CONTROL AND OPTIMIZATION
Volume 3, Number 4, December 2013 pp. 627–641
ERROR BOUNDS FOR SYMMETRIC CONE COMPLEMENTARITY PROBLEMS
Xin-He Miao
Department of Mathematics, School of Science, Tianjin University Tianjin 300072, China
Jein-Shan Chen
Department of Mathematics, National Taiwan Normal University Taipei 11677, Taiwan
Abstract. In this paper, we investigate the issue of error bounds for sym- metric cone complementarity problems (SCCPs). In particular, we show that the distance between an arbitrary point in Euclidean Jordan algebra and the solution set of the symmetric cone complementarity problem can be bounded above by some merit functions such as Fischer-Burmeister merit function, the natural residual function and the implicit Lagrangian function. The so-called R0-type conditions, which are new and weaker than existing ones in the lit- erature, are assumed to guarantee that such merit functions can provide local and global error bounds for SCCPs. Moreover, when SCCPs reduce to lin- ear cases, we demonstrate such merit functions cannot serve as global error bounds under general monotone condition, which implicitly indicates that the proposed R0-type conditions cannot be replaced by P -type conditions which include monotone condition as special cases.
1. Introduction. The symmetric cone complementarity problem (henceforth SC- CP) is to find a vector x ∈ V such that
x ∈ K, F (x) ∈ K and hx, F (x)i = 0, (1) where V is a Euclidean Jordan algebra, K ⊂ V is a symmetric cone (see Section 2 for details), h·, ·i denotes the usual Euclidean inner product and F is a continuous mapping from V into itself. When F reduces to a linear transformation L, i.e., F (x) = L(x) + q with q ∈ V, the above symmetric cone complementarity problem becomes
x ∈ K, L(x) + q ∈ K and hx, L(x) + qi = 0,
which is called a symmetric cone linear complementarity problem and denoted by SCLCP.
In this paper, we focus on the issue of error bounds for symmetric cone comple- mentarity problems. More specifically, we want to know, under what conditions,
2010 Mathematics Subject Classification. Primary: 65K10; Secondary: 90C33.
Key words and phrases. Error bounds, R0-type functions, merit function, symmetric cone complementarity problem.
The first author is supported by National Young Natural Science Foundation (No. 11101302) and The Seed Foundation of Tianjin University (No. 60302041). The second author is the corre- sponding author and his work is supported by National Science Council of Taiwan.
The reviewing process of the paper was handled by Naihua Xiu as the Guest Editor.
627
the distance between an arbitrary point x ∈ V and the solution set of SCCPs can be bounded above by a merit function. Recall that a function ψ : V → R is called a merit function for SCCPs if ψ(x) ≥ 0 for all x and ψ(x) = 0 ⇔ x solves SCCPs.
Error bounds for complementarity problems have received increasing attention in the recent literature because they play important roles in sensitivity analysis where the problem data is subject to perturbation, and convergence analysis of some well- known iterative algorithms [7,10,11,13] for solving the complementarity problems, see [16,20]. Usually, finding global error bound through the following merit function ψNR(x) := kφNR(x, F (x))k2 where φNR(x, y) = x − (x − y)+ ∀x, y ∈ V (2) is a popular way because it is easier to compute, where φNR is the natural residual complementarity function and z+ is the the metric projection of z ∈ V onto the symmetric cone K. There are other merit functions which can provide global er- ror bounds such as Fischer-Burmeister merit function and the implicit Lagrangian function. In particular, for symmetric cone complementarity problems, such merit functions are defined as follows.
ψFB(x) := 1
2kφFB(x, F (x))k2 where φFB(x, y) = (x + y) − (x2+ y2)12 ∀x, y ∈ V (3) and
ψMS(x) := 2αhx, F (x)i + {k(−αF (x) + x)+k2
−kxk2+ k(−αx + F (x))+k2− kF (x)k2}. (4) Here x2 = x ◦ x denotes the Jordan product of x and x, x12 is the unique element that satisfies (x12)2= x, k · k denotes the standard Euclidean norm, and α > 0 is a penalty parameter.
Issues regarding error bounds have been studied for classical linear or nonlinear complementarity problems. For linear complementarity problems (LCPs), ψNR(x), ψFB(x) and ψMS(x) are shown to be local error bounds for any LCPs [15, 16, 25], whereas ψNR(x) and ψMS(x) are shown to be global error bounds for LCPs under the condition of R0-matrix [15,19] or P -matrix [18]. In addition, Chen and Xiang [5] give a computable error bound for the P -matrix LCPs. In general, in order to obtain global error bounds for nonlinear complementarity problems (NCPs), ψNR(x) needs to satisfy some stronger conditions such as F being a uniform P -function and Lipschitz continuity, or F being a strongly monotone [4, 12]. Furthermore, the so-called R0-type conditions for NCPs are investigated by Chen in [2].
