Growth behavior of two classes of merit functions for symmetric cone complementarity problems

27  Download (0)

Full text

(1)

Journal of Optimization Theory and Applications, vol. 141, pp. 167-191, 2009

Growth behavior of two classes of merit functions for symmetric cone complementarity problems

Shaohua Pan1

School of Mathematical Sciences South China University of Technology

Guangzhou 510640, China E-mail: shhpan@scut.edu.cn

Jein-Shan Chen 2 Department of Mathematics National Taiwan Normal University

Taipei, Taiwan 11677 E-mail: jschen@math.ntnu.edu.tw

December 18, 2007 (revised March 29, 2008)

Abstract. In the solution methods of the symmetric cone complementarity problem (SCCP), the squared norm of a complementarity function serves naturally as a merit function for the problem itself or the equivalent system of equations reformulation. In this paper, we study the growth behavior of two classes of such merit functions, which are induced by the smooth EP complementarity functions and the smooth implicit La- grangian complementarity function, respectively. We show that, for the linear symmetric cone complementarity problem (SCLCP), both the EP merit functions and the implicit Lagrangian merit function are coercive if the underlying linear transformation has the P -property; for the general SCCP, the EP merit functions are coercive only if the un- derlying mapping has the uniform Jordan P -property, whereas the coerciveness of the implicit Lagrangian merit function requires an additional condition for the mapping, for example, the Lipschitz continuity or the assumption as in (46).

Key words. Symmetric cone complementarity problem, Jordan algebra, EP merit func- tions, implicit Lagrangian function, coerciveness.

1The author’s work is partially supported by the Doctoral Starting-up Foundation (B13B6050640) of GuangDong Province.

2Member of Mathematics Division, National Center for Theoretical Sciences, Taipei Office. The author’s work is partially supported by National Science Council of Taiwan.

(2)

1 Introduction

Given a Euclidean Jordan algebra A = (V, ◦, h·, ·i) with “◦” denoting the Jordan product and V being a finite-dimensional vector space over the real field R with the inner product h·, ·i. Let K be a symmetric cone in V and F : V → V be a continuous mapping. The symmetric cone complementarity problem (SCCP) is to find ζ ∈ V such that

ζ ∈ K, F (ζ) ∈ K, hζ, F (ζ)i = 0. (1)

The model provides a simple unified framework for various existing complementarity problems such as the nonlinear complementarity problem over nonnegative orthant cone (NCP), the second-order cone complementarity problem (SOCCP) and the semidefinite complementarity problem (SDCP), and hence has extensive applications in engineering, economics, management science, and other fields; see [1, 18, 26] and references therein.

When F (ζ) = L(ζ) + b with L : V → V being a linear transformation and b ∈ V, the SCCP becomes the linear complementarity problem over symmetric cones (SCLCP):

ζ ∈ K, L(ζ) + b ∈ K, hζ, L(ζ) + bi = 0. (2) Recently, there is much interest in the study of merit functions or complementarity functions associated with symmetric cones and the development of the merit function approach or the smoothing method for solving the SCCP. For example, Liu, Zhang and Wang [17] extended a class of merit functions proposed in [7] to the SCCP, Kong, Tun- cel and Xiu [16] studied the extension of the implicit Lagrangian function proposed by Mangasarian and Solodov [19] to symmetric cones; Kong, Sun and Xiu [14] proposed a regularized smoothing method by the natural residual complementarity function associ- ated with symmetric cones; and Huang and Ni [9] developed a smoothing-type algorithm with the regularized CHKS smoothing function over the symmetric cone.

A mapping φ : V × V → V is called a complementarity function associated with the symmetric cone K, if the following equivalence holds:

φ(x, y) = 0 ⇐⇒ x ∈ K, y ∈ K, hx, yi = 0. (3) By Propositions III.4.4–4.5 and Theorem V.3.7 of [5], the Euclidean Jordan algebra V and the corresponding symmetric cone K can be written as

V = V1× V2× · · · × Vm and K = K1× K2× · · · × Km, (4) where each Ai := (Vi, ◦, h·, ·i) is a simple Euclidean Jordan algebra and Ki is the sym- metric cone in Vi. Moreover, for x = (x1, . . . , xm), y = (y1, . . . , ym) ∈ V with xi, yi ∈ Vi,

x ◦ y = (x1◦ y1, . . . , xm◦ ym) and hx, yi = hx1, y1i + · · · + hxm, ymi.

(3)

Therefore, the characterization (3) of complementarity function is equivalent to

φ(x, y) = 0 ⇐⇒ xi ∈ Ki, yi ∈ Ki, hxi, yii = 0 for all i = 1, 2, . . . , m. (5) This means that, if φ is a complementarity function associated with the cone Ki, i.e.

