DOI:10.1051/cocv:2008003 www.esaim-cocv.org
SOME NEW EXISTENCE, SENSITIVITY AND STABILITY RESULTS FOR THE NONLINEAR COMPLEMENTARITY PROBLEM
∗Rub´ en L´ opez
1Abstract. In this work we study the nonlinear complementarity problem on the nonnegative or- thant. This is done by approximating its equivalent variational-inequality-formulation by a sequence of variational inequalities with nested compact domains. This approach yields simultaneously exis- tence, sensitivity, and stability results. By introducing new classes of functions and a suitable metric for performing the approximation, we provide bounds for the asymptotic set of the solution set and coercive existence results, which extend and generalize most of the existing ones from the literature.
Such results are given in terms of some sets called coercive existence sets, which we also employ for obtaining new sensitivity and stability results. Topological properties of the solution-set-mapping and bounds for it are also established. Finally, we deal with the piecewise affine case.
Mathematics Subject Classification.90C31, 90C33, 47J20, 49J40, 49J45.
Received July 12, 2006. Revised January 15, 2007.
Published online January 18, 2008.
Introduction
LetF :Rn+→Rnbe a given function, and letRn+be the set of vectors fromRnwith nonnegative components.
Thenonlinear complementarity problem in Rn, denoted by NCP(F), is the following:
find ¯x∈Rn+ such that F(¯x)∈Rn+ andF(¯x),x¯ = 0. (NCP) If we setF(x) =M x+q whereM ∈Rn×n andq∈Rn, then the problem is called thelinear complementarity problem (LCP). These problems are important in various equilibrium settings and has wide range of applications in Science and Engineering (see [3,8,15,19,22,32] and the references therein).
Problem (NCP) is known to be equivalent to the following variational inequality problem VIP(Rn+, F) (see Lem. 3.1 from [23]), which will serve as our main framework for its study.
find ¯x∈Rn+ such that F(¯x), x−x ≥¯ 0 ∀x∈Rn+. (VIP) Sensitivity and stability analysis of the NCP(F) is concerned with the study of the behavior of the solution(s) of this problem when the data are subject to change. This analysis provide valuable qualitative information
Keywords and phrases. Nonlinear complementarity problem, variational inequality, asymptotic analysis, sensitivity analysis.
∗This work has been supported by CONICYT-Chile through Proyecto FONDECYT de Iniciaci´on en Investigaci´on 11060015 and Proyecto DIN-03/2006 Universidad Cat´olica de la Sant´ısima Concepci´on.
1 Facultad de Ingenier´ıa, Universidad Cat´olica de la Sant´ısima Concepci´on, Concepci´on, Chile;rlopez@ucsc.cl
Article published by EDP Sciences c EDP Sciences, SMAI 2008
on the problem. In the literature we can find different approaches for obtaining sensitivity and stability results for the nonlinear complementarity problem and variational inequalities. Meggido [30], Tobin [35] and Kyparisis [26]
obtained some results on continuity of a locally unique solutionx(ε) of the parametric nonlinear complementarity problem NCP(F(·, ε)). All these papers assume differentiability of the functionF and they employ either the Implicit Function Theorem or the theory of generalized equations. Ha [18] used degree theory to derive sufficient conditions for the existence of solutions for small perturbations of the function. His approach does not require that the function F be differentiable. Dafermos [5] employed an approach based on geometric arguments considering the variational inequalities as an orthogonal projection on some set. Facchinei and Pang [7] studied the “total stability” of the variational inequality VIP(K, F) when both the function F and the set K are perturbed. They employed degree theory and variational analytic tools to establish a general stability result for a variational inequality with a bounded solution set.
Our approach is totally different from theirs; we use tools of variational convergence and asymptotic analysis.
We approximate the problem VIP(Rn+, F) by a sequence of problems VIP(Dk, Fk) (k ∈ N) where {Dk} is a sequence of nested compact convex sets converging to Rn+ and {Fk} is a sequence of continuous functions converging continuously to F. This approach allows us to develop a general theory yielding simultaneously existence, sensitivity and stability results. It is worth pointing out that our results are concerned with global sensitivity/stability analysis, which refers to the investigation of the change of the entire solution set SOL(F) when the functionF undergoes small perturbation, and not with isolated sensitivity analysis as in most of the papers cited before.
This paper is a continuation of the previous work [11] where this new approach was employed for studying the multivalued complementarity problem. This approach has been developed when studying (LCP) in [9]. The novelty of the present paper with respect to the former lies in the fact that we now deal with single-valued mappings and to this end we employ another metric instead of that characterizing the graphical convergence (see also Rem.1.9). The choice of this metric is natural and has the advantage that for the homogeneous type case it has an equivalent metric that allows us to obtain Lipschitzian properties for the solution-set-mapping for the piecewise affine case.
In Section 2, we list some preliminaries and introduce new classes of functions and the metrics we shall work with. In Section 3, we perform an asymptotic analysis of a sequence of normalized approximate solutions to (VIP) and obtain some bounds for the asymptotic set to the solution set. In Section 4, we study the nonemptiness and boundedness of the solution set for the new classes of functions and for the pseudomonotone ones. We organize, extend, and generalize most of the existing results from the literature. In Section 5, we obtain new sensitivity and stability results, study topological properties of the solution-set-mapping, and obtain some bounds for the solution set. Section 6 is devoted to study the piecewise affine case.