It is known that symmetric cone complementarity problems provide a unified framework for nonlinear complementarity problems (NCPs), semidefinite comple- mentarity problems (SDCPs) and second-order cone complementarity problems (SOCCPs). Along this line, there is some research work on error bounds for SCCP- s. For instance, Chen [3] gives some conditions towards error bounds and bounded level sets for SOCCPs; Pan and Chen [22] consider error bound and bounded level sets of a one-parametric class of merit functions for SCCPs; Kong, Tuncel and Xiu [14] study error bounds of the implicit Lagrangian ψMS(x) for SCCPs. In gener- al, one needs conditions such as F has the uniform Cartesian P -property and is Lipschitz continuous. Besides, Liu, Zhang and Wang [17] study error bounds of a class of merit functions for SCCPs, where the transformation F needs to be uniform P∗-property which is a more stringent condition. In this paper, motivated by [2],
we consider other conditions, which are indeed different from the aforementioned ones and called R0-type conditions, to find error bounds for SCCPs.
The paper is organized as follows. In section 2, some preliminaries on Euclidean Jordan algebra associated with symmetric cone are introduced. Moreover, we define a class of R0-type functions in Euclidean Jordan algebra V. In section 3, we show the same growth of Fischer-Burmeister merit function ψFB(x), the natural residual function ψNR(x) and the implicit Lagrangian function ψMS(x). In sections 4 and 5, we provide local and global error bounds for SCCPs or SCLCPs with R0-type conditions, respectively. Concluding remarks are given in section 6.
Throughout this paper, let R denote the space of real numbers. For an x ∈ V, (·)− be defined by x− := x+− x. In fact, x− is the metric projection of −x onto the symmetric cone K (see [26]). In this paper, we need the concept of BD-regular function [21]. For a locally Lipsctizian function H : V → V, the set
∂BH(x) = {lim ∇H(xk) : xk → x, xk ∈ DH}
is called the B-subdifferential of H at x, where DHdenotes the set of points where H is F -differentiable. The function H is said to be BD-regular at x if all the elements in ∂BH(x) are nonsingular. In addition, S denotes the solution set of SCCPs and we assume that S 6= ∅.
2. Preliminaries. In this section, we briefly review some basic concepts and back- ground materials on Euclidean Jordan algebra, which is a basic tool extensively used in the subsequent analysis. More details can be found in [6,26].
A triple (V, ◦, h·, ·i) (V for short) is called a Euclidean Jordan algebra where (V, h·, ·i) is a finite dimensional inner product space over R and (x, y) 7→ x ◦ y : V × V → V is a bilinear mapping satisfying
(i): x ◦ y = y ◦ x for all x, y ∈ V
(ii): x ◦ (x2◦ y) = x2◦ (x ◦ y) for all x, y ∈ V (iii): hx ◦ y, zi = hx, y ◦ zi for all x, y, z ∈ V
where x2 := x ◦ x, and x ◦ y is called the Jordan product of x and y. If a Jordan product only satisfies the conditions (i) and (ii) in the definition of Euclidean Jordan algebra V, the algebra V is said to be a Jordan algebra. Throughout the paper we assume that V is a Euclidean Jordan algebra with an identity element e and with the property x ◦ e = x for all x ∈ V. In a given Euclidean Jordan algebra V, the set of squares K := {x2: x ∈ V} is a symmetric cone [6, Theorem III.2.1]. This means that K is a self-dual closed convex cone and, for any two elements x, y ∈ int(K), there exists an invertible linear transformation Γ : V → V such that Γ(x) = y and Γ(K) = K. Consider the Euclidean Jordan algebra V and the convex cone K ⊂ V.
This K induces a partial order on V, i.e., for any x ∈ V, x ∈ K (x ∈ int(K)) ⇐⇒ x 0 (x 0).
An element c ∈ V such that c2 = c is called an idempotent in V; it is a primitive idempotent if it is nonzero and cannot be written as a sum of two nonzero idempo- tents. We say that a finite set {e1, e2, · · · , er} of primitive idempotents in V is a Jordan frame if
ei◦ ej= 0 for i 6= j, and
r
X
i=1
ei= e,
where r is called the rank of V. Now, we recall the spectral decomposition and Peirce decomposition of an element x in V.