φ(xi, yi) = 0 ⇐⇒ xi ∈ Ki, yi ∈ Ki, hxi, yii = 0, (6) then for any x = (x1, . . . , xm) ∈ V and y = (y1, . . . , ym) ∈ V with xi, yi ∈ Vi,

φ(x, y) = (φ(x1, y1), φ(x2, y2), . . . , φ(xm, ym))

is exactly a complementarity function associated with the cone K. Consequently, the SCCP can be reformulated as the following system of equations:

Φ(ζ) :=



φ(ζ1, F1(ζ)) ...

φ(ζm, Fm(ζ))

 = 0, (7)

which naturally induces a merit function f : V → R+ for the SCCP, defined as f (ζ) := 1

2kΦ(ζ)k2 = 1 2

Xm i=1

kφ(ζi, Fi(ζ))k2.

In the rest of this paper, corresponding to the Cartesian structure of V, we always write F = (F1, . . . , Fm) with Fi: V → Vi and ζ = (ζ1, . . . , ζm) with ζi ∈ Vi.

The merit function f is often involved in the design of the merit function methods or the equation reformulation methods for the SCCP. For these methods, the coerciveness of f plays a crucial role in establishing the global convergence results. In this paper, we will study the growth behavior of two classes of such merit functions, which respectively correspond to the EP-functions introduced by Evtushenko and Purtov [2] and the im- plicit Lagrangian function by Mangasarian and Solodov [19]. The EP-functions over the symmetric cone K were first introduced by Kong and Xiu [15], defined by

φα(x, y) := −x ◦ y + 1

2α[(x + y)]2 0 < α ≤ 1, (8) φβ(x, y) := −x ◦ y + 1

£(x)2 + (y)2¤

0 < β < 1 (9) where (·) denotes the minimum metric projection onto −K. They showed that φα and φβ are continuously differentiable and strongly semismooth complementarity functions associated with K. Later, Kong, Tuncel and Xiu [16] extended the implicit Lagrangian function to the symmetric cone K and studied its continuous differentiability and strongly semismoothness. The function is defined as follows:

φMS(x, y) := x ◦ y + 1

©[(x − αy)+]2− x2+ [(y − αx)+]2− y2ª

, (10)

(4)

where α > 0 (6= 1) is a fixed constant, and (·)+ denotes the minimum metric projection on K. Particularly, for the implicit Lagrangian merit function for the SCCP, they pre- sented a mild stationary point condition and proved that it can provide a global error bound under the uniform Cartesian P -property and Lipschitz continuity of F .

This paper is mainly concerned with the growth behavior of the merit functions induced by the above three types of smooth complementarity functions, that is,

fα(ζ) := 1

2α(ζ, F (ζ))k2 = 1 2

Xm i=1

αi, Fi(ζ))k2, (11)

fβ(ζ) := 1

2β(ζ, F (ζ))k2 = 1 2

Xm i=1

βi, Fi(ζ))k2, (12)

fMS(ζ) := 1

2MS(ζ, F (ζ))k2 = 1 2

Xm i=1

MSi, Fi(ζ))k2. (13)

Specifically, we show that for the SCLCP (2), the EP merit functions fα and fβ and the implicit Lagrangian function fMS are coercive only if the linear transformation L has the P -property; for the general SCCP, fα and fβ are coercive if the mapping F has the uniform Jordan P -property, but the coerciveness of fMS needs an additional condition of F , for example, the Lipschitz continuity or the assumption as in (46). When V = Rn and

“◦” denotes the componentwise product of the vectors, the obtaining results precisely reduce to those of [24, Theorems 2.1 and 2.3] and [11, Theorem 4.1]. However, for the general Euclidean Jordan algebra even the Lorentz algebra, to the best of our knowledge, similar results have not been established for these merit functions.

Throughout this paper, k · k represents the norm induced by the inner product h·, ·i and int(K) denotes the interior of the symmetric cone K. For a vector space V of finite dimension, we write its dimension as dim(V). For any x ∈ V, (x)+ and (x) denotes the metric projection of x onto K and −K, respectively, i.e., (x)+:= argminy∈K{kx − yk}.

2 Preliminaries

This section recalls some concepts and materials of Euclidean Jordan algebras that will be used in the subsequent analysis. More detailed expositions of Euclidean Jordan al- gebras can be found in Koecher’s lecture notes [13] and the monograph by Faraut and Kor´anyi [5]. Besides, one can find excellent summaries in the articles [21, 8, 22].