We shall use the following notation: x ≥0 (resp. x > 0) whenever x∈ Rn+ (resp. x∈ intRn+); B is the unit ball in Rn; I = {1, . . . , n}; ||y||∞-maximum norm, |y| = (|y1|, . . . ,|yn|), ||y||d = d,|y| is the d-norm, dmin= mini∈Idi, Δd={x≥0 :||x||d= 1}whenevery∈Rn andd >0;{ei:i∈I}is the canonical basis ofRn; SOL(F) (resp. FEA(F) ={x≥0 :F(x)≥0}, FEAs(F) ={x≥0 :F(x)>0}) is the solution (resp. feasibility, strictly feasibility) set of NCP(F); givend >0 the setA∞d ={v∈Rn : ∃ xk ∈A, ||xk||d→+∞, ||xxkk||
d →v} is the d-asymptotic set ofA; d(x, A) is the distance of a pointxto A; dI(A, B) is the integrated set distance betweenAandB;C={c:R++→R++ : c(0)≥0, limt→+∞c(t) = +∞}.
In what follows, bydandcwe shall denote a positive vector and a function fromC.
1. Preliminary facts
Here and in the subsequent sections we shall deal with functionsF defined onRn+. We recall some well-known definitions from the literature (see [8,19,34,37]). A functionF :Rn+→Rn is said to be:
• copositive ifF(x)−F(0), x ≥0 ∀x≥0;
• strictly copositive ifF(x)−F(0), x>0 ∀0=x≥0;
• strongly copositive if∃α >0 such thatF(x)−F(0), x ≥α||x||2 ∀x≥0;
• monotone ifF(x)−F(y), x−y ≥0 ∀x, y≥0;
• strictly monotone ifF(x)−F(y), x−y>0 ∀x, y≥0, x=y;
• strongly monotone if∃α >0 such thatF(x)−F(y), x−y ≥α||x−y||2 ∀x, y≥0;
• pseudomonotone ifF(x), y−x ≥0 =⇒ F(y), y−x ≥0 ∀x, y≥0;
• a G(d)-function (d–Garcia’s function), orF∈G(d), if SOL(F−F(0) +τ d) ={0} ∀τ >0;
• an R(d)-function (d–regular function), or F∈R(d), if SOL(F−F(0) +τ d) ={0} ∀τ ≥0;
• an R0-function, orF ∈R0, if SOL(F−F(0)) ={0};
• piecewise affine if Rn+can be represented as the union of finitely many polyhedral sets, relative to each of whichF(x) is given by an expression of the formM x+qforM ∈Rn×n and q∈Rn;
• Rn+–convex ifF(λx+ (1−λ)y)≤λF(x) + (1−λ)F(y) ∀x, y≥0, ∀λ∈[0,1].
If the inverse sign holds, we say thatF isRn+–concave;
• a Qb-function if SOL(F+q) is nonempty and compact for allq∈Rn.
We also recall some definitions concerning multifunctions, we shall need in what follows (see [1,17,20,33,34]).
LetX andY be two metric spaces. A multifunction orset-valued mapping Φ :X ⇒Y is said to be:
• (X =Rl,Y =Rm)piecewise polyhedral if its graph defined by gph Φ ={(x, y) :x∈dom Φ, y∈Φ(x)}
is expressible as the union of finitely many polyhedral sets;
• (X =Rl, Y =Rm)locally upper Lipschitzian at x¯ with modulus λ or locally UL(λ) at ¯xif there is a neighborhoodU of ¯xsuch that Φ(x)⊆Φ(¯x) +λx−x¯Bfor allx∈U. If in addition, Φ(¯x)=∅ then Φ is calledcalm at x;¯
• (X =Rl, Y =Rm)Lipschitzian if there exits a scalarλ >0 such that Φ(x)⊆Φ(y) +λx−yBfor allxand y;
• outer semicontinuous (OSC) (or closed) at x ∈ domΦ if whenever a sequence {(xk, yk)} ⊆ gph Φ converges to (x, y) theny∈Φ(x);
• inner semicontinuous (ISC) (oropen or lower semicontinuous (lsc)) atx∈domΦ if for anyy ∈Φ(x) and for any sequence{xk} ⊆domΦ such thatxk →xthere exists a sequence{yk}such thatyk ∈Φ(xk) for allkandyk →y;
• continuous at x∈domΦ if it is OSC and ISC at x.
We now introduce two new classes of functions which are the single-valued analogous to those defined in [11]
for the multivalued complementarity problem (see (b) of Ex.1.3below). As will be shown in the next example, such classes encompass various classes of functions from the literature that are important in the complementarity problem theory.
Definition 1.1. A functionF :Rn+→Rn is said to be:
• c-homogeneous (on Δd) ifF(λx)−F(0) =c(λ)[F(x)−F(0)] for allx∈Δd andλ >0;
• c-Mor´e (on Δd) ifF(λx)−F(0), x ≥c(λ)F(x)−F(0), xfor allx∈Δd andλ >0.