Theorem 2.1. (The Spectral Decomposition Theorem) [6, Theorem III.1.2] Let V be a Euclidean Jordan algebra. Then there is a number r such that, for every x ∈ V, there exists a Jordan frame {e1, e2, · · · , er} and real numbers λ1, λ2, · · · , λr with
x = λ1e1+ · · · + λrer.
Here, the numbers λi (i = 1, · · · , r) are the eigenvalues of x and the expression λ1e1+ · · · + λrer is the spectral decomposition (or the spectral expansion) of x.
In a Euclidean Jordan algebra V, let k · k be the norm induced by inner product kxk :=phx, xi for any x ∈ V. Corresponding to the closed convex cone K, let ΠK
denote the metric projection onto K, that is, for an x ∈ V, x∗= ΠK(x) if and only if x∗∈ K and kx − x∗k ≤ kx − yk for all y ∈ K. It is well known that x∗is unique.
For any x ∈ V, let x+denote the metric projection ΠK(x) of x onto K in this paper.
Combining the spectral decomposition of x with the metric projection of x onto K, we have the expression of metric projection x+ as follows [9]:
x+= ΠK(x) = max{0, λ1}e1+ · · · + max{0, λr}er, and
x−= ΠK(x) = max{0, −λ1}e1+ · · · + max{0, −λr}er.
Further, we have x = x+− x−, hx+, x−i = 0 and x+◦ x− = 0. Corresponding to each x ∈ V, let λi(x) (i = 1, 2, · · · , r) denote the eigenvalues of x. In the sequel, we write
ω(x) := max
1≤i≤rλi(x) and ν(x) := min
1≤i≤rλi(x).
With these notations, we note that
−x ∈ K ⇐⇒ ω(x) ≤ 0 and x ∈ K ⇐⇒ ν(x) ≥ 0.
We want to point out that different elements x, y have their own Jordan frames in spectral decomposition, which are not easy to handle when we need to do operations for x and y. Thus, we need another so-called Peirce decomposition to conquer such difficulty. In other words, in Peirce decomposition, different elements x, y share the same Jordan frame. We elaborate them more as below.
The Peirce decomposition: Fix a Jordan frame {e1, e2, · · · , er} in a Euclidean Jordan algebra V. For i, j ∈ {1, 2, · · · , r}, we define the following eigenspaces
Vii:= {x ∈ V | x ◦ ei= x} = Rei
and
Vij :=
x ∈ V | x ◦ ei= 1
2x = x ◦ ej
for i 6= j.
Theorem 2.2. [6, Theorem IV.2.1] The space V is the orthogonal direct sum of spaces Vij(i ≤ j). Furthermore,
Vij◦ Vij⊂ Vii+ Vjj, Vij◦ Vjk⊂ Vik, if i 6= k,
Vij◦ Vkl= {0}, if {i, j} ∩ {k, l} = ∅.
Hence, given any Jordan frame {e1, e2, · · · , er}, we can write any element x ∈ V as x =
r
X
i=1
xiei+X
i<j
xij,
where xi ∈ R and xij ∈ Vij. The expression Pr
i=1xiei+P
i<jxij is called the Peirce decomposition of x.
Given a Euclidean Jordan algebra V with dim(V) = n > 1, from Proposition III 4.4-4.5 and Theorem V.3.7 in [6], we know that any Euclidean Jordan algebra V and its corresponding symmetric cone K are, in a unique way, a direct sum of simple Euclidean Jordan algebras and the constituent symmetric cones therein, respectively, i.e.,
V = V1× · · · × Vm and K = K1× · · · × Km,
where every Vi is a simple Euclidean Jordan algebra (that cannot be a direct sum of two Euclidean Jordan algebras) with the corresponding symmetric cone Ki for i = 1, · · · , m, and n = Pm
i=1ni (ni is the dimension of Vi). Therefore, for any x = (x1, · · · , xm)T and y = (y1, · · · , ym)T ∈ V with xi, yi∈ Vi, we have
x ◦ y = (x1◦ y1, · · · , xm◦ ym)T ∈ V and hx, yi = hx1, y1i + · · · + hxm, ymi.
We end this section with some concepts on R0-type functions, which are crucial to establishing global error bounds. First, for any x ∈ V, let λi(x)(i = 1, · · · , r) denote the eigenvalues of x and
ω(x) := max
1≤i≤rλi(x).
