DOI 10.1007/s11590-010-0257-z O R I G I NA L PA P E R
The same growth of FB and NR symmetric cone complementarity functions
Shujun Bi · Shaohua Pan · Jein-Shan Chen
Received: 22 July 2010 / Accepted: 22 October 2010 / Published online: 3 November 2010
© Springer-Verlag 2010
Abstract We establish that the Fischer–Burmeister (FB) complementarity function and the natural residual (NR) complementarity function associated with the symmetric cone have the same growth, in terms of the classification of Euclidean Jordan alge- bras. This, on the one hand, provides an affirmative answer to the second open question proposed by Tseng (J Optim Theory Appl 89:17–37,1996) for the matrix-valued FB and NR complementarity functions, and on the other hand, extends the third impor- tant inequality of Lemma 3.1 in the aforementioned paper to the setting of Euclidean Jordan algebras. It is worthwhile to point out that the proof is surprisingly simple.
Keywords Symmetric cone· FB and NR complementarity functions · Growth
1 Introduction
LetA = (V, ◦, ·, ·V) be a Euclidean Jordan algebra (see Sect. 2for details). Let K be the set of all squares in V. Given the continuously differentiable mappings F, G : V→V, we consider the symmetric cone complementarity problem (SCCP): to
J.-S. Chen is Member of Mathematics Division, National Center for Theoretical Sciences, Taipei Office.
S. Bi· S. Pan
Department of Mathematics, South China University of Technology, Guangzhou 510640, China e-mail: [email protected]
S. Pan
e-mail: [email protected]
J.-S. Chen (
B
)Department of Mathematics, National Taiwan Normal University, Taipei 11677, Taiwan e-mail: [email protected]
find a vectorζ ∈ V such that
F(ζ ) ∈ K, G(ζ ) ∈ K, F(ζ ), G(ζ )V= 0. (1) This class of problems provides a unified framework for the classical nonlinear programming and complementarity problem [5] over the nonnegative orthant cone inRn, the second-order cone optimization and complementarity problem [1], and the semidefinite programming and complementarity problem [16,21], and becomes one of main research interests in the current optimization field; see, e.g., [4,6,10,12, 13,17–19,22].
Analogous to the three classes of special SCCPs above, the complementarity func- tion associated with the symmetric cone plays a crucial role in the development of merit function methods and smoothing (nonsmooth) Newton methods for solving the SCCPs. Recall that φ : V × V → V is called a symmetric cone complementary function if it satisfies the following equivalence:
φ(x, y) = 0 ⇐⇒ x ∈ K, y ∈ K, x, yV= 0. (2) With such a function, the SCCP can be reformulated as an unconstrained minimization
minζ∈V (ζ ) := 1
2 φ(F(ζ), G(ζ)) 2V, (3)
in the sense thatζ∗solves (1) if and only if it is a solution of (3) with zero optimal value, where · V denotes the norm induced by the inner product·, ·V. If is continuously differentiable, then the efficient unconstrained minimization methods can be applied for (3) to yield a solution of (1). This method is often known as the merit function approach.
Two most popular choices forφ are the natural residual (NR) symmetric cone com- plementarity functionφNR and the Fischer–Burmeister (FB) symmetric cone comple- mentarity functionφFB, respectively, defined as
φNR(x, y) := x − (x − y)+ ∀x, y ∈ V (4) and
φFB(x, y) := (x + y) − (x2+ y2)1/2 ∀x, y ∈ V, (5) where z+means the metric projection of z∈ V onto the symmetric cone K, x2= x ◦x denotes the Jordan product of x and itself, and x1/2means the unique square root of x∈ K, i.e., (x1/2)2= x. The squared norm of φFBinduces a smooth merit function with global Lipschitz continuous gradients (see [9,15]). This implies that finding solutions to (1) is equivalent to seeking solutions of the unconstrained smooth minimization problem
minζ∈V FB(ζ ) := 1
2φFB(F(ζ ), G(ζ ))2
V. (6)
However, in order to establish the convergence rate of the merit function method for the SCCPs based on (6), the key is to prove thatφFB andφNR has the same order of growth, i.e., to show that there exist constants c1 > 0 and c2 > 0 such that for all x, y ∈ V,
c1 φNR(x, y) 2V≤ φFB(x, y) 2V≤ c2 φNR(x, y) 2V. (7) WhenA is the Euclidean Jordan algebra R equipped with the multiplication of real numbers, Tseng showed in [20, Lemma 3.1] that inequality (7) holds with c1= 2−√
2 and c2= 2 +√
2; whenA is the Jordan spin algebra Ln(see Example 2.3 in the next section), Pan et al. [14] recently established inequality (7) by contradiction. We note that for the case whereA is the n × n real symmetric matrix algebra (see Example 2.2 in the next section), in 1998 Tseng [21] proposed an open question “whether the FB functionφFB(F(ζ ), G(ζ ))2
Vis bounded above and below by a constant multiple of the NR functionφNR(F(ζ ), G(ζ ))2
V”, which is equivalent to asking whether or not inequality (7) holds under this case. To our best knowledge, until now this open question is not resolved.