A Euclidean Jordan algebra is a triple (V, ◦, h·, ·iV), where (V, h·, ·iV) is a finite di- mensional inner product space over the real field R and (x, y) 7→ x ◦ y : V × V → V is a bilinear mapping satisfying the following three conditions:

(5)

(i) x ◦ y = y ◦ x for all x, y ∈ V;

(ii) x ◦ (x2◦ y) = x2◦ (x ◦ y) for all x, y ∈ V, where x2 := x ◦ x;

(iii) hx ◦ y, ziV = hy, x ◦ ziV for all x, y, z ∈ V.

We assume that there is an element e ∈ V such that x ◦ e = x for all x ∈ V and call e the unit element. Let ζ(x) be the degree of the minimal polynomial of x ∈ V, which can be equivalently defined as ζ(x) := min©

k : {e, x, x2, . . . , xk} are linearly dependentª

. Since ζ(x) ≤ dim(V), the rank of (V, ◦) is well defined by q := max{ζ(x) : x ∈ V}. In a Euclidean Jordan algebra A = (V, ◦, h·, ·iV), we denote K := {x2 : x ∈ V} by the set of squares. From [5, Theorem III.2.1], K is a symmetric cone. This means that K is a self-dual closed convex cone with nonempty interior int(K), and for any x, y ∈ int(K), there exists an invertible linear transformation T : V → V such that T (K) = K.

A Euclidean Jordan algebra is said to be simple if it is not the direct sum of two Euclidean Jordan algebras. By Propositions III.4.4 and III.4.5 and Theorem V.3.7 of [5], any Euclidean Jordan algebra is, in a unique way, a direct sum of simple Euclidean Jordan algebras. Moreover, the symmetric cone in a given Euclidean Jordan algebra is, in a unique way, a direct sum of symmetric cones in the constituent simple Euclidean Jordan algebras. Here are two popular examples of simple Euclidean Jordan algebras. One is the algebra Sn of n × n real symmetric matrices with the inner product hX, Y iSn := Tr(XY ) and the Jordan product X ◦ Y := (XY + Y X)/2, where Tr(X) is the trace of X and XY is the usual matrix multiplication of X and Y . In this case, the unit element is the identity matrix I in Snand the cone K is the set of all positive semidefinite matrices. The other is the Lorentz algebra Ln, also called the quadratic forms algebra, with V = Rn, h·, ·iV being the usual inner product in Rn and the Jordan product defined by

x ◦ y := (hx, yiRn, x1y2+ y1x2)T (14) for any x = (x1, x2), y = (y1, y2) ∈ R × Rn−1. Under this case, the unit element e = (1, 0), and the associate cone, called the Lorentz cone (or the second-order cone), is given by

K :=©

x = (x1, x2) ∈ R × Rn−1 : kx2k2 ≤ x1

ª.

Recall that an element c ∈ V is said to be idempotent if c2 = c. Two idempotents c and d are said to be orthogonal if c ◦ d = 0. One says that {c1, c2, . . . , ck} is a complete system of orthogonal idempotents if

c2j = cj, cj◦ ci = 0 if j 6= i, j, i = 1, 2, . . . , k, and Pk

j=1cj = e.

A nonzero idempotent is said to be primitive if it cannot be written as the sum of two other nonzero idempotents. We call a complete system of orthogonal primitive idempotents a Jordan frame. Then, we have the following spectral decomposition theorem.

(6)

Theorem 2.1 [5, Theorem III.1.2] Suppose that A = (V, ◦, h·, ·iV) is a Euclidean Jordan algebra with rank q. Then for each x ∈ V, there exist a Jordan frame {c1, c2, . . . , cq} and real numbers λ1(x), λ2(x), . . . , λq(x) such that x = Pq

j=1λj(x)cj.

The numbers λj(x) (counting multiplicities), which are uniquely determined by x, are called the eigenvalues of x. In the sequel, we write the maximum eigenvalue and the minimum eigenvalue of x as λmax(x) and λmin(x), respectively. Furthermore, the trace of x, denoted by tr(x), is defined as tr(x) :=Pq

j=1λj(x).

By [5, Proposition III.1.5], a Jordan algebra A = (V, ◦) over R with a unit element e ∈ V is Euclidean if and only if the symmetric bilinear form tr(x ◦ y) is positive definite.

Therefore, we may define an inner product h·, ·i on V by hx, yi := tr(x ◦ y), ∀ x, y ∈ V.

Let k · k be the norm on V induced by the inner product h·, ·i, namely, kxk := p

hx, xi =³Pq

j=1λ2j(x)

´1/2

, ∀ x ∈ V.