Definition 2.3. A function F : V → V is called
(a): an Rs0-function if for any sequence {xk} that satisfies kxkk → ∞, (−xk)+
kxkk → 0, (−F (xk))+
kxkk → 0, we have
lim inf
k→∞
ω(φNR(xk, F (xk))) kxkk > 0;
(b): an Rs01-function if for any sequence {xk} that satisfies kxkk → ∞, (−xk)+
kxkk → 0, (−F (xk))+
kxkk → 0, we have
lim inf
k→∞
hxk, F (xk)i kxkk > 0;
(c): an R02s -function if for any sequence {xk} that satisfies kxkk → ∞, (−xk)+
kxkk → 0, (−F (xk))+
kxkk → 0, we have
lim inf
k→∞
ω(xk◦ F (xk)) kxkk > 0.
From the property hx, yi ≤ ω(x ◦ y)kek2 (see [26, Proposition 2.1(ii)]) and the above concepts, it is not hard to see that Rs01 ⇒ Rs02. In addition, by applying the Peirce Decomposition Theorem, the following lemma shows another implication Rs0⇒ Rs02.
Lemma 2.4. If the function F : V → V is an Rs0-function, then F is an Rs02- function.
Proof. For the sake of simplicity, for any x, y ∈ V, we let
x u y := x − (x − y)+, x t y := y + (x − y)+.
It is easy to verify that x t y := y + (x − y)+= x + (y − x)+. Moreover, these are commutative operations with
(x u y) ◦ (x t y) = x ◦ y, x u y + x t y = x + y and
x t y − x u y = |y − x| ∈ K.
If we consider the element x u y = x − (x − y)+∈ V and apply Spectral decompo- sition (Theorem2.1), there exist a Jordan frame {e1, e2, · · · , er} and real numbers λ1, λ2, · · · , λr such that
x u y = λ1e1+ · · · + λrer.
On the other hand, considering the element x t y = x + (y − x)+ ∈ V and applying Peirce decomposition (Theorem2.2), we know
x t y =
r
X
i=1
xiei+X
i<j
xij
with xi∈ R and xij ∈ Vij. Without loss of generality, let λ1= ω(x u y). To proceed the arguments, we first establish an inequality:
x1≥ λ1. Note that
(x t y − x u y) =
r
X
i=1
(xi− λi)ei+X
i<j
xij ∈ K.
Thus, it follows that
hx t y − x u y, e1i = (x1− λ1)ke1k2≥ 0,
which yields x1 ≥ λ1. Now suppose Rs0 condition holds. Take a sequence {xk} satisfying the required condition in Definition2.3(c), i.e.,
kxkk → ∞, (−xk)+
kxkk → 0, (−yk)+
kxkk → 0, where yk:= F (xk). From Rs0condition, we have
lim inf
k→∞
ω(xku yk)
kxkk = lim inf
k→∞
λ1
kxkk > 0 and λ1> 0. (5) For the element xk◦ yk∈ V, applying Spectral decomposition (Theorem2.1) again, there exist a Jordan frame {f1, f2, · · · , fr} and real numbers µ1, µ2, · · · , µr with µ1≥ µ2≥ · · · ≥ µrsuch that
xk◦ yk = µ1f1+ · · · + µrfr. Then, we have ω(xk◦ yk) = µ1. On the other hand,
xk◦ yk = (xku yk) ◦ (xkt yk)
= (λ1e1+ · · · + λrer) ◦ (
r
X
i=1
xiei+X
i<j
xij)
=
r
X
i=1
λixiei+
r
X
i=1
λiei◦ (X
i<j
xij)
=
r
X
i=1
λixiei+
r
X
i=1
λi 2
X
i<j
xij. Hence,
λ1x1he1, e1i = hxk◦ yk, e1i
= µ1hf1, e1i + µ2hf2, e1i + · · · + µrhfr, e1i
≤ µ1hf1, e1i + µ1hf2, e1i + · · · + µ1hfr, e1i
≤ rµ1θ,
where θ = max{hf1, e1i, · · · , hfr, e1i}. This leads to µ1
kxkk ≥ λ1x1he1, e1i rθkxkk , which combining with the formula (5) implies that
lim inf
k→∞
ω(xk◦ yk)
kxkk = lim inf
k→∞
µ1
kxkk ≥ lim inf
k→∞
λ1x1he1, e1i rθkxkk > 0,
where the second inequality holds due to x1 ≥ λ1 > 0 and he1rθ,e1i > 0. Therefore, the implication Rs0⇒ R02s holds.