In this paper, we show that (7) holds with c1= 2 −√
2 and c2= 2 +√
2, which does not only offer an affirmative answer to the open question of [21], but also extends the results of [20, Lemma 3.1] and [14] to the setting of symmetric cones. Particularly, the proof is surprisingly simpler than that of [20, Lemma 3.1] and [14]. As a direct consequence of (7), we also establish the global error bound property of the FB merit function for SCCPs.
2 Preliminaries
This section recalls some results on Euclidean Jordan algebras that will be used in the next section. More detailed expositions of Euclidean Jordan algebras can be found in the monograph by Faraut and Korányi [3] and Koecher’s lecture notes [7].
A Euclidean Jordan algebra is a triple(V, ◦, ·, ·V) where (V, ·, ·V) is a finite dimensional inner product space over the real number fieldR and (x, y) → x ◦ y : V × V → V is a bilinear mapping satisfying the following three conditions:
(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) x ◦ y, zV= y, x ◦ zVfor all x, y, z ∈ V.
Henceforth, we assume thatA = (V, ◦, ·, ·V) is a Euclidean Jordan algebra with an element e ∈ V (called the unit element) such that x ◦ e = x for all x ∈ V. By [3, Theorem III. 2.1], the set of squaresK :=
x2| x ∈ V
is a symmetric cone. In the following, we present three common examples of Euclidean Jordan algebras.
Example 2.1 ConsiderRnwith the (usual) inner product and Jordan product defined respectively as
x, y =
n i=1
xiyi and x◦ y = x ∗ y ∀x, y ∈ Rn
where xi denotes the i th component of x, etc., and x∗ y denotes the component- wise product of vectors x and y. Then, Rn is a Euclidean Jordan algebra with the nonnegative orthantRn+as its cone of squares.
Example 2.2 The algebraSn of n× n real symmetric matrices. Let Sn×nbe the space of all n× n real symmetric matrices with the trace inner product and Jordan product, respectively, defined by
X, Y T:= Tr(XY ) and X ◦ Y := 1
2(XY + Y X) ∀X, Y ∈ Sn×n.
Then,(Sn×n, ◦, ·, ·T) is a Euclidean Jordan algebra, and we write it as Sn. The cone of squaresSn+×ninSnis the set of all positive semidefinite matrices inSn×n. Example 2.3 The Jordan spin algebraLn. Consider Rn (n > 1) with the inner product·, · and Jordan product
x◦ y :=
x, y
x0¯y + y0¯x
for any x = (x0; ¯x), y = (y0; ¯y) ∈ R×Rn−1. We denote the Euclidean Jordan algebra (Rn, ◦, ·, ·) by Ln. The cone of squares, called the Lorentz cone (or the second-order cone), is given byLn+:=
(x0; ¯x) ∈ R × Rn−1| x0≥ ¯x . For x ∈ V, let m(x) := min
k: {e, x, x2, . . . , xk} are linearly dependent and define the rank ofA by r := max{m(x) : x ∈ V}. Recall that an element c ∈ V is idempotent if c2 = c, and it is a primitive idempotent if it is nonzero and cannot be written as a sum of two nonzero idempotents. One says that a finite set{c1, c2, . . . , ck} of primitive idempotents inV is a Jordan frame if
cj◦ ci = 0 if j = i for all j, i = 1, 2, . . . , k, and k
j=1cj = e.
