are restricted to the class of linearly constrained convex programs [21]-[23].In essence, such a monotonicity assumption (or P0-property) is needed to ensure thenonsingularity of a key matrix that is used to define the main computational stepof the methods. The major objective of this paper is to propose an interior-pointmethod for solving a general, nonmonotone NCP. In a subsequent paper, we studythe specialization of this method to the Karush-Kuhn-Tucker optimality conditionsof a general nonlinear program formulated as a mixed NCP.

Our proposed method is an infeasible interior-point potential reduction algorithm;it involves two major ideas. One is to maintain the positivity of the iterates while acertain merit function is being decreased; this joint task is accomplished by means ofthe modified Armijo technique described in [22]. The other idea is to make use of theiteration function in the recent NE/SQP method [2], [27] to define a suitable meritfunction. The resulting algorithm does not rely on any monotonicity assumption ofthe problem. Instead, a key condition, called s-regularity, plays an important role inthe convergence analysis.

This interior-point method, which we call a positive algorithm for the NCP to sig-nify that the iterates are positive vectors but not necessarily feasible to the problem,consists of solving a sequence of linear equations each defined by a symmetric positivedefinite matrix; the unique solutions of these equations, if nonzero, provide descentdirections for a special logarithmic merit function, which is a combination of theNE/SQP iteration function and the positivity conditions. The NE/SQP method alsomaintains the nonnegativity of the iterates; but it does so by imposing this require-ment as constraints in the direction-finding subproblems that are then either (convex)quadratic [27] or linear programs [2]. Consequently, the positive algorithm may beconsidered as providing an interior-point approach to alleviate the direction-findingtask in the NE/SQP method.

2. Preliminaries. Given a function f" Rn Rn, which is assumed to be con-tinuously differentiable in an open set containing R the nonlinear complementarity+problem, denoted NCP (f), is to find a vector x Rn such that

x>_O, f(x) >_O, xTf(x)-O.The reader is referred to [5] for a comprehensive review of the theory and applicationsof this problem. In [27], the NE/SQP method was proposed as a solution procedurefor this problem. Clearly, the NCP (f) is equivalent to the following nonnegativelyconstrained system of nonsmooth equations:

(1) H(x) :: min(x, f(x)) O, x >_ O,

where min denotes the componentwise minimum of two vectors. The NE/SQP methodgenerates a sequence of nonnegative iterates by successively solving a sequence ofnonnegatively constrained least-squares subproblems whose solutions provide descentdirections for the merit function

O(x) H(x)TH(x).

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A POSITIVE ALGORITHM FOR THE NCP 131

Exploiting the idea of staying in the positive orthant, we describe an iterative al-gorithm for solving the NCP (f) in which each direction-finding step requires thesolution of a single system of linear equations.

It would be useful to summarize the key properties of the norm function 0. Clearly,0 is nonnegative; its zeros are precisely the solutions of the NCP (f). The function 0 isgenerally not Frchet differentiable at an arbitrary vector, but it has a strong Frchetderivative at all its zeros [25, Prop. 1]. Moreover, 0 is directionally differentiableeverywhere with the directional derivative at a vector x along the direction d givenby [24]

O’(x,d)= E xidi+ E ximin(di,Vfi(x)Td)+ E fi(x)Vfi(x)Td"

Motivated by this expression, we define three fundamental index sets associated withan arbitrary vector z E Rn"

{i: < {i: {i: >For notational convenience, we let Jl(z) Ix(z) U Ie(z). We note immediately thatif x Rn is nonnegative, then for all vectors d Rn,

O’(x, d) <_ E xidi + E fi(x)Vfi(x)Td"iJi(x) ielf(x

This inequality is the key to the descent step in the algorithm to be described later.To write the inequality in a more compact form, we define the n x n matrix A(x)whose ith column is given by

e ifieJf(x),A(x)i-

Vf(x) if/

where e is the ith coordinate vector. In terms of this matrix, the above inequalitybecomes

O’(x, d) <_ H(x)TA(x)Td.It is important to note that the directional derivative O(x, d) is generally not a

continuous function in x for a fixed but arbitrary d and neither is the matrix-valuedfunction A(x). However, the next result gives an important generalization of theinequality (2). A variant of this result can be found in [27, Lem. 4]. For the sake ofcompleteness, we give a simpler and more direct proof of this result here.

PROPOSITION 2.1 Let x* Rn be arbitrary. Then for any sequence {xk} c++R+ converging to x* and any sequence {hk} C_ Rn\{O) converging to O, there holds

(3) limsupO(xk W hk) O(xk) H(xk)TA(xk)Thk < O.

k-oo [[hk[[

Proof. For each i 1,..., n, let

1 1 2(xk Hi(xk)[A(xk)i]ThkAHi(xk, hk) =-- -H2i (xk + hk) -H1 hk 1Afi(xk,hk) =_ -f(xk + )-- -f(Xk) fi(xk)Vfi(xk)Thk,1 1

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132 R.D. C. MONTEIRO, J.-S. PANG, AND T. WANG

We claim that for all k sufficiently large,

(4) AHi(xt, ht) < max(Afi(xk, ht:)

Indeed, there are three cases to consider: whether i e I(x*), e Ix(x*), or E Ie(x*).Assume first that i If(x*), that is fi(x*) < x. Using the fact that both sequences{x} and {xk + ha} converge to x* and a simple continuity argument, we obtain

k kk and fi(xk + ha) ( x -F h for all k sufficiently large. Using the definitionf (x <of the functions g(.) and g(.), we then obtain

AHi(xa, hk) Afi(xa, ha);

hence (4) holds. For the case in which e Ix(x*), that is fi (x*) > x, we can similarlyshow that

AHi(xk, hk) Axi(xk, ha);

so (4) also holds. Consider now the case in which i e I(x*), that is fi(x*) x > O.k kThen, for all k sufficiently large, we have fi(xa + ha) > 0 and xi + h > 0. This

implies that

H(xa + hk) min{f(xk + ha), (xk + hk)2}.