Next, we introduce the so-called weak R0-type functions, which will be used to establish bounded level sets for SCCPs.
Definition 2.5. A function F : V → V is called an R0w-function if for any sequence {xk} that satisfies
kxkk → ∞, lim sup
k→∞
ω((−xk)+) < ∞, lim sup
k→∞
ω((−F (xk))+) < ∞, we have
ω(xku F (xk)) → ∞.
When the mapping F is a linear mapping, that is, F (x) = L(x) + q for q ∈ V, Rs0-function and Rw0-function are equivalent to R0-property (or R0-matrix) of L (i.e., the SCLCP with q = 0 has a unique zero solution). Those proofs are similar to proofs for [2, Proposition 2.2]. Hence, we omit them. Moreover, by the definition 2.3and2.5, we have the following relation between Rs0and Rw0.
Theorem 2.6. For the function F : V → V, we have R0s=⇒ Rw0.
Proof. Suppose Rs0 condition holds. Take a sequence {xk} satisfying the required condition in Definition2.5, i.e.,
kxkk → ∞, lim sup
k→∞
ω((−xk)+) < ∞, lim sup
k→∞
ω((−F (xk))+) < ∞.
It follows that
kxkk → ∞, (−xk)+
kxkk → 0, (−yk)+ kxkk → 0.
By the definition of Rs0, we have lim inf
k→∞
ω(xku yk) kxkk > 0.
Combining with kxkk → ∞ implies that
ω(xku yk) → ∞.
Therefore, the implication Rs0⇒ Rw0 holds.
3. Growth behavior of ψFB, ψNR and ψMS. In this section, we show results that indicate the same growth of merit functions ψFB, ψNR and ψMS. The following theorem tells us that ψNR and ψMS have the same growth behavior.
Theorem 3.1. Let ψNR and ψMS be defined as in (2) and (4), respectively. For each α > 1, the following holds
2(α − 1)ψNR(x) ≤ ψMS(x) ≤ 2α(α − 1)ψNR(x) ∀x ∈ V. (6) Proof. To begin the proof, we denote
f (x, α) := − hαF (x), (x − αF (x))+− xi − 1
2k(x − αF (x))+− xk2.
We want to point out that there is another expression for f (x, α) as given below, see [8, Thm 3.1].
f (x, α) = max
y∈K−
αF (x) + 1
2(y − x), y − x
= −
αF (x) + 1
2((x − αF (x))+− x), (x − αF (x))+− x
(7)
≥ −
αF (x) + 1
2((x − F (x))+− x), (x − F (x))+− x
. Now, we compute
1
αf (x, α) = −hF (x), (x − αF (x))+− xi − 1
2αk(x − αF (x))+− xk2
= hx, F (x)i + 1
αhx − αF (x), (x − αF (x))+i
− 1
2αk(x − αF (x))+k2− 1 2αkxk2
= hx, F (x)i + 1
2α k(x − αF (x))+k2− kxk2 . Likewise,
f (x, 1) = −
F (x) +1
2((x − F (x))+− x), (x − F (x))+− x
and
αf (x,1
α) = − 1
2α k(−αx + F (x))+k2− kF (x)k2 . Combining the above two equations, we obtain an identity for ψMS(x)
ψMS(x) = 2α 1
αf (x, α) − αf (x,1 α)
. (8)
To show the desired two inequalities, we proceed by two steps. The first step is to verify the left-hand side of (6). To see this,
ψMS(x) = 2α 1
αf (x, α) − αf (x,1 α)
= 2α 1
αf (x, α) − f (x, 1)
+ 2α
f (x, 1) − αf (x,1 α)
≥ 2α
− hF (x), (x − F (x))+− xi − 1
2αk(x − F (x))+− xk2 +hF (x), (x − F (x))+− xi + 1
2k(x − F (x))+− xk2
+2α
f (x, 1) − αf (x,1 α)
= 2αα − 1
2α ψNR(x) + 2α
f (x, 1) − αf (x,1 α)
= (α − 1)ψNR(x) + 2α
− hF (x), (x − F (x))+− xi
−1
2k(x − F (x))+− xk2+F (x), (x − 1
αF (x))+− x +α
2k(x − 1
αF (x))+− xk2
≥ (α − 1)ψNR(x) + 2αα − 1 2α ψNR(x)
= 2(α − 1)ψNR(x),
where the first inequality follows from (7). Next, we verify the right-hand side of (6). To this end, we observe two things:
1
αf (x, α) − f (x, 1)
= −hF (x), (x − αF (x))+− xi − 1
2αk(x − αF (x))+− xk2 +hF (x), (x − F (x))+− xi +1
2k(x − F (x))+− xk2
= α − 1
2α ψNR(x) + 1