Then, from equation (12) of [20], it follows that

kx ◦ yk ≤ kxk · kyk, ∀ x, y ∈ V. (15)

For a given x ∈ V, we define the linear operator L : V → V by L(x)y := x ◦ y for every y ∈ V.

Since the inner product h·, ·i is associative by the associativity of tr(·) (see [5, Proposition II.4.3]), i.e., for all x, y, z ∈ V, it holds that hx, y ◦ zi = hy, x ◦ zi, the linear operator L(x) for each x ∈ V is symmetric with respect to h·, ·i in the sense that

hL(x)y, zi = hy, L(x)zi, ∀ y, z ∈ V.

We say that elements x and y operator commute if L(x) and L(y) commute, i.e., L(x)L(y) = L(y)L(x).

Let ϕ : R → R be a real-valued function. Then, it is natural to define a vector-valued function associated with the Euclidean Jordan algebra A = (V, ◦, h·, ·i) by

ϕV(x) := ϕ(λ1(x))c1+ ϕ(λ2(x))c2+ · · · + ϕ(λq(x))cq, (16) where x ∈ V has the spectral decomposition x = Pq

j=1λj(x)cj. The function ϕV is also called L¨owner operator in [22] and shown to inherit many properties from ϕ. Especially,

(7)

when ϕ(t) is chosen as max{0, t} and min{0, t} for t ∈ R, respectively, ϕV becomes the metric projection operator onto K and −K:

(x)+ :=

Xq j=1

max©

0, λj(x)ª

cj and (x) :=

Xq j=1

min©

0, λj(x)ª

cj. (17)

It is easy to verify that x = (x)++ (x), |x| = (x)+− (x)and kxk2 = k(x)+k2+ k(x)k2. An important part in the theory of Euclidean Jordan algebras is the Peirce decom- position theorem which is stated as follows.

Theorem 2.2 [5, Theorem IV.2.1] Let A = (V, ◦, h·, ·i) be a Euclidean Jordan algebra with rank q and {c1, c2, . . . , cq} be a Jordan frame in V. For i, j ∈ {1, 2, . . . , q}, define

Vii:=©

x ∈ V : x ◦ ci = xª

, Vij :=©

x ∈ V : x ◦ ci = 1

2x = x ◦ cjª

, i 6= j.

Then the space V is the orthogonal direct sum of subspaces Vij (i ≤ j). Furthermore, (a) Vij ◦ Vij ⊆ Vii+ Vjj;

(b) Vij ◦ Vjk ⊆ Vik if i 6= k;

(c) Vij ◦ Vkl = {0} if {i, j} ∩ {k, l} = ∅.

To close this section, we recall the concepts of the P -property and the uniform Jordan P -property for a linear transformation and a nonlinear mapping.

Definition 2.1 A linear transformation L : V → V is said to have the P -property if ζ and L(ζ) operator commute

ζ ◦ L(ζ) ∈ −K

¾

⇒ ζ = 0.

Definition 2.2 A mapping F = (F1, . . . , Fm) with Fi : V → Vi is said to have

(i) the uniform Cartesian P -property if there is a positive scalar ρ such that for any ζ, ξ ∈ V, there is an index ν ∈ {1, 2, . . . , m} such that

ν − ξν, Fν(ζ) − Fν(ξ)i ≥ ρkζ − ξk2.

(ii) the uniform Jordan P -property if there is a positive scalar ρ such that for any ζ, ξ ∈ V, there is an index ν ∈ {1, 2, . . . , m} such that

λmax[(ζν− ξν) ◦ (Fν(ζ) − Fν(ξ))] ≥ ρkζ − ξk2.

Unless otherwise stated, in the subsequent analysis, we assume that A = (V, ◦, h·, ·i) is a simple Euclidean Jordan algebra of rank q and dim(V) = n.

(8)

3 Coerciveness of f

α

and f

β

In this section, we study under what conditions the EP merit functions fα and fβ are coercive. For this purpose, we first present several technical lemmas.

Lemma 3.1 [15, Lemma 3.1] For a given Jordan frame {c1, c2, . . . , cq}, if z ∈ V can be written as

z = Xq

i=1

zici+ X

1≤i<j≤q

zij

with zi ∈ R for i = 1, 2, . . . , q and zij ∈ Vij for 1 ≤ i < j ≤ q, then

z+ = Xq

i=1

sici + X

1≤i<j≤q

sij, z = Xq

i=1

wici + X

1≤i<j≤q

wij,

where si ≥ (zi)+ ≥ 0, 0 ≥ (zi) ≥ wi with si+ wi = zi for i = 1, . . . , q, and sij, wij ∈ Vij

with sij + wij = zij for 1 ≤ i < j ≤ q.