Remark 1.2. (a) It is important to point out that our notions of c-homogeneous on Δd and c-Mor´e on Δd functions are given on the compact set Δd instead of being given on the wholeRn+ as is done in the literature (see Ex.1.3). Henceforth, we shall consider that all the functions arec-homogeneous andc-Mor´e on the set Δd unless otherwise is specially stated.
(b) If the function F is: copositive type, monotone type, G(d), R(d), R0, piecewise-affine, Qb, Rn+-convex, Rn+-concave,c-homogeneous, orc-Mor´e, then so too is F+q for all q ∈Rn. This property does not hold for pseudomonotone functions as is shown in Example1.7below.
(c)We may impose that the relationships definingc-homogeneous orc-Mor´e functions hold for allλsufficiently large (see [37]) or for all λ≥1 (see [11]) respectively. It is important to point out that our existence results hold under those assumptions. However, for obtaining also bounds for the solution set, sensitivity and stability results in a unified manner, our definitions are more suitable.
Example 1.3. (a)A functionH :Rn+→Rn is said to be (positively)generalized homogeneous (see [37]), if for somecthe equalityH(λx) =c(λ)H(x) holds for allx≥0 andλ >0. In particular, ifc(λ) =λγ, thenH is said to be (positively) homogeneous of degree γ > 0. For instance, the functionsH1(x) =M x, where M ∈Rn×n;
H2(x) = (h21(x), . . . , h2n(x)), whereh2i(x) = max{wij, x:j∈Λi} withwij ∈Rn and Λi being a finite index set, are homogeneous of degree 1. The functionH3(x) =||x||M xis homogeneous of degree 2.
IfF−F(0) is generalized homogeneous (for somec), thenFisc-homogeneous on Δdfor anyd. The functions F1(x) =xif 0≤x≤1 and F1(x) = 3x−2 ifx≥1,F2(x) =xif 0≤x≤1 and F2(x) =x2 ifx≥1, and F3(x, y) = ex+y
x2+y2(x, y) are c-homogeneous on Δd for d = 1, c1(λ) = F1(λ), c2(λ) = F2(λ), and d= (1,1),c3(λ) =λ2eλ−1 respectively. However,Fi(x)−Fi(0) (i= 1,2,3) are not generalized homogeneous.
(b) A multifunction Φ : Rn+ ⇒ Rn is said to be c-subhomogeneous, if 0∈ Φ(0) and for some c it holds that Φ(λx)⊆c(λ)Φ(x) for allx∈Δdandλ >0 (see [11]). If Φ is single-valued, then Φ isc-homogeneous. Therefore, for single-valued mappings the classes ofc-subhomogeneous andc-homogeneous mappings coincide.
(c)A function F :Rn+→Rn such thatF(λx)−F(0), x ≥c(λ)F(x)−F(0), xfor somec and for allx≥0 and λ >0 (see [37]) is c-Mor´e on Δd for anyd. For instance,F such thatF(λx)−F(0)≥c(λ)[F(x)−F(0)]
for some c and for allx≥0 andλ > 0 (see [31]) satisfies such a condition. The function F(x, y) = (x3, y4) satisfies the latter inequality for ˜c(λ) = min{λ3, λ4} but it is notc-homogeneous on Δd for anycand d.
(d) The functionsF4(x) =M x+q forM ∈Rn×n, q∈Rn, andF5(x) =PC(x)-projection function onto the polyhedral set C ⊆Rn are piecewise affine (see Ex. 12.31 from [34]). If T : Rn ⇒Rn is maximal monotone piecewise polyhedral and f : Rn →R is proper lsc convex piecewise linear-quadratic, then the resolvent and proximal mappingsRλ(x) = (I+λT)−1andPλf(x) = arg minw
f(w) +2λ1||w−x||2
are piecewise affine for everyλ >0 (see Props. 12.29 and 12.30 from [34] respectively).
(e)IfFis copositive (resp. strictly copositive), thenF∈G(d) (resp.F ∈R(d)) for alld. Indeed, ifx∈SOL(F− F(0)+τ d) forτ >0 (resp.τ≥0), then 0≤x⊥F(x)−F(0)+τ d≥0, therefore, 0≤ F(x)−F(0), x=−τd, x a contradiction ifx= 0.
The following definition generalizes that used for linear mappings in [21] and is the single-valued variant of that used for multifunctions in [11].
Definition 1.4. LetF :Rn+→Rn be a continuous function. The d-numerical range ofF is by definition the set ω(F) :={F(x)−F(0), x: x∈Δd}.We defineMF := maxω(F) andmF := minω(F).
The next result is similar to Propositions 2.5 and 4.5 from [11] and complements them. Part (a) generalizes Corollary 3.3 from [31].
Proposition 1.5. Let F :Rn+→Rn be a continuous function.