Now we may state the second version of the spectral decomposition theorem.
Theorem 2.1 ([3, Theorem III.1.2]) LetA be a Euclidean Jordan algebra with rank r.
Then, for every x ∈ V, there exist a Jordan frame {c1, c2, . . . , cr} and real numbers λ1(x), λ2(x), . . . , λr(x), arranged in the decreasing order λ1(x) ≥ · · · ≥ λr(x), such that
x= λ1(x)c1+ λ2(x)c2+ · · · + λr(x)cr.
The numbersλj(x) (counting multiplicities), which are uniquely determined by x, are called the eigenvalues of x, and tr(x) =r
j=1λj(x) is called the trace of x.
Letφ : R → R be a scalar valued function. Then, it is natural to define a vector valued functionφV: V → V associated with the Euclidean Jordan algebra A [8,19]
by
φV(x) := φ(λ1(x))c1+ φ(λ2(x))c2+ · · · + φ(λr(x))cr,
where x ∈ V has the spectral decomposition x = r
j=1λj(x)cj. This function is also called Löwner’s operator in recognition of Löwner’s contribution. Whenφ(t) = t+:= max{0, t} for t ∈ R, φV(x) becomes the metric projector operator over K:
x+= (λ1(x))+c1+ (λ1(x))+c2+ · · · + (λr(x))+cr ∀x ∈ V;
whileφ(t) = t−:= min{0, t} for t ∈ R, it is the metric projector operator over −K
x−= (λ1(x))−c1+ (λ1(x))−c2+ · · · + (λr(x))−cr ∀x ∈ V.
In the sequel, we let|x| be Löwner’s operator induced by φ(t) = |t| for t ∈ R. Then,
|x| = x+− x−= 2x+− x = x − 2x− ∀x ∈ V. (8)
Recall that a Euclidean Jordan algebra is said to be simple if it is not the direct sum of two Euclidean Jordan algebras. It is easy to see thatSnandLnare simple Euclidean Jordan algebras, whereas the Euclidean Jordan algebra in Example 2.1 is not simple. LetHn×ndenote the space of n×n complex Hermitian matrices, Qn×nthe space of n× n quaternion Hermitian matrices, and O3×3the space of 3× 3 octonion Hermitian matrices.
Theorem 2.2 ([3, Theorem V.3.7]) Suppose that A = (V, ◦, ·, ·V) is a simple Euclidean Jordan algebra of rank r ≥ 3. Then, A is isomorphic to one of the following
(i) The algebraSnof n× n real symmetric matrices given by Example 2.2;
(ii) The algebraHnof all n× n complex Hermitian matrices with trace inner prod- uct x, yT := Tr(xy∗) and Jordan product x ◦ y := 12(xy + yx) for any x, y ∈ Hn×n;
(iii) The algebraQnof all n× n quaternionic Hermitian matrices with trace inner productx, yT:= Tr(xy∗) and Jordan product x ◦ y := 12(xy + yx) for any x, y ∈ Qn×n;
(iv) The algebraO3 of all 3× 3 octonionic Hermitian matrices with trace inner productx, yT:= Tr(xy∗) and Jordan product x ◦ y := 12(xy + yx) for any x, y ∈ O3×3;
(v) The Jordan spin algebraLngiven by Example 2.3.
where the notation “∗” means the conjugate transpose, Tr(xy) denotes the trace of x y which is the multiplication of matrices x and y, anda means the real part of a.
Unless otherwise stated, in the rest of this paper, we assume thatA = (V, ◦, ·, ·V) is a simple Euclidean Jordan algebra, and denote · V, · and · Tthe norm induced by the inner product·, ·V, ·, · and ·, ·T, respectively. We also write xKy (respec- tively, x Ky) to mean x− y ∈ K (respectively, x − y ∈ intK).
3 Main result
To establish the main result of this paper, the following lemma plays an important role.
Lemma 3.1 (a) For any x, y ∈ V, if x K 0, y K0 and x K y, then x1/2K y1/2.
(b) For any u, v, w ∈ V, if w K0 and 2w2= u2+ v2, then there holds that
w K 1
2(u + v).