The following lemma summarizes some important inequalities involved in the maxi- mum eigenvalue and the minimum eigenvalue for any x ∈ V. Since their proofs can be found in [21, Lemma 14] and [23, Proposition 2.1], we here omit them.

Lemma 3.2 For any x, y ∈ V, the following inequalities always hold:

(a) λmin(x)kck2 ≤ hx, ci ≤ λmax(x)kck2 for any nonzero idempotent c;

(b) µ

λmax(x + y) − λmax(x)

≤ kyk and µ

λmin(x) − λmin(x + y)

≤ kyk;

(c) λmax(x + y) ≤ λmax(x) + λmax(y) and λmin(x + y) ≥ λmin(x) + λmin(y).

Using Lemmas 3.1–3.2, we may establish a lower bound for kφα(x, y)k and kφβ(x, y)k.

Lemma 3.3 Let φα and φβ be given by (8) and (9), respectively. Then, for any x, y ∈ V, α(x, y)k ≥ 2α − α2

max n

[(λmin(x))]2, [(λmin(y))]2 o

, (18)

β(x, y)k ≥ 1 − β2 max

n

[(λmin(x))]2, [(λmin(y))]2 o

. (19)

(9)

Proof. Suppose that x has the spectral decomposition x =Pq

i=1xici with xi ∈ R and {c1, c2, . . . , cq} being a Jordan frame. From Theorem 2.2, y ∈ V can be expressed by

y = Xq

i=1

yici+ X

1≤i<j≤q

yij, (20)

where yi ∈ R for i = 1, 2, . . . , q and yij ∈ Vij. Therefore, for any l ∈ {1, 2, . . . , q},

hcl, x ◦ yi = hcl◦ x, yi =

* xlcl,

Xq i=1

yici+ X

1≤i<j≤q

yij +

= xl

* cl,

Xq i=1

yici +

+ xl

*

cl, X

1≤i<j≤q

yij +

= xlyl, (21)

where the last equality is since hcl,P

1≤i<j≤qyij®

= 0 by the orthogonality of Vij (i ≤ j).

We next prove the inequality (18). From (20) and the spectral decomposition of x,

x + y = Xq

i=1

(xi+ yi)ci+ X

1≤i<j≤q

yij.

which together with Lemma 3.1 implies that

(x + y) = Xq

i=1

uici+ X

1≤i<j≤q

uij,

where ui ≤ (xi+ yi)≤ 0 for i = 1, 2, . . . , q and uij ∈ Vij. By this, we can compute D

cl, [(x + y)]2E

=

* cl

à q X

i=1

uici+ X

1≤i<j≤q

uij

!

, (x + y) +

=

* ulcl+

Ã

cl X

1≤i<j≤q

uij

! ,

Xq i=1

uici+ X

1≤i<j≤q

uij +

= u2l + ul

*

cl, X

1≤i<j≤q

uij +

+

* X

1≤i<j≤q

uij, cl Xq

i=1

uici +

+

*

cl X

1≤i<j≤q

uij, X

1≤i<j≤q

uij +

= u2l +

* cl,

à X

1≤i<j≤q

uij

!2+

, ∀ l = 1, 2, . . . , q, (22)

(10)

where the last equality is since hcl,P

1≤i<j≤quij®

= 0 by the orthogonality of Vij (i ≤ j).

Now, using equations (21)–(22), we obtain that hcl, −φα(x, y)i =

¿

cl, x ◦ y − 1

2α[(x + y)]2 À

= xlyl 1

u2l +

* cl,

à X

1≤i<j≤q

uij

!2+

≤ xlyl 1

2α[(xl+ yl)]2, ∀ l = 1, 2, . . . , q, (23) where the inequality is due to the following facts

ul≤ (xl+ yl) ≤ 0 and

* cl,

à X

1≤i<j≤q

uij

!2+

≥ 0.

On the other hand, from Lemma 3.2 (a) we have that

hcl, −φα(x, y)i ≥ λmin(−φα(x, y))kclk2 = λmin(−φα(x, y)), ∀ l = 1, 2, . . . , q. (24) Thus, combining (23) with (24), it follows that

2αλmin(−φα(x, y)) ≤ 2αxlyl− [(xl+ yl)]2, ∀ l = 1, 2, . . . , q.

Let λmin(x) = xν with ν ∈ {1, 2, . . . , q}. Then, we particularly have that

2αλmin(−φα(x, y)) ≤ 2αλmin(x)yν − [(λmin(x) + yν)]2. (25) We next proceed the proof by the two cases: λmin(x) ≤ 0 and λmin(x) > 0.