Using this relation and considering whether e If(xa) or i e J(xa), we can easilyverify that (4) holds. We can now complete the proof of (3). Indeed, using (4) weobtain that

Since

limsupAHi(xk’ ha) < 0 Vi 1,...,n.

k- Ilhall

n

O(xk + ha)- O(Xk) H(xk)TA(xk)Thk Z AHi(xk’hk)’i---1

relation (3) follows. El

3. Some important functions. The merit function to be used in the algorithmis defined as follows. Let c > 0 and > n be given scalars. Define

and let

{x e R 0}++ O(x) >

n

c(X) c log O(x) -F log(0(x) -}- eTx) log xii--1

Vx E

where e is the vector of all ones. The scalar c is a penalty parameter that will bechanged in the algorithm if it is deemed to be too small; unlike c, is fixed throughout.The third term in the function c is the logarithmic barrier function to prevent theiterates from reaching the boundary of the nonnegative orthant. The middle term isused to balance the third term; this is analogous to the potential function introduced in

[30] for linear programs that have been used extensively in many primal-dual interior-point methods; see also [15] for a related merit function.

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Clearly, we have

log O(x) + ( n)log(0(x) + eTx) Vx e

n

c(x)

_(c + if) log O(x) E log xi

i--1

VxE.

These inequalities have two important implications that we summarize in the resultbelow.

PROPOSITION 3.1. For a fixed c > O, the following statements hold:(a) if {xk } C_ f and limk-o c(xk) --oc, then limk_.o O(xk) 0;(b) for any a > 0 and t E R, there exist constants a > 0 and b > 0 such that

n[xER++,c(x)<_t, 0(x)>_a] a <_ xi <_ b Vi l,...,n.

The function c is directionally differentiable everywhere with the directionalderivative at the vector x along the direction d Rn given by

cO’(xd)+

n

(O’(x, d) +O(x) + ex i=l xi

Recalling the inequality (2), we define the forcing function

z(x,d) H(x)TA(x)Td +n

(A(x)H(x) + e)Td- E d-AO(X) J- eTx

i=l Xi

whose role in the algorithm will become obvious momentarily. Clearly, by (2), wehave

X(5) c( d) <_ Zc(X, d)

for all x R_+ with O(x) > 0 and all d R. Consequently, given such a vectorx, if we can find a vector d such that zc(x, d) < 0, then d is a descent direction forthe function at x. To generate such a direction, we consider the system of linearequations

(6) (A(x)A(x)T + X-2)d + w(x) O,

where

wc(x) (x) A(x)H(x) + (A(x)H(x) + e) x-1(x) + ex

with X being the diagonal matrix with xi’s on the diagonal and x-1 being the vectorwhose ith component is equal to 1/xi. The system of linear equations (6) is equivalentto the least-squares problem

minimize (llA(x)Tdll2 + IIX-dll) + zc(x,d),

where the minimization is over all vectors d Rn with the vector x fixed.

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134 R.D.C. MONTEIRO, J.-S. PANG, AND T. WANG

Noting that the matrix

(7) M(x) A(x)A(x)T + Z-2

is symmetric positive definite, we let dx be the unique solution of (6). Then we have

zc(x, dx) -(IIA(x)TdxlI22 + IIX-ldll) <_ 0;

moreover, zc(x, dx) < 0 if and only if wc(x) O. Consequently, if wc(x) 0, thendx is a descent direction for the function at the point x. Nevertheless, if Wc(X) isequal to zero, then d 0 and we need to generate an alternate direction. In thiscase, we double the penalty constant c (actually, any scaling exceeding 1 suffices) andsolve another system of the form (6) with the modified c.

With a (nonzero) descent direction dx successfully computed, we then perform aline search starting at x with the objective of decreasing the merit function c by asufficient amount while preserving the positivity of the next iterate. Details of such aline search step can be found in [22]. The main procedure is then repeated with thenew iterate replacing the old one if termination with regard to a prescribed rule stillhas not occurred.

We close this section with an immediate consequence of Proposition 2.1.PROPOSITION 3 2 Let x* E be arbitrary. Then for any sequence {xk} c Rn

converging to x*, any sequence {dk} with {(Zk)-ldk} bounded, and any sequence{/k } of positive scalars converging to zero, there holds

(8) limsupc(Xk + Akdk) -Pc(x)- AkZc(xk’d) < O.

k-* "kk k k k/Xki)Proof. Since {(Xk)-ldk} is bounded, {/kk} -- 0 and x +Akd x (l+)kd

it follows that xk + kdk is positive for all k sufficiently large. Using the fact that

log(1 + s) s + o(s), where lim o(s___) O,8--0 8

we obtain that for each 1,..., n,

limlg(xk + )kdki) --lgxi

limlog(1 + ,kkdk/xk) )kdk/Xk

Using this limit, Proposition 2.1 and the definition of c(’), we can now easily derive(8). 0

4. The positive algorithm. Summarizing the ideas outlined in the previoussection, we now present the details of the long awaited algorithm for solving the NCP(f), where f is an arbitrary continuously differentiable function.