Proof (a) This is result of [6, Prop. 8], which is also implied by [8].
(b) Since u2+ v2− 2u ◦ v = (u − v) ◦ (u − v) ∈ K, using 2w2= u2+ v2yields
w2=1
2(u2+ v2) K 1
4(u2+ v2) +1
2u◦ v = 1
4(u + v)2.
From part(a) andw K0, this implies thatw K 12|u + v| K 21(u + v).
Proposition 3.1 LetLnbe the Euclidean Jordan algebra in Example 2.3. Then, (2 −√
2) φNR(x, y) ≤ φFB(x, y) ≤ (2 +√
2) φNR(x, y) , ∀x, y ∈ Ln. Proof Fix any x, y ∈ V. If φNR(x, y) = 0, then we also have φFB(x, y) = 0, and the desired result is immediate. Therefore, in the following arguments we assume that φNR(x, y) = 0. Using Eq. (8) and the definition ofφNR, it is not hard to see that
φNR(x, y) = 1
2[(x + y) − |x − y|].
This together with the definition ofφFBgives
φFB(x, y) = 2φNR(x, y) + |x − y| − (x2+ y2)1/2
= 2φNR(x, y) + z(x, y) (9)
where z(x, y) ≡ |x − y| − (x2+ y2)1/2. By Eq. (9) and the triangle inequality, it suffices to argue z(x, y) ≤√
2 φNR(x, y) , that is,
z(x, y) 2≤ 1
2 x + y − |x − y| 2. (10)
Substituting the expression of z(x, y) into (10), we obtain that (10) is equivalent to
|x − y| − (x2+ y2)1/22≤ 1
2 x + y − |x − y| 2
⇐⇒ x − y 2− 2|x − y|, (x2+ y2)1/2 + (x2+ y2)1/2 2
≤ 1 2
x + y 2+ x − y 2− 2|x − y|, x + y
⇐⇒ c(x, y) 2− 2c(x, y) − (x + y), |x − y| − 2x, y ≤ 0 (11) where c(x, y) ≡ (x2+ y2)1/2. Thus, to prove the desired result, it suffices to argue that inequality (11) holds. Indeed, since
c(x, y) 2= c(x, y)2, e = x2+ y2, e = x 2+ y 2,
we have
c(x, y) 2− 2c(x, y) − (x + y), |x − y| − 2x, y
= x − y 2− 2c(x, y) − (x + y), |x − y|
= |x − y|, −2c(x, y) + (x + y) + |x − y|. (12)
Applying Lemma3.1withw = c(x, y), u = (x + y) and v = |x − y|, we know
−2c(x, y) + (x + y) + |x − y| ∈ −Ln+.
This, along with|x − y| ∈ Ln+and Eq. (12), shows that inequality (11) holds.
Proposition 3.2 Suppose thatA = (V, ◦, ·, ·V) is a simple Euclidean Jordan alge- bra with the rank r ≥ 3. Then, it holds that
(2 −√
2) φNR(x, y) T≤ φFB(x, y) T≤ (2 +√
2) φNR(x, y) T ∀x, y ∈ V.
Proof By Theorem 2.2, it suffices to prove that the desired result holds forA = Sn, or Hn, or Qn, orO3. Fix any x, y ∈ V with V = Sn×n, or Hn×n, orQn×n, or O3×3. IfφNR(x, y) = 0, the result is direct. Thus, it suffices to consider the case of φNR(x, y) = 0. Note that for the simple Euclidean Jordan algebra Sn, orHn, orQn, orO3, we still have
φFB(x, y) = 2φNR(x, y) + z(x, y)
with z(x, y) ≡ |x − y| − (x2+ y2)1/2. By the triangle inequality, it suffices to prove
z(x, y) 2T≤ 2 φNR(x, y) 2T =1
2 x + y − |x − y| 2T. (13)
Using the definition of · 2Tand noting thatTr(uv∗) = Tr(uv) = Tr(vu) for all u, v ∈ V, an elementary computation yields that (13) is equivalent to
Tr
(x − y)2+ c(x, y)2− 2|x − y|c(x, y)
≤ 1 2Tr
(x + y)2+ (x − y)2− 2|x − y|(x + y)
⇐⇒ Tr
−|x − y|(2c(x, y) − (x + y)) + (x − y)2
≤ 0
⇐⇒1 2Tr
|x − y|
c(x, y) − (x + y) + |x − y|
2
≥ 0
⇐⇒1 2
|x − y|, c(x, y) −(x + y) + |x − y|
2
T
≥ 0 (14)
where c(x, y) ≡ (x2+ y2)1/2. Applying Lemma3.1withw = c(x, y), u = (x + y) andv = |x − y| yields that c(x, y) − ((x + y) + |x − y|)/2 K0. This together with
|x − y| K0 implies that inequality (14) holds. Thus, we complete the proof.