2αψNR(x) − 1
2αk(x − αF (x))+− xk2 +hF (x), (x − F (x))+− (x − αF (x))+i
= α − 1
2 ψNR(x) −(α − 1)2 2α ψNR(x)
− 1
2αk(x − αF (x))+− (x − F (x))+k2+ 1
αk(x − F (x))+− xk2
−1
αh(x − αF (x))+− x, (x − F (x))+− xi +hF (x), (x − F (x))+− (x − αF (x))+i
= α − 1 2 ψNR(x)
− 1
2αk(α − 1)(x − (x − F (x))+) + (x − αF (x))+− (x − F (x))+k2
− h(x − F (x))+− x + F (x), (x − αF (x))+− (x − F (x))+i
≤ α − 1 2 ψNR(x) and
f (x, 1) − αf (x,1 α)
= −hF (x), (x − F (x))+− xi −1
2k(x − F (x))+− xk2 +hF (x), (x − 1
αF (x))+− xi +α 2
(x − 1
αF (x))+− x
2
= max
y∈K−hF (x) + 1
2(y − x), y − xi + α min
y∈Kh1
αF (x) +1
2(y − x), y − xi
≤ −
F (x) +1
2((x − F (x))+− x), (x − F (x))+− x
+D
F (x) +α
2((x − F (x))+− x), (x − F (x))+− xE
= α − 1
2 k(x − F (x))+− xk2
= α − 1
2 ψNR(x).
The above two expressions together with the identity (8) yield ψMS(x) ≤ 2α α − 1
2 ψNR(x) +α − 1 2 ψNR(x)
= 2α(α − 1)ψNR(x).
Thus, the proof is complete.
The same growth of the ψNR and ψFB is already proved in [1, Proposition 3.1]
that we present it as below theorem.
Theorem 3.2. Let ψNR and ψFB be defined as in (2) and (3), respectively. Then,
2 −√
2
kφNR(x, y)k ≤ kφFB(x, y)k ≤ 2 +√
2
kφNR(x, y)k for any x, y ∈ V.
Combining Theorem3.1and Theorem3.2, we can reach the conclusion that ψFB, ψNR, and ψMS have the same growth.
4. Local Error Bounds. This section contains the proofs of boundedness of level set and local error bounds for SCCPs. To obtain such properties, we first present the definition of the local error bound and two lemmas that play important roles in the following analysis.
Definition 4.1. For the residual function r(x) = kφNR(x, F (x))k, the function r(x) is a local error bound if there exist constants c > 0 and δ > 0 such that for each x ∈ {x ∈ V | d(x, S) ≤ δ}, there holds
d(x, S) ≤ cr(x),
where S denote the solution set of the problem (1) and d(x, S) = infy∈Skx − yk.
Lemma 4.2. Let φFB be defined as in (3). Then, for any x, y ∈ V, k(φFB(x, y))+k2≥ 1
2 k(−x)+k2+ k(−y)+k2 . Proof. This is the result of [22, Lemma 5.2].
Lemma 4.3. Let φNR be defined as in (2). Then, for any x, y ∈ V, there is a constant β > 0 such that
kφNR(x, y)k2≥β
2 k(−x)+k2+ k(−y)+k2 .
Proof. By applying Theorem 3.2 and Lemma 4.2, the desired result is obtained immediately.
For simplicity, we denote r(x) := kφNR(x, F (x))k in the remaining part of this paper and call it a residual function. It is trivial that r(x) = (ψNR(x))12 for any x ∈ V.
Theorem 4.4. Consider the residual function r(x) = kφNR(x, F (x))k. If F is an Rw0-function, then the level set L(γ) := {x ∈ V| r(x) ≤ γ} is bounded for all γ ≥ 0.
Proof. Suppose there is an unbounded sequence {xk} ⊆ L(γ) for some γ > 0. If lim sup ω((−xk)+) = ∞, then (through a subsequence) k(−xk)+k → ∞, by Lemma 4.3, which implies that r(xk) → ∞. This contradicts the boundness of L(γ). A similar contradiction ensues if lim sup ω((F (−xk))+) = ∞. Thus, for the specified unbounded sequence {xk} satisfing the condition in Definition 2.5, by Definition 2.5, we also obtain that ω(φNR(xk, F (xk))) → ∞. With r(xk) = kφNR(xk, F (xk))k, it is easy to see that r(xk) → ∞. This leads to a contradiction. Consequently, the level set L(γ) := {x ∈ V| r(x) ≤ γ} is bounded for all γ ≥ 0.