Step O. (Initialization) Let > n, 6 > 0, and a, a, p E (0, 1) be given constants.Choose a scalar co > 0 and a vector x > 0 arbitrarily. Set k 0.

Step 1. (Direction generation) Compute wk wc (xk). If liTk II <-- 6, set

Ck+l 2Ck and Xk+l Xk.

Replace k by k + 1 and return to the beginning of this step. If Ilwkll > 6, let dk bethe unique solution to the system of linear equations:

(9) M(xk)d + wk O.

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Step 2. (Armijo line search) Determine the maximum stepsize

T --sUp{T" Xk -I-Tdk_

0}

and let

o’r if T <k > 1/lldkll if Tk

Let mk be the smallest nonnegative integer m such that

(10)

Set

c (xk + kpmdk) c (xk) < akpmzc (x, d).

C+I=Ck and xk+l xk + -kp"d.Step 3. (Termination check) If x+1 fails a termination check (for example, if

O(xk+l) > e for a prescribed tolerance e > 0), return to step 1 with k replaced byk+l.

We make several remarks about the above algorithm. First, the use of the scalar5 to guard against a zero vector wc (xk) is an extension of the discussion in the lastsection. As we shall see from the convergence analysis in 5, it is not enough to justcheck whether the vector Wc (xk) is zero or not; we actually need to ensure that thisvector is not too small in norm for the direction dk to be useful. Second, the maximumstepsize T nd the scalar a (0, 1) together will ensure that the next iterate xk+

remains a positive vector. Finally, a standard argument in an Armijo line search andthe inequality (5) will ensure that the integer mk cn be determined in a finite numberof trils; this proof is omitted. Moreover, in ce m 1, we must have

(11) c (xk + kPm-dk) c (xk) akPm- (xk, dk)Zc

by the definition of mk.It would be useful to compare the positive algorithm with the framework proposed

in [15] for solving the NCP (f). For this purpose, we consider this problem beingdefined by the following conditions:

u- 0,

(13) x_O,

(14) y >_ 0,

(15) xTy--O.

The positive algorithm generates a sequence of iterates {x }, which induces a corre-sponding sequence {yk} via the relation y f(xk) for all k. Hence, the combinedsequence {(xk,yk)} satisfies the conditions (12) and (13) but not necessarily (14) or

(15); in fact, the latter two conditions are the goal of the positive algorithm. Onthe other hand, the methods described in [15] generate a sequence {(xk, yk)} thatsatisfies (13) and (14) but not necessarily (12) or (!5), which is the goal of these otherinfeasible interior-point methods for solving the NCP (f).

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5. Convergence analysis. In this section, we analyze the limiting propertiesof an infinite sequence (xk} generated by the positive algorithm. By the infinitenature of this sequence, we have (xk) > 0 for all k. We divide the analysis into twocases, depending on whether the penalty constant Ck is updated infinitely often oronly finitely many times. We first take up the latter case.

Finite update of c. The next result analyzes the case in which Ilwkll > 5 for allindices k sufficiently large.

THEOREM 5.1. Suppose that the penalty sequence {Ck} is updated finitely manytimes in the positive algorithm. Then,

(16) lim 0(xk) 0.

Consequently, every accumulation point of {xk}, if it exists, must be a solution of theNCP (f).

Proof. Since {ck} is updated finitely many times, there exists an index k0 > 0and a constant c > 0 such that ck c for all k >_ k0. Assume by contradiction that(16) does not hold. Then there exist a constant > 0 and a subsequencesuch that K c_ {ko, ko + 1,...} and

(17) inf O(xk) >_ .k6/C

Inequality (10) and the fact that zc(x,d) < 0, for all k > ko, imply thatk > k0} is decreasing. Moreover, (17) and Proposition 3.1(a) imply that {c(xk 6 K:} is bounded below. Hence, this sequence converges and, by (10), we have

(8) lim kpmz(xk, d) O.

We next show that

infzc (xk’ dk)l > O.(19)

Indeed, we know that {x}keC C {x R (x) < t 0(x) > }, where t =_ (xk)++By Proposition 3.1(b), {xk}ketC is bounded and for all k ],

>a, i=1,.., n(20)

for some constant a > 0. Using this fact, we can easily show that

(21) IIM(xk)ll IIM(xk)-lll < T Vk e 1C

for some constant T > 0. Hence, using the fact that IIw(xa)ll > 5, we obtain that forall k 6 K:,

]Zc(Xk, dk)l IWc(Xk)Tdk IIW(xk)TM(xk)-W(Xk)IIIldkll Ildkll IIM(x)-xw(x)ll

IlM(xk)ll-lllw(xk)ll 2 II(xk)ll

Hence, (19) holds. Combining (18) and (19), we obtain

(22) lim ttdkll o,

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We next show that

(23) inf ’k dk > O.

Indeed, if k E E is such that T o, then

-klldkll >_ 1

by the definition of k. On the other hand, if Tk < cx, then using (20) and the factthat xk + Tdk has some component equal to zero, we can easily deduce that

(24) Tlldkll >_ a.