(a) if F isc-Mor´e, thenmF||x||dc(||x||d)≤ F(x)−F(0), x for allx≥0;
(b) if F isc-homogeneous, thenF isc-Mor´e and F(x)−F(0), x ≤MF||x||dc(||x||d)for allx≥0;
(c) if F is strongly copositive, thenF is¯c-Mor´e on Δd for anydandc(λ) =¯ Mαλ
F||d||2;
(d) ifF is strictly copositive, thenmF >0. If in addition, F is homogeneous of degree1, thenF is strongly copositive;
(e) if F isc-Mor´e andmF >0, thenF is strictly copositive;
(f) if F is Rn+-convex, then it holds that F(λx)−F(0) ≥ λ[F(x)−F(0)] for all x ≥0 and λ ≥ 1, and F(λx)−F(0)≤λ[F(x)−F(0)] for allx≥0 andλ∈[0,1];
if in addition, F is strictly copositive, then F is ˜c-Mor´e on Δd for any d and ˜c(λ) =λ if λ≥1 and
˜
c(λ) = minx∈ΔdF(λx)−F(0),x
F(x)−F(0),x if 0≤λ≤1;
(g) if F is pseudomonotonec-homogeneous andFEA(F)= 0, thenF is copositive.
Proof. (a)-(b): Are trivial.
(c): If x ∈ Δd and λ > 0, then by hypothesis and since it holds that 1 = ||x||d ≤ ||d|| ||x|| we obtain F(λx)−F(0), λx ≥αλ2||x||2≥ ||d||αλ2M2FF(x)−F(0), x.
(d): IfF is strictly copositive, then by Weierstrass theoremmF >0. Moreover, by the homogeneity of degree 1 and (a), we obtainF(x)−F(0), x ≥mF||x||2d≥mFd2min||x||2 for all 0=x≥0.
(e): It follows from (a).
(f): If x ≥ 0, then by hypothesis F(x) = F(λ1(λx) + (1−λ1)0) ≤ λ1F(λx) + (1−λ1)F(0) for λ ≥ 1, thus,
F(λx)−F(0)≥λ[F(x)−F(0)]. The proof forλ∈[0,1] runs as before. The rest of the proof is immediate.
(g): If ¯x∈FEA(F) andx >0, then there existstx>0 such that ||x||t
dx−x¯≥0 andF(¯x),||x||t
dx−x ≥¯ 0 for allt > tx. Byc-homogeneity we haveF(||x||t
dx)−F(0) = c(||x||c(t)
d)[F(x)−F(0)] and by using the last inequality and the pseudomonotonicity hypothesis we conclude thatF(0) + c(||x||c(t)
d)[F(x)−F(0)], ||x||t
dx−x ≥¯ 0. On dividing the previous inequality byc(t)tand taking the limitt→+∞, we getF(x)−F(0), x ≥0 for allx >0.
Ifxlies on the boundary ofRn+, then there exists{xk} ⊆intRn+ such thatxk →xandF(xk)−F(0), xk ≥0.
By the continuity ofF we obtainF(x)−F(0), x ≥0.
Our approach consists of the approximation of (VIP) by a sequence of solvable problems and the later study of the asymptotic properties of a sequence of normalized solutions of such approximations. To do that we must introduce a suitable metric space.
The metric space (C(Rn+;Rn),D): where C(Rn+;Rn) is the set of continuous functions F = (f1, . . . , fn) defined onRn+ andDis the metric defined by:
D(F, G) := max
i∈I dIepi(fi, gi) + max
i∈I dIhyp(fi, gi)
where dIepi and dIhyp are the metrics which characterize the epi-convergence (→) ande hypo-convergence (→)h respectively and forf, g∈C(Rn+;R) are defined by dIepi(f, g) := dI(epif,epig) and dIhyp(f, g) := dI(hypf,hypg) where the right-hand terms are the integrated set distances between the epigraphs and hypographs off andg respectively (see Chaps. 5 and 7 from [34]).
In order to obtain some properties for this metric, we recall a well-known type of convergence. A sequence of functions {Fk}from Rn toRn converges continuously to the functionF if Fk(xk)→F(x) wheneverxk→x.
In Proposition 7.2 from [34] it is proved that for{fk}andf being functions fromRn toRit holds that:
• fk e→f if and only if at each pointxone has: •fk h→f if and only if at each pointxone has:
lim infkfk(xk)≥f(x) for every sequencexk→x lim supkfk(xk)≤f(x) for some sequencexk→x
lim infkfk(xk)≥f(x) for some sequencexk →x lim supkfk(xk)≤f(x) for every sequencexk →x.
Thus, by joining both assertions we conclude that fk e→f andfk h→f iff {fk} converges continuously tof (see Th. 7.11 from [34]). Consequently, D(Fk, F)→0 iff {Fk}converges continuously toF.
Proposition 1.6. Let {Fk} andF be functions such thatD(Fk, F)→0. If the sequence{Fk} is from one of the following classes: c-homogeneous,c-Mor´e, copositive, or monotone, then so too isF.
Proof. Let{Fk}be from the first class of functions and letx∈Δd andλ >0 be given. By hypothesis we have Fk(λx)−Fk(0) =c(λ)[Fk(x)−Fk(0)]. Taking limit and by the continuous convergence of{Fk}to F we get F(λx)−F(0) =c(λ)[F(x)−F(0)], thus, F isc-homogeneous. The rest of the proof runs as before.
Example 1.7. Each functionFk(x, y) = (y+k1,0)is pseudomonotone andD(Fk, F)→0 forF(x, y) = (y,0), but F is not pseudomonotone sinceF(e1), e2−e1= 0 and F(e2), e2−e1<0.
We now introduce a new metric space, which plays an important role when dealing with c-homogeneous functions.