Case (i): λmin(x) ≤ 0. Under this case, we will prove the following inequality:

2αλmin(x)yν− [(λmin(x) + yν)]2 ≤ −(2α − α2)[(λmin(x))]2, (26) which, together with (25), immediately implies that

α(x, y)k ≥ |λmin(−φα(x, y))| ≥ (2α − α2)

[(λmin(x))]2. (27) In fact, if λmin(x) + yν ≥ 0, then we can deduce that

2αλmin(x)yν− [(λmin(x) + yν)]2 = 2α(λmin(x))(yν)+

≤ −(2α − α2)[(λmin(x))]2; and otherwise we will have that

2αλmin(x)yν − [(λmin(x) + yν)]2 = 2αλmin(x)yν − [(λmin(x) + yν)]2

≤ −(2α − α2)[λmin(x)]2

= −(2α − α2)[(λmin(x))]2.

(11)

Case (ii): λmin(x) > 0. For this case, the inequality (27) clearly holds.

Summing up the above discussions, the inequality (27) holds for any x, y ∈ V. In view of the symmetry of x and y in φα(x, y), we also have that

α(x, y)k ≥ (2α − α2)

[(λmin(y))]2

for any x, y ∈ V. Thus, the proof of the inequality (18) is completed.

We next prove the inequality (19). By the spectral decomposition of x, we have that (x)2 =Pq

i=1[(xi)]2ci, which in turn implies that

hcl, (x)2i = [(xl)]2, ∀ l = 1, 2, . . . , q. (28) In addition, from Lemma 3.1 and the expression of y given by (20), it follows that

y= Xq

i=1

vici+ X

1≤i<j≤q

vij,

where vi ≤ (yi) ≤ 0 for i = 1, . . . , q and vij ∈ Vij. By the same arguments as (22),

hcl, (y)2i = v2l +

* cl,

à X

1≤i<j≤q

vij

!2+

, ∀ l = 1, 2, . . . , q. (29)

Now, from equations (21), (28) and (29), it follows that hcl, −φβ(x, y)i =

¿

cl, x ◦ y − 1

£(x)2+ (y)2¤À

= xlyl 1

((xl))2+ vl2+

* cl,

à X

1≤i<j≤q

vij

!2+

≤ xlyl 1

£((xl))2+ (vl)2¤

≤ xlyl 1

£((xl))2+ ((yl))2¤

, ∀ l = 1, 2, . . . , q, where the first inequality is due to the nonnegativity of hcl, (P

1≤i<j≤qvij)2i, and the second one is since vl ≤ (yl)≤ 0. On the other hand, by Lemma 3.2 (a),

hcl, −φβ(x, y)i ≥ λmin(−φβ(x, y))kclk2 = λmin(−φβ(x, y)), ∀ l = 1, 2, . . . , q.

Combining the last two inequalities immediately leads to λmin(−φβ(x, y)) ≤ xlyl 1

£((xl))2+ ((yl))2¤

, ∀ l = 1, 2, . . . , q.

(12)

Let λmin(x) = xν with ν ∈ {1, . . . , q} and suppose that λmin(x) ≤ 0. Then, λmin(−φβ(x, y)) ≤ λmin(x)yν 1

£((λmin(x)))2+ ((yν))2¤

= [(λmin(x))][(yν)] − 1

£((λmin(x)))2+ ((yν))2¤

= − [β(λmin(x))− (yν)]2− (1 − β2)[(λmin(x))]2

≤ −(1 − β2)

[(λmin(x))]2, which in turn implies that

β(x, y)k ≥ |λmin(−φβ(x, y))| ≥ (1 − β2)

[(λmin(x))]2. (30) If λmin(x) = xν > 0, then the inequality (30) is obvious. Thus, (30) holds for any x, y ∈ V.

In view of the symmetry of x and y in φβ(x, y), we also have β(x, y)k ≥ |λmin(−φβ(x, y))| ≥ (1 − β2)

[(λmin(y))]2 for any x, y ∈ V. Consequently, the desired result follows. 2

The following proposition characterizes an important property for the smooth EP- complementarity functions φα and φβ under a unified framework.

Proposition 3.1 Let φα and φβ be given as in (8) and (9), respectively. Let {xk} ⊂ V and {yk} ⊂ V be the sequences satisfying one of the following conditions:

(i) either λmin(xk) → −∞ or λmin(yk) → −∞;

(ii) λmin(xk), λmin(yk) > −∞, λmax(xk), λmax(yk) → +∞ and kxk◦ ykk → +∞.

Then, kφα(xk, yk)k → +∞ and kφβ(xk, yk)k → +∞.