Consequently, (23) holds since k aT for all k _> 0. Combining (22) and (23), weobtain limk(ec)--, pm O. Hence, mk

_1 for all k E K: sufficiently large. Since

{xk}kiE is bounded, we may take K:’ c K: such that limk(ec,)_o x x* Rn++

and mk _> 1 for all k KT. By (11), we obtain

zc(xk,dk)

or equivalently,

(25) c(xk + AkP) c(Xk) )kWc(Xt)Tpk > --(1 a) Zc(Xt’ dk) > (1 a)6,kt [[dk[[ T

where Ak =-- -Pm-lldll and pk =_ d/lldll" Note that relation (22) implies that

limk(e:)--,o Ak 0. Also, it is easy to verify that {(Xk)-lpk} is bounded. Hence, byProposition 3.2, we know that the limsup of the left-hand side of (25) is nonpositiveand this violates (25). We have thus obtained a contradiction and therefore (16) musthold.

We point out that other choices for the matrix M(x) used in the computation ofthe search direction are possible. In addition to the symmetry and positive definitenessof M(x), all that is required is that the condition number cond(M(x)) of the matrixM(x), defined as

cond(M(x))- IIM(x)ll IIM(x)-ll,

be uniformly bounded on any compact subset of R_+; cf. (21). The above convergenceproof of Theorem 5.1 (and thus the limit (16)) remains valid under this condition.Note that this condition is much weaker than the requirement that cond(M(x)) beuniformly bounded on the whole Rn Our choice of M(x) in (7) satisfies the first++.condition but not the latter one.

It is important to point out that Theorem 5.1 is established under absolutely noassumption on the function f other than its continuous differentiability. Note alsothat the theorem does not require the boundedness of the sequence {xk}; indeed,as the following example shows, this sequence may be unbounded if no restriction isimposed on the function f.

Example. Let

(x) e-, x e R.

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For x > 0 sufficiently large, it is easy to obtain the functions H and we as follows:

S(x) e-x and

we(x) -2c+ 1-e-xx_.e_2x X

Note that lim_, w(x) -2c; hence, provided that 2c > 5, we must have Iwc(x)[ >5 for x > 0 sufficiently large. Consequently, corresponding to such an x, the searchdirection at x is

-v(x)dx e_2X + x-2 > > 0

e-2x -t- x-2

Thus, if we initiate the positive algorithm with x sufficiently large and the constantco > 5/2, the algorithm will generate an increasing sequence {xk} with Ck remainingconstant. This sequence cannot be bounded for otherwise its limit point would bea strictly positive solution of the NCP (f); but the only solution to the NCP (f) is

Boundedness of iterates. The above example is not surprising because generally,if the NCP (f) has no solution, then although the limiting value of the sequence{0(xk)} is zero, the sequence of iterates {xk} must be unbounded. Consequently,some conditions on f must be needed for the latter sequence to be bounded. Thefollowing discussion pertains to this boundedness issue of {xk }.

We recall some properties of a vector-valued mapping. A mapping F Rn -- Rnis norm-coercive on a set X C Rn if

lim IIF(x) II-

F is coercive on X in the Hadamard sense if

lim IIx * F(x)

where a, b denotes the Hadamard product of two vectors a, b E Rn, i.e., (a, b)i aibifor all i. It is easy to see that a mapping F is norm-coercive on X if and only if forall t _> 0, the level set

(x e x IIF(x)ll

_t}

is bounded; moreover, coercivity in the Hadamard sense implies norm-coercivity. Ex-amples of mappings that are coercive on Rn in the Hadamard sense include the/

strongly copositive mappings and the uniform P-functions. The former are thosemappings F for which there exists a constant "1 > 0 such that

max xiFi(x) > for all x e Rn(i(n "

and the latter are those mappings F for which there exists a constant "2 > 0 suchthat

max (xi- yi)(Fi(x)- Fi(y)) > 211x- yl12 for all x, y e R.l<i<n

Given a mapping F Rn --, R, a principal subfunction of F is defined as follows.For an arbitrary index set ( C_ { 1,..., n} with cardinality k and complement and an

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arbitrary vector a Rn-k, the function G" Rk Rk defined by G(x) Fa(x, a)is a k-dimensional principal subfunction of F.

COROLLARY 5.2. Suppose that the penalty parameter ck is updated finitely manytimes in the positive algorithm. Then the sequence {xk} is bounded under any one ofthe following conditions:

(a) there exists a scalar t > 0 such that the level set

Lt := fxeRn 0(x) < t}++

is bounded;(b) f is (globally) Lipschitzian on Rn and every k-dimensional principal subfunc-+

Rn-ktion fa(" a) of f is norm-coercive on R for every fixed vector a E +(c) f is (globally) Lipschitzian and coercive in the gadamard sense on R_.Proof. By Theorem 5.1, the sequence {0(xk)} converges to zero. Hence, for all

k sufficiently large, xk is contained in the level set Lt. Consequently, (a) implies theboundedness of {zk}. By the proof of [24, Lem. 4], it follows that (b) implies (a).Finally, we show that (c) implies (b). But this is an easy. consequence of the identity

a, I,(x.. I,(a.. + a, a,) a,))

and the Lipschitzian property of f, which implies that

limsup Ilae (f (xa, as)- f (aa, ae))l <

Thus, by the coercivity of f on R_ in the Hadamard sense, it follows that the principalsubfunction fa(.,a) must be coercive on R_ in the Hadamard sense. As a conse-quence of an observation preceding this corollary, it follows that this subfunction isnorm-coercive on R. D

Our next result concerns the LCP that corresponds to the NCP (f) in whichf is an affine mapping. The proof of this result requires a fundamental continuityproperty of the solution set of the LCP regarded as a multifunction of the constantvector of the problem [1, Thm. 7.2.1]. To explain the latter property, consider theLCP defined by the vector q Rn and matrix M Rnxn:

(26) x >_ O, q + Mx >_ O, xT(q + Mx) O.