Theorem 4.4 says that r(x) has property of bounded level set under R0-type condition. However, r(x) cannot serve as local error bound under R0-type condition only, even for NCP case which is a special case of SCCPs. An example is given in [2] that illustrates r(x) cannot be a local error bound for an R0-type NCP (F is R0-type function). More specifically, consider F : R → R with F (x) = x3, it is easy to verify that F is an Rs0-function, and the corresponding NCP has a bounded solution set S = {0}. However, r(x) cannot be a local error bound. A question arises here: Under what additional condition, can r(x) be a local error bound for SCCPs?
The following theorem answers this question by providing a sufficient condition for SCCPs.
Theorem 4.5. Consider the residual function r(x) = kφNR(x, F (x))k. Suppose that the solution set S of SCCPs is nonempty and that φNR is BD-regular at all solutions of SCCPs. Then, r(x) is a local error bound if it has a local bounded level set.
Proof. Since r(x) has a local bounded level set, there exists ε > 0 such that the level set L(ε) = {x| r(x) ≤ ε} is bounded. Thus the set L(ε) = {x| r(x) ≤ ε}
is compact. Suppose that the conclusion is wrong. Then, there exists a sequence {xk} ⊂ L(ε) such that
r(xk)
dist(xk, S) → 0 as k → ∞.
Here dist(xk, S) denotes the distance between xk and S. Thus, r(xk) → 0 and it follows from compactness of L(ε) that there is a convergent subsequence. Without loss of generality, let {xk} be a convergent sequence, and ¯x be its limit, that is, xk→ ¯x ∈ L(ε). Then, r(¯x) = 0, which implies ¯x ∈ S. It turns out that
r(xk)
kxk− ¯xk→ 0 as k → ∞. (9)
From [24], we know that φNR(x, F (x)) is semismooth. By applying [21, Proposition 3] and BD-regular property of φNR(x, F (x)), there exist constants c > 0 and δ > 0 such that r(x) ≥ ckx − ¯xk for any x with kx − ¯xk < δ. This contradicts (9).
Consequently, the residual function r(x) is a local error bound for SCCPs.
Results analogous to Theorem4.5can be stated for the other two merit functions, with Theorem3.1and3.2, we may conclude that ψFBand ψMSare local error bounds for SCCPs.
5. Global Error Bound. In this section, we find a global error bound for SCCPs by using an R0-type condition and a BD-regular condition. To achieve these results, we present the following definition and a technical lemma.
Definition 5.1. For the residual function r(x) = kφNR(x, F (x))k, the function r(x) is a global error bound if there exist constant c > 0 such that for each x ∈ V,
d(x, S) ≤ cr(x),
where S denote the solution set of the problem (1) and d(x, S) = infy∈Skx − yk.
Lemma 5.2. Let {xk} be any sequence such that kxkk → ∞. If F is an Rs0- function, then
lim inf
k→∞
r(xk) kxkk > 0.
Proof. Suppose that the result is false. There exists a subsequence xnkwith kxnkk →
∞ such that
r(xnk)
kxnkk → 0. (10)
From Lemma4.3, it follows that (−xnk)+
kxnkk → 0 and (−F (xnk))+
kxnkk → 0.
This together with the definition of Rs0-function implies lim inf
k→∞
ω(φNR(xnk, F (xnk))) kxnkk > 0,
which contradicts the formula (10). Consequently, we have the desired result.
Theorem 5.3. Suppose that F is an Rs0-function and that φNR is BD-regular at all solutions of SCCPs. Then, there exists a κ > 0 such that for any x ∈ V
dist(x, S) ≤ κr(x),
where S is the solution set of SCCPs, dist(x, S) denotes the distance between x and S.
Proof. By the definition of R0s-function, Theorem 4.4 and Theorem 4.5, we claim that r(x) is a local error bound so there exist c > 0 and δ > 0 such that
r(x) < δ ⇒ d(x, S) ≤ cr(x).