Proof. If Case (i) is satisfied, then the assertion is direct by Lemma 3.3. In what follows, we will prove the assertion under Case (ii). Notice that in this case the sequences {xk}, {yk} and {xk+ yk} are all bounded below since λmin(xk), λmin(yk) > −∞ and

λmin(xk+ yk) ≥ λmin(xk) + λmin(yk) > −∞.

Therefore, the sequences©

[(xk+ yk)]2ª

, {((xk))2} and {((yk))2} are bounded. Since kxk◦ ykk → +∞, we must have λmin(xk◦ yk) → −∞ or λmax(xk◦ yk) → +∞.

(13)

If λmin(xk◦ yk) → −∞ as k → ∞, then by Lemma 3.2(b) there hold that λmin(−φα(x, y)) = λmin

·

(xk◦ yk) − 1

2α((xk+ yk))2

¸

≤ λmin(xk◦ yk) + 1

°°((xk+ yk))2°

° , λmin(−φβ(x, y)) = λmin

·

(xk◦ yk) − 1

¡((xk))2+ ((yk))2¢¸

≤ λmin(xk◦ yk) + 1

°°((xk))2+ ((yk))2°

° ,

which together with the boundedness of k((xk+yk))2k and k((xk))2+((yk))2k implies that λmin(−φα(xk, yk)) → −∞ and λmin(−φβ(xk, yk)) → −∞. Since

α(xk, yk)k ≥ |λmin(−φα(x, y))| and kφβ(xk, yk)k ≥ |λmin(−φβ(x, y))|, we immediately obtain that kφα(xk, yk)k → +∞ and kφβ(xk, yk)k → +∞.

If λmax(xk◦ yk) → +∞ as k → ∞, from Lemma 3.2 (c) it then follows that λmax(−φα(x, y)) = λmax

·

(xk◦ yk) − 1

2α((xk+ yk))2

¸

≥ λmax(xk◦ yk) − 1

°°((xk+ yk))2°

° , λmax(−φβ(x, y)) = λmax

·

(xk◦ yk) − 1

¡((xk))2+ ((yk))2¢¸

≥ λmax(xk◦ yk) − 1

°°((xk))2+ ((yk))2°

° ,

which, by the boundedness of k((xk + yk))2k and k((xk))2 + ((yk))2k, implies that λmax(−φα(xk, yk)) → +∞ and λmax(−φβ(xk, yk)) → +∞. Noting that

α(xk, yk)k ≥ |λmax(−φα(xk, yk))| and kφβ(xk, yk)k ≥ |λmax(−φβ(xk, yk))|, we readily obtain that kφα(xk, yk)k → +∞ and kφβ(xk, yk)k → +∞. 2

When V = Rn with “◦” being the componentwise product of the vectors, the condi- tion kxk◦ ykk → +∞ automatically holds if λmax(xk), λmax(yk) → +∞, and Proposition 3.1 reduces to the result of [10, Lemma 2.5] for the NCPs. But for the general Euclidean Jordan algebra, this condition is necessary as illustrated by the following example.

Example 3.1. Consider the Lorentz algebra Ln = (Rn, ◦, h·, ·iRn) introduced in Section 2. Assume that n = 3 and take the sequences {xk} and {yk} as follows:

xk =

k k 0

 and yk=

k

−k 0

for each k.

(14)

It is easy to verify that λmin(xk) = 0, λmin(yk) = 0, λmax(xk), λmax(yk) → +∞, but kxk◦ykk 9 +∞. For such {xk} and {yk}, by computation we have that kφα(xk, yk)k = 0 and kφβ(xk, yk)k = 0, i.e. the conclusion of Proposition 3.1 does not hold.

In addition, in the subsequent analysis, we also use the continuity of Jordan product stated by the following lemma. Since the proof can be found in [9] or [20], we omit it.

Lemma 3.4 Let {xk} and {yk} be the sequences such that xk → ¯x and yk → ¯y when k → ∞. Then, we have that xk◦ yk → ¯x ◦ ¯y.

Now we are in a position to establish the coerciveness of fα and fβ. Assume that A = (V, ◦, h·, ·i) is a general Euclidean Jordan algebra. We first consider the SCLCP case.

Theorem 3.1 Let fα and fβ be given by (11) and (13), respectively. If F (ζ) = L(ζ) + b with the linear transformation L having the P -property, then fα and fβ are coercive.

Proof. Let {ζk} be a sequence such that kζkk → +∞. We only need to prove that fαk) → +∞ and fβk) → +∞. (31) By passing to a subsequence if necessary, we assume that ζk/kζkk → ¯ζ, and consequently (L(ζk) + b)/kζkk → L(¯ζ). If λmink) → −∞, then from Proposition 3.1 it follows that αk, L(ζk) + b)k, kφβk, L(ζk) + b)k → +∞, which in turn implies (31).