We let SOLM(q) denote the (possibly empty) solution set of this problem. As amultifunction in q, SOLM is locally upper Lipschitzian in the following sense: for afixed but arbitrary q, there exist a constant L > 0 and a neighborhood V of q suchthat for all q V,

SOLM(q’) C_ SOLM(q) -b LIIq-

where B denotes the (closed) unit ball in Rn with the Euclidean norm. There aretwo immediate consequences of this result: one is that if SOLM(qk) is nonempty fora sequence of vectors {qk} converging to q, then SOLM(q) is nonempty; moreover, ifthe latter solution set is bounded, then all solution sets SOLM(qI) with ql sufficientlyclose to q are uniformly bounded; see [1].

COROLLARY 5.3. Suppose that the penalty parameter ck is updated finitely manytimes when the positive algorithm is applied to the LCP (26). Then SOLM(q) must

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be nonempty. Moreover, if this solution set is bounded, then there exists a constantL > 0 such that for all k sufficiently large,

d(xk, SOLM (q)) _< L’II min(xk, q +where d(x, S) denotes the distance from the vector x to the set S; in particular, thesequence {xk} is bounded.

Proof. Let yk min(xk, q / Mxk). Then Theorem 5.1 implies that the sequence(yk} converges to zero. The definition of yk implies that the vector zk xk -yk ESOLM(qk) where

qk q + Myk yk

which clearly converges to q. Hence, the desired conclusions follow easily from theaforementioned consequences of the locally upper Lipschitzian property of the solutionset of an LCP. [3

Remark. The boundedness conclusion of the sequence {xk } in Corollary 5.3 doesnot follow from Corollary 5.2. The reason is that the solution set SOLM(q) is equalto the level set

{x Rn n+ (x) < 0} { e R+’(x)= 0},

which is different from any of the sets Lt with t _> 0. Clearly, the polyhedrality natureof the LCP has much to do with the validity of Corollary 5.3.

Infinite update of c. We return to the NCP (f) and analyze the other case of thepositive algorithm, namely, when Ilwck (xk)ll > 5 fails for infinitely many k. Our goalhere is to demonstrate that if x* is the limit of any subsequence {xk k } forwhich

and if x* is an s-regular vector in the sense defined in [27], then x* solves the NCP(f). For the sake of clarity, we review this concept in the definition below.

DEFINITION. A vector x >_ 0 is said to be s-regular if the following system oflinear inequalities has a solution in the variable d:

where

xi +di =0

fi(x) + Vfi(x)Td 0

xi+di >_0

fi(x) + Vfi(x)Td >_ 0

fo e () o (x),o -(x),fo e (),o (x),

xi + di <_ 0 for Ie+(x),fi(x) + Vfi(x)Td <_ 0 for e I+(x),

Z](x) {i. f(x) < > o),

+/-y(x) {i f,(x) < x 0),Z$(x) {i f(x) x > o),

o (x) { f(x) o).

In [27, Prop. 3], a sufficient condition for a nonnegative vector x to be s-regularwas established in terms of certain matrix-theoretic properties of the Jacobian matrix

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Vf(x). We will postpone the discussion of this condition until the next subsection. Inwhat follows, we proceed to establish a convergence result that complements Theorem5.1. Given a subset X c_ Rn and a point x E X, we recall the set of feasible directionsat x with respect to X"

(x, X) =_ {d e Rn 5 > 0such that x + ad e X for all a e [0, ] }.

LEMMA 5.4. Suppose that the penalty sequence {ck} is updated infinitely oftenand that x* is the limit of a subsequence {xk k a} for which

]lwkll < 5 for all k a.

Then, the sequence {u k a}, where u =- A(x)U(x) has an accumulation pointand any accumulation point u of this sequence satisfies

(27) dTu >_ 0 Vd e (x*,R).

Proof. Since xk - x* and there is only a finite number of distinct index setsIf(xk), it is easy to see that the sequence {uk k } has an accumulation point.Assume that u is one such accumulation point and let us show that (27) holds.Indeed, let B {ilx > 0} and N =_ {ilx 0}. Noting that ’(x*, R_) {d eR dN >_ 0}, we conclude that (27) is equivalent to the condition

(28) u 0, u >_ 0.

We now show that (28) holds. Indeed, using the assumption that Ilwkll <_ 5 for allk a and the definition of w, we obtain

(29)

where

tk - Vk

O(xk) (xk)-Iak

<_5t(__,xkj

for all k E a,Ck

ve(x) (e + u

ck (O(xk) / eTxk)for all k.

Observe that the sequence vk 0 since ck oc Hence, from the fact that x ken

x > 0 and c c, and relation (29), we obtain that u limkeU 0.