The metric space (C(Rn+;Rn)c,Do): where C(Rn+;Rn)c is the set of c-homogeneous continuous functions F = (f1, . . . , fn) defined onRn+ andDo is the metric defined by:
Do(F, H) := max
x∈Δd
||F(x)−F(0)−H(x) +H(0)||∞+||F(0)−H(0)||∞.
It is not difficult to prove thatDo(Fk, F)→0 iff {Fk} converges uniformly toF on Δd andFk(0)→F(0).
Proposition 1.8.
(a) If F, H∈C(Rn+;R)nc, then||F(x)−H(x)||∞≤c(||x||d)Do(F, H) +||F(0)−H(0)||∞ for allx≥0;
(b) ifcis nondecreasing and{Fk},F are fromC(Rn+;Rn)c, thenDo(Fk, F)→0 iff D(Fk, F)→0. Hence, DandDo are equivalent metrics on C(Rn+;Rn)c.
Proof. (a): Forx= 0 the assertion is trivial. If 0=x≥0, then by thec-homogeneity we have
||F(x)−H(x)||∞≤ ||F(x)−F(0)−[H(x)−H(0)]||∞+||F(0)−H(0)||∞≤c(||x||d)Do(F, H) +||F(0)−H(0)||∞. (b): “Only if part”: If {xk} converges tox, then there exists anr >0 such that ||xk||d ≤r for allk, thus, by hypothesisc(||xk||d)≤c(r). By (a), the continuity ofF, and the following inequalities
||Fk(xk)−F(x)||∞ ≤ ||Fk(xk)−F(xk)||∞+||F(xk)−F(x)||∞
≤ c(r)Do(Fk, F) +||Fk(0)−F(0)||∞+||F(xk)−F(x)||∞
we conclude thatFk(xk)→F(xk). HenceD(Fk, F)→0.
“If part”: The hypothesis implies that{Fk}converges uniformly toFon all compact subsets ofRn+(see Th. 7.14 from [34]). In particular, this holds on Δd and{0}. HenceDo(Fk, F)→0.
The last part follows from the fact that two metrics are equivalent iff they induce the same convergence.
Remark 1.9. By Corollary 5.45 from [34] it is known that{Fk}converges continuously toF iff{Fk}converges graphically toFand the sequence{Fk}is eventually locally bounded at each ¯x,i.e., there exist a neighborhoodV of ¯x, k0 ∈N, and a bounded setB such thatFk(V)∈B for all k≥k0. Therefore, if we approximate (VIP) by using the continuous convergence, then the approximation is also by using the graphical convergence and the results from [11] hold (in terms of the metric characterizing the graphical convergence). However, if we approximate (VIP) by using the graphical convergence we cannot capture all the results of this paper (unless an additional assumption is assumed).
In the next three sections we shall employ the metricDfor obtaining our results. In the last section we shall employ the equivalent metricDoto deal withc-homogeneous piecewise affine functions forcbeing nondecreasing.
2. Asymptotic analysis
We approximate (VIP) by the following sequence of problems:
findxk∈Dk : Fk(xk), x−xk ≥0 ∀x∈Dk (PVIPk) whered >0,{σk}is an increasing sequence of positive numbers converging to +∞,Dk={x∈Rn+:||x||d≤σk} andD(Fk, F)→0.
The existence of solutionsxk to this problem is a consequence of the following result (see [25]).
Theorem 2.1(Hartman-Stampacchia). LetC⊂Rn be a compact convex nonempty set and letF :C→Rn be a continuous function. There exists a vectorx¯∈C such that F(¯x), x−x ≥¯ 0 for all x∈C.
Henceforth, we assume that all the functions we shall deal with are from C(Rn+;Rn).
We now introduce some sets calledcoercive existence sets, which are fundamental in our study. The impor- tance of these sets lies in the fact that they provide valuable information on (NCP): the accumulation points of any sequence of normalized approximate solutions belong to these sets (see Lem. 2.3), they bound the as- ymptotic cones/sets of the solution set (see Cor.2.4), and our main existence, stability, and sensitivity results
are given in terms of a property satisfied by these sets (the property of being equal to{0}):
W(F) = {v≥0 : τv =−F(v)−F(0), v ≥0, F(v)−F(0) +τvd≥0}
Wq(H) = {v≥0 :H(v)−H(0)≥0,H(v)−H(0), v= 0,H(0) +q, v ≤0}
U(F) = {v≥0 : F(v)−F(0), v ≤0}
Uq(H) = {v≥0 : H(v)−H(0), v= 0,H(0) +q, v ≤0}
V(F) = Rn+∩[−F(Rn+)]∗.
Two of these sets can be written by means of the set SOL(H−H(0)), termedcomplementary kernel of NCP(H).
Wq(H) = SOL(H−H(0))∩ {−H(0)−q}∗ and W−H(0)(H) = SOL(H−H(0)) (2.1) and ifτv =−F(v)−F(0), v ≥0 as above, then
[v∈W(F)and ||v||d= 1] =⇒v∈SOL(F−F(0) +τvd). (2.2) Proposition 2.2.