Suppose r(x) does not have the global error bound property. Then, there exists xk such that for any fixed ¯x ∈ S,
kxk− ¯xk ≥ dist(xk, S) > kr(xk) (11)
for all k. Clearly, the inequality r(xk) < δ cannot hold for infinitely many k0s, else kr(xk) < d(xk, S) ≤ cr(xk) implies that k ≤ c for infinitely many k0s. Therefore, r(xk) ≥ δ for all large k. Now,
kxk− ¯xk ≥ d(xk, S) ≥ kr(xk) ≥ kδ
for infinitely many k0s. This implies that kxkk → ∞. Now divide the inequality and take the limit k → ∞, we have
1 = lim
k→∞
kxk− ¯xk kxkk ≥ lim
k→∞kr(xk) kxkk → ∞,
where the last implication holds because F is an Rs0-function and Lemma5.2. This clearly is a contradiction.
Adopting Theorem 3.1, Theorem 3.2 and Theorem 5.3, we have the following corollary for SCCPs.
Corollary 1. Under the same conditions as in Theorem5.3, both the merit function ψFB(x) and the implicit Lagrangian function ψMS(x) are global error bounds for SCCPs.
When F : V → V is a linear mapping, that is, F (x) = L(x) + q with q ∈ V, if L has R0-property, then r(x) being a local error bound can be improved as being a global error bound for SCLCPs, which is shown in the following theorem.
Theorem 5.4. Suppose that r(x) is a local error bound for SCLCPs and the linear transformation L has R0-property. Then, there exists k > 0 such that dist(x, S) ≤ kr(x) for every x ∈ V.
Proof. Suppose that the conclusion is false. Then, for any integer k > 0, there exists an xk ∈ Rn such that dist(xk, S) > kr(xk). Let z(xk) denote the closest solution of SCLCPs to xk. Choosing a fixed solution x0∈ S, we have
kxk− x0k ≥ kxk− z(xk)k ≥ dist(xk, S) > kr(xk). (12) Since r(x) is a local error bound, it implies that there exist some integer K > 0 and δ > 0 such that for all k > K, r(xk) > δ. If not, then for every integer K > 0 and any δ > 0, there exist some k > K such that r(xk) ≤ δ. By property of local error bound of r(x), we have
δ
kkxk− z(xk)k > δr(xk) ≥ kxk− z(xk)k.
Thus, we obtain δk > 1. As k goes to infinity, this leads to a contradiction. Conse- quently, r(xk) > δ. This together with (12) implies that kxk−x0k ≥ kxk−z(xk)k >
kδ which says that kxkk → ∞ as k → ∞. Now, we consider the sequence {kxxk
kk}.
There exist a subsequence {xki} such that
i→∞lim xki
kxkik = x.
Hence, it follows from (12) that 1 = lim
i→∞
kxki− x0k kxkik
≥ lim
i→∞kir(xki) kxkik
= lim
i→∞ki
xki
kxkik−
xki
kxkik− L xki
kxkik − q kxkik
+
= lim
i→∞kikx − (x − L(x))+k.
This implies that kx − (x − L(x))+k = 0, which shows that x is a nonzero solution of SCLCPs with q ∈ V. It contradicts the R0-property of L. Then, the proof is complete.
There is one thing worthy of pointing it out. If we replace the condition of R0- property into the monotonicity for the linear transformation L, the conclusion of Theorem 5.4 may not hold. This can be illustrated by the following example by using the implicit Lagrangian function ψMS.
Example 5.1. Let L : R2→ R2 be defined as L :=
" 1
√2 −√1
2
−√1
2
√1 2
#
and q :=
2 0
.
It is easy to prove that the symmetric cone is R2+and the corresponding SCLCP has a unique solution x∗ = (0, 0)T. Choosing xk =
√k 2,√k
2
T
, k ≥ 0 gives F (xk) = L(xk) + q = (2, 0)T. Then, for any k > 2√
2α with α > 1, we have ψMS(xk) = 4α
k
√2
+
−2α + k
√2
2 +
k
√2
2
− 2
k
√2
2
− 4
= 4 α2− 1 .
However, dist(xk, S) = kxkk = k. This implies dist(xk, S) > ψMS(xk) as k → ∞, which explains that ψMS(x) cannot serve as global error bound for SCLCPs.
6. Concluding Remarks. In this paper, we have established some local and global error bounds for symmetric cone complementarity problems under the so- called R0-type conditions. These new results on error bounds are based on the Fischer-Burmeister merit function, the natural residual function, and the implic- it Lagrangian function. For symmetric cone linear complementarity problems, we have pointed out that global error bound do not exist under the condition of lin- ear transformation L being monotone, which implicitly indicates that the proposed R0-type conditions cannot be replaced by monotone condition as special cases.
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Received May 2013; 1strevision June 2013, final revision August 2013.
E-mail address: xinhemiao@tju.edu.cn E-mail address: jschen@math.ntnu.edu.tw