Moreover, from (29) and the fact that XN 0, we obtain that Uv / Vv _> 0 for allk e a sufficiently large. Hence,

+ > 0,k

and the result follows. DObserve that condition (27) can be viewed as a weak stationarity condition for

the point x*. If the accumulation point x* is nondegenerate; that is, it satisfies

x fi(x*) for all 1,..., n, then we can conclude that x* is a stationary point forthe function 0(x) as the following corollary states.

COROLLARY 5.5. Let the assumptions of Lemma 5.4 hold and assume furtherthat x* is nondegenerate. Then, we have

(30) 0’ (x*, d) >_ 0 for all d e JZ(x*, Rn+).

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Proof. Using the fact that x* is nondegenerate, we conclude that Ix(xk) Ix(x*)and Ii(x) Is(x*) for all k sufficiently large. It is now straightforward to see that thesequence {uk k E } converges and that its limit point u satisfies dTu 0 (x*, d)for all d Rn. The result now follows from Lemma 5.4.

The above two results say nothing about x* being a solution of NCP (f) or,equivalently, that x* satisfies 0(x*) 0. To conclude that x* is a solution of NCP(f), we need to assume that x* is an s-regular vector. Before showing this fact, westate a preliminary result that gives several conditions that are related to s-regularity.

LEMMA 5 6. Let x Rn be given and for all d Rn defineming (d) =_ (x,d) f(x)Vf(x)Vd + min{f(x)Vf(x)Td, xd)

iell(x) iele(x)

+ Z xidi,

+i() i() ()

Let gx (x,R) -- n be any homogeneous function of degree 1 (i.e., gx(Ad)g(d), for all > 0 and d JZ(x,R)) satisfying

_max(d)<gx(d)<g (d) for alldeg(x,R),

and consider the following conditions on x:(a) x is an s-regular vector;(b) there exists a d e (x, R) such that

f/(x) [f/(x) + Vfi(x)Td] < 0 for all e Ii(x),(31) max {xi[xi + di] f/(x)[f/(x) + Vf(x)Td]) < 0 for all e Ie(x),

x[ + d] < o ]o atZ e/(x);

(c) there exist scalar / > 0 and vector d e f(x, R) such that 7O(x) + gx(d) < 0;(d) if (x) > 0 then there exists d e f(x,R) such that gx(d) < O.

Then, the following implications hold:

(a) ==v (b) (c) == (d).

Proof. The implication (a) (b) follows from the definition of s-regularity andthe fact that d ’(x, R_) if and only if di >_ 0 for all such that xi O. To showthe second implication, assume that (b) holds. Summing the relations in (31) over alli 1,..., n, we obtain that

20(x) + gaX(d) _< 0,

which obviously implies (c) with 7-- 2. The equivalence of (c) and (d) can be easilyproved using the fact that gx is homogeneous of degree 1.

We are now ready to state the main result of this subsection.THEOREM 5.7. Suppose that the penalty sequence {ck) is updated infinitely often.

If x* is the limit of a subsequence (xk k e to) for which

IIwll 6 or all k t

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and x* is s-regular, then O(x*) O.Proof. Assume for contradiction that 0(x*) 0. Let u be an accumulation

point of the sequence (uk k E a} as defined in the statement of Lemma 5.4. Then, itis easily seen that the function gx(d) =- dTu satisfies the assumptions of Lemma 5.6.Hence, since x* is an s-regular vector and O(x*) O, it follows from Lemma 5.6(d)that there exists a vector d (x*, R) such that dTu < O. Since this contradictsLemma 5.4, we must have O(x*) O.

The class of s-regular functions. We may combine Theorems 5.1 and 5.7 to givea unifying convergence result for the positive algorithm. For this purpose, we definethe following class of functions.

DEFINITION. A function f" Rn --. R is said to be s-regular if every nonnegative+vector is s-regular.

A function f with the property that Vf(x) is a P-matrix for all x >_ 0 must bes-regular; in particular, any uniform P-function f on Rn (and hence, any stronglymonotone function) is s-regular. The proof of these observations hinges on the fol-lowing result, which is a paraphrase of Proposition 3 in [27]. (The reader may wantto consult [1] for discussion of the various matrix classes involved here.)

PROPOSITION 5.8. Let x be a nonnegative vector. Suppose that (i) the principalsubmatrix

(32) VI) f) (x)

is nonsingular, and (ii) the Schur complement of this matrix in

(33)

is an S-matrix.

-v &+ (x) -v f (x) v f (x)

(Here, the index sets are all evaluated at the given x.) Then x is an s-regular vector.We observe that if x _> 0 is such that Vf(x) is a P-matrix, then conditions (i) and

(ii) of Proposition 5.8 are satisfied. Indeed, if Vf(x) is a P-matrix then the matrices

(32) and (33) are both P-matrices. In particular, (32) is nonsingular, showing thatcondition (i) holds. Moreover, since the Schur complement of any principal submatrixof a P-matrix is a P-matrix and since every P-matrix is an S-matrix, condition (ii)follows.

An interesting example of an s-regular function that is neither P nor monotoneis the negative identity function. Indeed, if f(x) -x, then I+ (x) I(x) forall x > 0. Consequently, any nonnegative vector is s-regular and f is an s-regularfunction. More generally, if f is a function for which these two index sets are emptyfor all nonnegative vectors and whose Jacobian matrix Vf(x) is nondegenerate for allx >_ 0, then f must be s-regular.