(a) if F−F(0)is homogeneous of degree γ >0, then SOL(F−F(0))# = int SOL(F −F(0))∗. Moreover, q∈int SOL(F−F(0))∗ iff Wq(F) ={0};
(b) if FEAs(F)=∅, then V(F) ={0}. The inverse implication holds ifF is pseudomonotone;
(c) if F is copositive, thenU−F(0)(F) = U(F). Moreover, F(0) +q∈U(F)# iff Uq(F) ={0}; (d) if F isc-homogeneous, thenSOL(F−F(0))is a cone.
Proof. (a): The equality follows from Exercise 6.22 of [34] and the remainder is obvious.
(b): If ¯x∈FEAs(F) andv∈V(F), thenF(¯x), v ≤0, a contradiction ifv= 0. For the inverse implication see Theorem 2.4.4 from [8].
(c): It is obvious.
(d): Let 0=x∈SOL(F −F(0)) be given. By hypothesis c(t||x||c(||x||d)
d)[F(x)−F(0)] =F(tx)−F(0) for allt >0.
Multiplying F(x)−F(0)≥0 and F(x)−F(0), x= 0 by c(t||x||c(||x||d)
d) and c(t||x||c(||x||d)t
d) respectively, we obtain that
tx∈SOL(F−F(0)) for allt >0.
Lemma 2.3(Basic Lemma). Let{xk}be a sequence of solutions to(PVIPk)such thatxkd=σk and xσk
k →v.
(a) If each Fk isc-homogeneous, thenv∈W(F)∩SOL(F−F(0) +τvd)∩Δd. (i) IfF is also a G(d)-function, thenv∈SOL(F−F(0))∩Δd;
(ii) if eachFk =Hk+qk is also copositive andF(x) =H+q, thenv∈Wq(H)∩Δd; (b) if each Fk isc-Mor´e, thenv∈U(F)∩Δd.
If each Fk =Hk+qk is also copositive andF =H+q, then v∈Uq(H)∩Δd; (c) if each Fk is pseudomonotone, thenv∈V(F)∩Δd.
Proof. (a): By hypothesis Fk(xk)−Fk(0) = c(σk)[Fk(xσk
k)−Fk(0)]. Replacing this in (PVIPk), dividing by c(σk)σk, and taking limit for x = 0 and x = σk y y
d with 0 = y ≥ 0 respectively, by the continuous convergence of {Fk} to F we obtain F(v)−F(0), v ≤ 0 and F(v)−F(0), y ≥ ydF(v)−F(0), v for ally≥0. By takingy=eifori∈I, we getv∈W(F) and by (2.2) we conclude thatv∈SOL(F−F(0) +τvd).
(i): IfF ∈G(d), then necessarilyτv= 0.
(ii): If each Fk =Hk+qk is copositive, then so too areHk and H (see Prop.1.6). Clearly, H ∈ G(d) and by (i) we getτv = 0, thus, 0 ≤v⊥H(v)−H(0)≥0. By taking x= 0 in (PVIPk) and by the copositivity of Hk we obtain 0≥ Hk(xk) +qk, xk=Hk(xk)−Hk(0), xk+Hk(0) +qk, xk ≥ Hk(0) +qk, xk.On dividing byσk and taking limit we obtainH(0) +q, v ≤0.
(b): By hypothesis we have Fk(xk)−Fk(0), xk ≥ c(σk)Fk(xσk
k)−Fk(0), xk. Replacing this inequality
in (PVIPk) forx= 0 we get 0≥ Fk(xk)−Fk(0), xk+Fk(0), xk ≥c(σk)Fk(xσk
k)−Fk(0), xk+Fk(0), xk. On dividing byc(σk)σkand taking limit, by the continuous convergence of{Fk}toFwe getF(v)−F(0), v ≤0.
If eachFk =Hk+qk is copositive, then we proceed similarly as in (a).
(c): If x≥0 is arbitrary, then there exists kx ∈Nsuch that (PVIPk) holds for all k ≥kx and by hypothesis we get Fk(x), x−xk ≥0. Dividing byσk and taking limit, by the continuous convergence of{Fk}to F we
obtainF(x), v ≤0.
We can characterize the boundedness of a nonempty setA⊆Rn by means of its asymptotic cone A∞ or its d-normalized asymptotic set A∞d as follows: Ais bounded iffA∞ ={0} iffA∞d =∅ (see [2,11]). Therefore, for studying the boundedness of the solution set to (NCP) we now obtain bounds for the asymptotic sets/cones of this set. These bounds are related to those for affine variational inequality problems from [8,32] and for multivalued complementarity problems from [11].
Corollary 2.4.
(a) ForH beingc-homogeneous:
q∈RnSOL(H+q)∞⊆SOL(H−H(0)).
If in addition, H is copositive, then SOL(H+q)∞⊆Wq(H).
(b) ForH beingc-Mor´e:
q∈RnSOL(H+q)∞d ⊆U(H)∩Δd.
If in addition, H is copositive, then SOL(H+q)∞d ⊆Uq(H)∩Δd. (c) If F is pseudomonotone andSOL(F)=∅, thenSOL(F)∞= V(F).
Proof. (a): Letq∈Rn be fixed andv∈SOL(H+q)∞. Ifv = 0, the assertion is trivial. Ifv= 0, there exists {xk} ⊆SOL(H +q) and tk ↓ 0 such thattkxk → v. Clearly, H(xk)−H(0) =c(||xk||d)[H(||ttkxk
kxk||d)−H(0)].