Now assume that {ζk} is bounded below. We argue that the sequence {L(ζk) + b} is unbounded by contradiction. Suppose that {L(ζk) + b} is bounded. Then,

L(¯ζ) = lim

k→∞

L(ζk) + b

kk = 0 ∈ K.

Since {ζk} is bounded below and λmaxk) → +∞ by kζkk → +∞, there is an element d ∈ V such that (ζ¯ k− ¯d)/kζk− ¯dk ∈ K for each k. Noting that K is closed, and we have

k→∞lim

ζk− ¯d

k− ¯dk = lim

k→∞

k− ¯d)/kζkk

°°ζk/kζkk − ¯d/kζkk°

° = ζ¯

k¯ζk = ¯ζ ∈ K.

Thus, ¯ζ ∈ K, L(¯ζ) ∈ K and ¯ζ ◦ L(¯ζ) = 0. From Proposition 6 of [8], it follows that ¯ζ and L(¯ζ) operator commute. This together with ¯ζ ◦ L(¯ζ) = 0 ∈ −K and the P -property of L implies that ¯ζ = 0, yielding a contradiction to k¯ζk = 1. Hence, the sequence {L(ζk) + b}

is unbounded. Without loss of generality, assume that kL(ζk) + bk → +∞.

If λmin(L(ζk) + b) → −∞, then using Proposition 3.1 yields the desired result of (31).

We next assume that the sequence {L(ζk) + b} is bounded below. We prove that ζk

kk L(ζk) + b

kk 9 0. (32)

(15)

Suppose that (32) does not hold, then from Lemma 3.4 it follows that ζ ◦ L(¯¯ ζ) = lim

k→+∞

ζk− d

kk L(ζk) + b − d

kk = 0 ∀d ∈ V. (33)

Since {ζk} and {L(ζk) + b} are bounded below and λmaxk), λmax(L(ζk) + b) → +∞, there is an element ˜d such that ζk− ˜d ∈ K and L(ζk) + b − ˜d ∈ K for each k. Therefore,

ζk− ˜d

kk ∈ K and L(ζk) + b − ˜d

kk ∈ K, ∀ k.

Noting that K is closed and ¯ζ = limk→∞ ζk− ˜kkd and L(¯ζ) = limk→∞ L(ζk)+b− ˜kk d, we have

ζ ∈ K and L(¯¯ ζ) ∈ K. (34)

From (33) and (34) and Proposition 6 of [8], it follows that ¯ζ and L(¯ζ) operator commute.

Using the P -property of L and noting that ¯ζ ◦ L(¯ζ) = 0 ∈ −K, we then obtain ¯ζ = 0, which clearly contradicts k¯ζk = 1. Therefore, (32) holds. Since kζkk → +∞, we have kζk◦ (L(ζk) + b)k → +∞. Combining with λmink), λmin(L(ζk) + b) > −∞ and kk, kL(ζk) + bk → +∞, it follows that the sequences {ζk} and {L(ζk) + b} satisfy condition (ii) of Proposition 3.1. This means that the result of (31) holds. 2

Theorem 3.2 Let fα and fβ be defined as in (11) and (13), respectively. If the mapping F has the uniform Jordan P -property, then fα and fβ are coercive.

Proof. The proof technique is similar to that of [11, Theorem 4.1]. For completeness, we include it. Let {ζk} be a sequence such that kζkk → +∞. Corresponding to the Cartesian structure of V, we write ζk= (ζ1k, . . . , ζmk) with ζik ∈ Vi for each k. Define

J :=©

i ∈ {1, 2, . . . , m} | {ζik} is unboundedª .

Clearly, the set J 6= ∅ since {ζk} is unbounded. Let {ξk} be a bounded sequence with ξk= (ξ1k, . . . , ξmk) and ξik ∈ Vifor i = 1, 2, . . . , m, where ξikfor each k is defined as follows:

ξik =

½ 0 if i ∈ J;

ζik otherwise, i = 1, 2, . . . , m.

Since F has the uniform Jordan P -property, there is a constant ρ > 0 such that ρkζk− ξkk2 max

i=1,...,mλmax£

ik− ξik) ◦ (Fik) − Fik))¤

= λmax£

ζνk◦ (Fνk) − Fνk))¤

≤ kζνk◦ (Fνk) − Fνk))k

≤ kζνkkkFνk) − Fνk)k, (35)

Figure

Updating...

References

Related subjects :