Combining Propositions3.1with3.2, we readily obtain the main result of this paper.
Theorem 3.1 Suppose thatA = (V, ◦, ·, ·V) is a simple Euclidean Jordan algebra.
LetφNRandφFBbe defined by (4) and (5), respectively. Then, it holds that (2 −√
2) φNR(x, y) V≤ φFB(x, y) V≤ (2 +√
2) φNR(x, y) V ∀x, y ∈ V.
By Theorem3.1, we may establish the global error bound property for the FB merit function of SCCPs under the jointly uniform Cartesian P-property of F and G. To this end, we next assume thatA is a direct product of simple Euclidean Jordan algebras:
A = A1× A2× · · · × Am,
where each Ai = (Vi, ◦, ·, ·Vi) is a simple Euclidean Jordan algebra with
m
i=1dim(Vi) = dim(V). Then, K = K1×K2×· · ·×KmwithKibeing a symmetric cone inVi. For any x, y ∈ V, we write x = (x1, . . . , xm), y = (y1, . . . , ym) with xi, yi ∈ Vi. Then,
x◦ y = (x1◦ y1, . . . , xm◦ ym) and x, yV= x1, y1V1+ · · · + xm, ymVm. Consequently, the SCCP (1) is equivalent to finding a vectorζ ∈ V such that
Fi(ζ ) ∈ Ki, Gi(ζ ) ∈ Ki, Fi(ζ ), Gi(ζ )Vi = 0, i = 1, 2, . . . , m (15) where F= (F1, . . . , Fm) and G = (G1, . . . , Gm) with Fi, Gi: V → Vi.
Definition 3.1 [2] The mappings F and G are said to have the jointly uniform Cartesian P-property if there exists a constantρ > 0 such that for any ζ, ξ ∈ V, there is an indexν ∈ {1, . . . , m} such that
Fν(ζ ) − Fν(ξ), Gν(ζ ) − Gν(ξ)Vν ≥ ρ ζ − ξ 2V.
Theorem 3.2 Suppose that F and G have the jointly uniform Cartesian P-property and are globally Lipschitz continuous with constants L1> 0 and L2> 0, respectively.
If the SCCP (1) has an optimal solution, sayζ∗, then 2−√
2
(2L1+ L2)2FB(ζ ) ≤ ζ − ζ∗ 2V≤ (2 +√
2)(L1+ L2)2
ρ2 FB(ζ ) ∀ζ ∈ V where the constantρ is same as in Definition3.1.
Proof Fix anyζ ∈ V. Let R(ζ ) ≡
φNR(F1(ζ ), G1(ζ )), . . . , φNR(Fm(ζ ), Gm(ζ ))
∈ V. Then, using Theorem 3.1 and noting that FB(ζ ) ≡ 12m
i=1 φFB(Fi(ζ ), Gi(ζ )) 2Vi, we get
2−√ 2
2 R(ζ ) 2V≤ FB(ζ ) ≤ 2+√ 2
2 R(ζ ) 2V. In addition, using the same arguments as in [9, Theorem 6.3], we have
1
2L1+ L2 R(ζ ) V≤ ζ − ζ∗ V≤ L1+ L2
ρ R(ζ ) V.
From the last two inequalities, we immediately obtain the desired result.
Acknowledgments S. Pan’s work was supported by National Young Natural Science Foundation (No.
10901058) and Guangdong Natural Science Foundation (No. 9251802902000001) and the Fundamental Research Funds for the Central Universities (SCUT). J.-S. Chen’s work was supported by National Science Council of Taiwan.
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