The following theorem is immediate from the previous results.THEOREM 5.9. If f is an s-regular function, then every accumulation point of a

sequence of iterates produced by the positive algorithm is a solution of the NCP (f).A referee of this paper correctly points out that the above theorem does not pro-

vide conditions under which a sequence of iterates produced by the positive algorithmwill have at least one accumulation point. The difficulty with this deficiency of the

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theorem lies in the case where the scalar c is updated infinitely often; indeed, in thepresent version of the algorithm, whenever ck is updated, we do not change the iteratexk, and essentially, do nothing. It might be necessary to modify the algorithm to yielda more desirable result.

If the function f has the property that the function 0 has bounded level sets,then the only way for the sequence {xk} to have no accumulation point is thatlimk(xk) cx. Although we cannot rule out this possibility when {ck} isunbounded, it does seem rather unlikely to occur in practice. To substantiate thisstatement, we have implemented the positive algorithm for solving a variety of comple-mentarity problems. The results are reported in the next section. In all the numericMtests we have conducted, the P-values at termination were consistently substantiallysmaller than the %values at initiation, even in cases when the positive algorithm failedto solve a particular problem.

6. Numerical results. We have carried out some numerical experiments withthe positive algorithm applied to two sets of complementarity problems: one experi-ment consists of NCPs arising from various equilibrium models that are documentedin detail in [27]; the other is a set of randomly generated LCPs. For the first set ofNCPs, we also compared the positive algorithm with the NE/SQP method describedin [27] because the latter method was also based on the formulation (1) of the NCPand was highly successful in terms of robustness and speed. Our experiments showthat the positive algorithm is faster than the NE/SQP method on the test problemsbut less robust. The improved speed is not surprising since in calculating each searchdirection, the positive algorithm solves only one system of linear equations whereasthe NE/SQP method solves a convex quadratic programming with lower bound con-

straints; indeed, the simplicity of the direction generation is the single most importantfeature of the positive algorithm.

The positive algorithm was implemented in a FORTRAN-77 computer code andthe experiments were conducted on a SUN SPARCStation IPX with 16 megabytes ofmemory and one CPU processor. Double precision arithmetic was employed in thecalculations. In each iteration, we solved the system of linear equations (9) for thesearch direction by using the LU decomposition; the subroutines in [29] were used.We terminated the algorithm when the 0-value was less than 10-12 The parametersof the algorithm were set as follows:

n + 1, 5 10-6, a p 0.5, and c 0.95.

We set the initial penalty parameter co 103. The numerical results for the positivealgorithm applied to the equilibrium problems are summarized in Table 1.

In Table 1, n denotes the dimension of the NCP, niter the number of iterations,and aver.nls the average number of steps needed in the Armijo line search. In thecolumn of niter, the numerators are the numbers of systems of linear equations solvedby the positive algorithm, and the denominators are the numbers of quadratic pro-grams solved by the NE/SQP method as reported in [27]. The column of 0-valuesgives some indication of the speed of the positive algorithm at the tail of the iter-ations. In these runs, all the starting points were chosen to be the vector of ones,except for the Hansen-Koopmans problem where the NE/SQP method started from(xl,... ,xl0, y, Y2, z, z2) (0.3,..., 0.3, 0, 0, 0, 0) while the positive algorithm startedfrom (Xl,..., Xl0, y, y2, z,.z2) (0.3,..., 0.3, 0.1, 0.1, 0.1, 0.1); the reason for this de-viation is that the positive algorithm must start from a positive vector.

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TABLE 1

Problem n niter aver.nlsKojima-Shindo 4 8/7 2.000

Mathieson 4 6/3 1.167Nash-Cournot 10 9/9 1.000

Hansen-Koopmans 14 22/10 1.667Spatial price 42 21/20 1.050’Traffic equi. 50 28/20 3.357

last three 0-values8.01D-08/2.10D-12/5.05D- 173.59D-08/8.24D-11/1.91D-134.62D-02/2.23D-06/6.07D-151. lSD-0S/1.S5D-12/2.01D-161.14D- 11/1.57D-12/2.16D-132.36D-09/6.10D-12/8.34D,15

The column of niter demonstrates the relative efficiency of the positive algorithmversus the NE/SQP method. Since one single system of linear equations was solved inthe former algorithm, versus a quadratic program in the latter, the advantage of thepositive algorithm in terms of speed should be evident. Indeed, we have comparedthese two algorithms on the largest of these problems, the traffic equilibrium problem,on the VAX 6000 computer at the Homewood Computing Facility Center at The JohnsHopkins University. The positive algorithm and the NE/SQP method used 3.94 and8.35 CPU seconds, respectively. For the NE/SQP method, the FORTRAN packageQPSOL [3] was used to solve each quadratic subprogram.

There are three test problems reported in [27] that were not included in the presentexperimentation. These are the PIES model, the Walrasian equilibrium problem withproduction, and the generalized von Thiinen model. These problems are mixed NCPsas defined in [1]; the present version of the positive algorithm needs to be modified todeal with this class of problems. This is a topic for further study.

The set of LCPs to which the positive algorithm was applied can be classifiedinto four types, each according to the properties of the matrix M. Specifically, thegeneration of M was as follows. In each case, M was completely dense.

1. M ATA. Each entry of the matrix A was generated uniformly from theinterval (0,50), with a probability of 0.5 for the entry to be given a negative sign. Thismatrix M is symmetric positive semidefinite.