By replacing this equality inH(xk) +q≥0 andH(xk) +q, xk= 0, dividing byc(||xk||d) andc(||xk||d)||xk||d
respectively, and taking limit we obtain 0≤ ||v||vd ⊥H(||v||v
d)−H(0)≥0. By Proposition2.2(d) we conclude that v∈SOL(H−H(0)).
If H is also copositive, from 0 = H(xk)−H(0), xk+H(0) +q, xk ≥ H(0) +q, xk we conclude that H(0) +q, v ≤0.
(b): Let q ∈Rn be fixed andv ∈ SOL(H+q)∞d . There exists a sequence {xk} such that||xk||d → +∞ and
xk
||xk||d →v. By setting σk :=||xk||d and since eachxk is solution to (PVIPk) forFk =H+q for allk, by the first part of (b) of the Basic Lemma for suchFk, we conclude thatv∈U(H)∩Δd.
IfH is also copositive, then the second bound follows by using the last part of (b) of the Basic Lemma.
(c): IfF is pseudomonotone, then it is well-known that (see Lem. 2.1 from [22]) SOL(F) =
x≥0
¯
x≥0 :F(x), x−x¯ ≥0 . (2.3)
Since each right-hand set is closed and convex and SOL(F)=∅, by properties of asymptotic cones (see Props. 3.9 and 3.23 from [34]) we conclude that SOL(F)∞=
x≥0{x¯≥0 : F(x), x−x¯ ≥0}∞= V(F).
3. Main existence results
In this section, we obtain coercive existence results for (NCP). To do this, in (PVIPk) we consider that Fk=F for allk,i.e., we approximate (VIP) by the following sequence of problems
findxk∈Dk : F(xk), x−xk ≥0 ∀x∈Dk. (VIPk) We now recall an existence theorem for (VIP), which is related to this approximation (see Th. 4.2 from [25]).
We give the proof for reader’s convenience.
Proposition 3.1 (Hartman-Stampacchia). A necessary and sufficient condition that there exist a solution to(VIP) is that there exists a numberksuch that ||xk||d < σk.
Proof. If there exists a solutionxto (VIP), thenxis a solution to (VIPk) whenever||x||d< σk.
If we suppose that||xk||d< σk for somek, thenxk is also a solution to (VIP). Indeed, givenx∈Rn+, we get w=xk+ε(x−xk)∈Dk forε >0 sufficiently small. Thus, 0≤ F(xk), w−xk=εF(xk), x−xk, and since
xis arbitrary we conclude thatxk∈SOL(F).
From this proposition we conclude that all the sequences{xk} of solutions to (VIPk) satisfying||xk||d=σk for all k must be specially studied. We deal with such sequences by using the Basic Lemma. To this end, by following the line of reasoning from [11] we give the next definition.
Definition 3.2. We denote byW the set of sequences{xk}inRn+, such that for eachk∈N,
xk solves problem (VIPk) and xkd=σk. (3.1) We point out that the second requirement of (3.1) is verified if SOL(F) is either empty or unbounded. Indeed, if SOL(F) =∅then||xk||d =σk for allkby the above reasoning. If SOL(F) is nonempty and unbounded, then for anykthere existsxk ∈SOL(F) such that||xk||d≥k, and then we putσk=||xk||d.
IfW is an empty set, then SOL(F) is nonempty and closed. The latter follows from the continuity ofF. The next result generalizes Theorem 3.8.6 from [3], which is a well-known existence theorem for copositive matrices sinceq∈int SOL(M)∗ iff Wq(M) ={0}(see Prop.2.2), and Theorem 1 from [12] for homogeneous of degreeγ >0 functions.
Theorem 3.3. Let F bec-homogeneous.
(a) If W(F) ={0}, thenF ∈Qb.
(b) If F =H+q is copositive, and Wq(H) ={0}, thenSOL(F)is nonempty and compact.
Proof. (a): Letq∈Rn be an arbitrary vector and suppose that there exists {xk} ∈ W for the functionF +q.
There is a vectorv such that up to subsequences xσk
k →v, and by (a) of the Basic Lemma for Fk =F+qfor allk, we conclude that 0=v∈W(F+q) = W(F), a contradiction. Thus, SOL(F+q) is nonempty and closed.
Its boundedness follows from Corollary2.4(a). Therefore,F ∈Qbsinceqwas arbitrary.
(b): Suppose that there exists{xk} ∈ W. There is a vectorv such that up to subsequences xσk
k →v. By (a) of the Basic Lemma for Fk =F for all k, we conclude that 0=v ∈ Wq(H), a contradiction. Thus, SOL(F) is
nonempty and closed. Its boundedness follows from Corollary2.4(a).
As a consequence of this theorem we extend and generalize various coercive existence results from the lit- erature: for generalized homogeneous functions (see [37]), for d-regular or strictly copositive functions being homogeneous of degreeγ >0 (see [19] and [31] respectively), and for lineard-Garc´ıa functions (see [9]). To do this, similarly as in [37], this time forF beingc-homogeneous, we define the constantsμ∗inf andμ∗sup as follows:
0≤μ∗inf= lim inf
||x||d→+∞
x≥0
|F(x), x|
||x||dc(||x||d) and μ∗sup= lim sup
||x||d→+∞
x≥0
|F(x), x|
||x||dc(||x||d) <+∞.