2. M is diagonally dominant. We set M A, where A was generated in thesame way as above except that each diagonal entry was set to be one plus the sum ofthe absolute values of the off-diagonM entries in the same row.

3. M A2. Here A was generated as before. This matrix M is indefinite.4. M is a positive matrix. Each entries of M was generated uniformly from the

interval (1,50).For each matrix M generated, we constructed a solvable LCP as follows. First we

generated a random number from the interval (0,1); if this number was greater than0.5, then we set x’ 0.0, otherwise we generated a number from the interval (0,50)and set x* to be that number. Therefore, roughly half of the components of x* were

* -Mi, x*zeros. We then formed the vector q as follows. If x > 0, then q whereM,. denotes the ith row of M; otherwise q -Mi,.x* + r, where r is a randomnumber in (0,50) scaled by 0.3. Clearly the resulting LCP (q, M) has at least one

solution, namely, x*. For M of the first two types, x* is the unique solution.After each LCP was generated, we scaled the problem in the following way: let

s be the sum of all entries of M and q, we scaled M and q by the factor 50Is. Thepositive algorithm was then applied to the scaled LCP. It turned out that such ascaling was quite useful in ensuring the effectiveness of the algorithm.

The parameters of the algorithm were given the same values as before. We sum-marize the numerical results in Tables 2 and 3 for the cases M ATA and M

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TABLE 2M ATA, n 100.

Problem 1 2 3 4 5nter 28 33 30 32 26

aver. nls 1.037 1.031 1.034 1.032 1.040Initial 0 40308.4 35554.7 36078.4 43694.5 32757.5Final 0 2.0D-14 3.8D-14 3.9D-14 1.5D-14 3.9D-13

Problem 6 7 8 9 10niter 41 35 23 31 25

aver. ’nls 1.025 1.029 1.045 1.033 1.041Initial 31538.1 35382.4 29677.3 45676.6 31909.8Final 0 1.0D-13 4.1D-14 6.5D-13 9.2D-15 7.0D-13

TABLE 3M is diagonally dominant, n-- 100.

Problem 1 2 3 4 5nier 11 9 10 10 11

aver. nls 1.300 1.125 1.222 1.444 1.300’Initial 0 46418.3 42027.2 47092.5 53837.2 36841.5

Final 0 1.6D-14 3.4D-13 2.6D-13 1.9D-14 3.1D-14

Problem 6 7 9 10niter 13 10

aver. nls 1.583 1.333Initial 0 45390.3 44541.4Fihal 0 180D-14 2’.3D-13

1.11136948.61.0D-14

10 91.111 1.125

4860’1.9 43534.82.5D-14 8.3D-13

diagonally dominant, respectively. The entries in the tables are self-explanatory (n isthe dimension of M). As we have mentioned, the matrix M is completely dense; thisis the reason why we have not attempted to solve problems of larger size in these twocases; in other words, data storage has imposed a restriction on our ability to use theSun workstation for solving larger problems of this kind.

Observe that in the above two cases, all 10 problems in each group were success-fully solved; more importantly, the computational statistics were very encouraging.In contrast, the results for the remaining two cases were not as good. In each of thesecases, we ran LCPs of size n 10, 20, 30. Ten problems were tested in each category.When M A2, the following results were obtained. For n 10, seven problems weresolved to satisfaction; for n 20, one; and for n 30, two. When M is positive, weobtained the following results. For n 10, six problems were solved to satisfaction;for n 20, four; and for n 30, three. Invariably, when a successful run occurred,the results were good (i.e., small number of iterations, good speed at the tail, andsmall number of line searches). When an unsuccessful run occurred, it was due to theexcessive number of iterations (80 was the maximum we set) and the small magni-tude of the search directions; for these failed runs, the 0-values were consistently inthe range of 10-4 and 10-s, which were small but not enough for successful termi-nation according to our rule. There was good reason to believe that the iterates attermination of these unsuccessful runs were not s-regular vectors for the functions.

In summary, our computational results suggest that the positive algorithm holdspromise in practice for solving complementarity problems that satisfy certain regu-larity conditions. For problems that do not necessarily satisfy the latter conditions,the algorithm requires further study and modification is needed.

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7. Some concluding remarks. In this paper, we have presented and tested apositive algorithm for solving the general nonlinear complementarity problem. Somelimiting properties of this algorithm were derived. At this time, although some theo-retical issues remain with the algorithm, the computational experience we have gath-ered suggests that this algorithm is quite competitive with a previous algorithm ona set of equilibrium problems and has the potential for solving certain nonmonotoneNCPs effectively.

REFERENCES

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[11]

[12]

[13]

[14]

[1] R. COTTLE, J. S. PANG, AND R. E. STONE, The Linear Complementarity Problem, AcademicPress, Boston, 1992.

[2] S. GABRIEL AND J. PANG, A trust region method for constrained nonsmooth equations withapplications, in Large-Scale Optimization: State of the Art, W. Hager, D. Hearn, andP. Pardalos, eds., Academic Publishers B.V., Boston, 1994, pp. 159-186.

[3] P. GILL, W. MURRAY, M. SAUNDERS, AND M. WRIGHT, User’s guide for qpsol (version 3.2): Afortran package for quadratic programming, Tech. Report SOL 84-6, Systems OptimizationLaboratory, Dept. of Operations Research, Stanford University, Stanford, CA, 1984.

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a positive algorithm for the nonlinear complementarity...

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