Corollary 3.4. Let F bec-homogeneous. ThenF ∈Qbunder any of the following conditions:
(a) the equation F(x)−F(0), x=−μhas no solutions (x, μ)∈Δd×[μ∗inf, μ∗sup];
(b) F ∈R(d) = G(d)∩R0.
Proof. (a): If 0=v ∈W(F), thenF(v)−F(0), v ≤ 0. By hypothesis we may assume thatvd = 1. Since μ∗inf ≤ |F(v)−F(0), v|=−F(v)−F(0), v ≤μ∗sup,we conclude that (v,F(v)−F(0), v) solves the equation in part (a), a contradiction. Thus, W(F) ={0} and the result follows from Theorem3.3.
(b): Letq∈Rn be an arbitrary vector and suppose that there exists{xk} ∈ Wfor the functionF+q. There is a vectorvsuch that up to subsequences xσk
k →v. By (a) of the Basic Lemma forFk =F+qfor allk, and since F+q∈G(d) we get 0=v∈SOL(F−F(0)), a contradiction to F ∈R0. Therefore, SOL(F+q) is nonempty and closed. Its boundedness follows from Corollary2.4(a). Thus,F ∈Qbsinceqwas arbitrary.
It is worth pointing out that part (a) holds if F is copositive andμ∗inf >0 whereas part (b) holds if F is strictly copositive (see Ex. 1.3(e)). Moreover, as a consequence of this corollary we generalize Corollary 2.1 from [10] given for the linear complementarity problem.
Corollary 3.5. Let F be ac-homogeneousG(d)-function. It holds that: F ∈Qb⇐⇒F ∈R0⇐⇒F ∈R(d).
Proof. If F ∈ Qb, then by Proposition 2.2(d) we get SOL(F−F(0)) = {0}, thus, F ∈ R0 and F ∈ R(d) = G(d)∩R0. IfF ∈R(d), thenF ∈R0 and by the above corollary we conclude thatF ∈Qb. Theorem 3.6. Let F bec-Mor´e.
(a) If U(F) ={0}, thenF ∈Qb.
(b) If F =H+q is copositive, and Uq(H) ={0}, thenSOL(F)is nonempty and compact.
Proof. Proceed similarly as in the proof of Theorem 3.3, this time using (b) of the Basic Lemma and (b) of
Corollary2.4.
As a consequence of this theorem we deduce two coercive existence results from the literature for strongly monotone functions (see [19]) and for strongly copositive functions (see [23]). We also extend a result for strictly copositive functions satisfying the assumption of Ex.1.3(c) (see [37]).
Corollary 3.7. Consider the following statements:
(a) F is strongly monotone;
(b) F is strongly copositive;
(c) F isc-Mor´e and strictly copositive;
(d) F ∈Qb.
It holds that (a) =⇒(b) =⇒(c) =⇒(d).
Proof. (a)⇒(b): It is obvious.
(b)⇒(c): Use Proposition1.5(c).
(c)⇒(d): By hypothesis we have U(F) ={0}. The result follows from Theorem3.6(a).
We next result summarizes some well-known coercive existence results concerning the pseudomonotone case.
Condition (a) appears in [24]. Condition (b) is calledCrouzeix’s condition and appears in [4]. Condition (d) is called Karamardian’s condition, appears in [23], and implies the nonemptiness and compactness of SOL(F) without the pseudomonotonicity assumption. We give a proof by using our approach.
Theorem 3.8. Let F be pseudomonotone. The following four statements are equivalent:
(a) FEAs(F)=∅; (b) V(F) ={0};
(c) SOL(F)is a nonempty compact convex set;
(d) there exists a compact convex setK⊆Rn+ such that ∀x∈Rn+\K ∃z∈K: F(x), z−x<0.
Proof. (a)⇔(b): See Proposition2.2(b).
(b)⇒(c): Suppose that there exists {xk} ∈ W. There is a vector v such that up to subsequences xσk
k → v.
By (c) of the Basic Lemma forFk=F for allk, we conclude that 0=v∈V(F), a contradiction. Thus, SOL(F) is nonempty and closed. Its boundedness and convexity follow from Corollary2.4(c) and (2.3) respectively.
(b)⇐(c): It follows from Corollary2.4(c).
(c)⇒(d): TakeK= SOL(F).
(d)⇒(c): By hypothesisW=∅, thus, SOL(F) is nonempty and closed. By (d) we also get that SOL(F)⊆K.
Corollary 3.9. If F is strictly monotone andFEA(F)=∅, thenNCP(F)has a unique solution.
Proof. If 0=v ∈ V(F), then v ≥0 and F(x), v ≤0 for all x≥0. If ¯x∈ FEA(F), then F(¯x), v ≥0 and henceF(¯x), v= 0. By hypothesisF(¯x+v)−F(¯x),(¯x+v)−x¯ >0, thus,F(¯x+v), v>0, a contradiction.
Therefore, V(F) ={0}and SOL(F)=∅by Theorem3.8. For the uniqueness see Proposition 3.2 from [19].
