2012
where x2 = x◦x, and x1/2denotes the unique square root of x∈ K, i.e., x1/2◦x1/2= x. Compared with the function ϕscNR, this function has a remarkable advantage, namely, its squared norm induces a continuously differentiable merit function, and furthermore, the merit function has a globally Lipschitz continuous gradient; see [10, 12] for details. This will greatly facilitate the globalization of nonsmooth Newton methods based on ϕscFB.
It is known that when V is the space of all n × n symmetric matrices with a specific Jordan product,K corresponds to positive semidefinite cone, whereas when V is the IRnspace with a specific Jordan product,K corresponds to the Lorentz cone (also known as second-order cone), see [2]. Moreover, it was shown in [13] that ϕscFB is strongly semismooth under the aforementioned two cases. Whether such property holds for general Euclidean Jordan algebra has been an open question thereafter.
In this paper, we provide an almost-complete answer for it and explain why the incomplete part occurs.
2. Preliminaries
This section recalls some results on Euclidean Jordan algebras that will be used in subsequent analysis and definition of semismoothness . More detailed expositions of Euclidean Jordan algebras can be found in Koecher’s lecture notes [3] and the monograph by Faraut and Kor´anyi [2].
Let V be an n-dimensional vector space over the real field R, endowed with a bilinear mapping (x, y) 7→ x ◦ y from V × V into V. The pair (V, ◦) is called a Jordan algebra if
(i) x◦ y = y ◦ x for all x, y ∈ V,
(ii) x◦ (x2◦ y) = x2◦ (x ◦ y) for all x, y ∈ V.
Note that a Jordan algebra is not necessarily associative, i.e., x◦ (y ◦ z) = (x ◦ y) ◦ z may not hold for all x, y, z ∈ V. We call an element e ∈ V the identity element if x◦ e = e ◦ x = x for all x ∈ V. A Jordan algebra (V, ◦) with an identity element e is called a Euclidean Jordan algebra if there is an inner product,⟨·, ·⟩V, such that
(iii) ⟨x ◦ y, z⟩V =⟨y, x ◦ z⟩V for all x, y, z∈ V.
Given a Euclidean Jordan algebraA = (V, ◦, ⟨·, ·⟩V), we denote the set of squares as K :={
x2 | x ∈ V} .
By [2, Theorem III.2.1], K is a symmetric cone. This means that K is a self-dual closed convex cone with nonempty interior and for any two elements x, y ∈ int(K), there exists an invertible linear transformationT : V → V such that T (K) = K and T (x) = y.
For any given x∈ A, let ζ(x) be the degree of the minimal polynomial of x, i.e., ζ(x) := min
{
k :{e, x, x2,· · · , xk} are linearly dependent} .
Then the rank of A is defined as max{ζ(x) : x ∈ V}. In this paper, we use r to denote the rank of the underlying Euclidean Jordan algebra. Recall that an element c∈ V is idempotent if c2 = c. Two idempotents ci and cj are said to be orthogonal if ci ◦ cj = 0. One says that {c1, c2, . . . , ck} is a complete system of orthogonal idempotents if
c2j = cj, cj◦ ci = 0 if j̸= i for all j, i = 1, 2, · · · , k, and ∑k
j=1cj = e.
An idempotent is primitive if it is nonzero and cannot be written as the sum of two other nonzero idempotents. We call a complete system of orthogonal primitive idempotents a Jordan frame. Now we state the second version of the spectral decomposition theorem.
Theorem 2.1 ([2, Theorem III.1.2]). Suppose that A is a Euclidean Jordan algebra with the rank r. Then for any x∈ V, there exists a Jordan frame {c1, . . . , cr} and real numbers λ1(x), . . . , λr(x), arranged in the decreasing order λ1(x) ≥ λ2(x) ≥
· · · ≥ λr(x), such that
x = λ1(x)c1+ λ2(x)c2+· · · + λr(x)cr.
The numbers λj(x) (counting multiplicities), which are uniquely determined by x, are called the eigenvalues and tr(x) =∑r
j=1λj(x) the trace of x.
Since, by [2, Prop. III.1.5], a Jordan algebra (V, ◦) with an identity element e ∈ V is Euclidean if and only if the symmetric bilinear form tr(x◦ y) is positive definite, we may define another inner product on V by ⟨x, y⟩ := tr(x ◦ y) for any x, y ∈ V.
The inner product ⟨·, ·⟩ is associative by [2, Prop. II. 4.3], i.e., ⟨x, y ◦ z⟩ = ⟨y, x ◦ z⟩
for any x, y, z∈ V. Every Euclidean Jordan algebra can be written as a direct sum of so-called simple ones, which are not themselves direct sums in a nontrivial way.
In finite dimensions, the simple Euclidean Jordan algebras come from the following five basic structures.
Theorem 2.2 ([2, Chapter V.3.7]). Every simple Euclidean Jordan algebra is iso- morphic to one of the following.
(i) The Jordan spin algebraLn.
(ii) The algebra Sn of n× n real symmetric matrices.
(iii) The algebraHn of all n× n complex Hermitian matrices.
(iv) The algebra Qn of all n× n quaternion Hermitian matrices.
(v) The algebraO3 of all 3× 3 octonion Hermitian matrices.
Next we provide the definition of semismoothness [14] for readers’ convenience.
LetX and Y be two finite dimension inner product spaces over the field R. Suppose that Φ :X → Y is a locally Lipschitz continuous function, by Rademacher’s theo- rem, Φ is almost everywhere differentiable. We denote DΦ the set of points where Φ is differentiable and denote Φ′(x) the derivative of Φ at x if Φ is differentiable at x. The B-subdifferential of Φ at x, denoted by ∂BΦ(x), is the set of V such that V ={limk→∞Φ′(xk)}, where {xk} ∈ DΦ is a sequence converging to x. The Clarke’s generalized Jacobian of Φ at x is the convex hull of ∂BΦ(x), denoted by
∂Φ(x) = conv∂BΦ(x). In fact, there are several equivalent ways for defining the concept of semismoothness. Here is one of them.
Definition 2.3. Suppose that Φ :X → Y is a locally Lipschitz continuous function.
We say that Φ is semismooth at x if
(i) Φ is directionally differentiable at x;
(ii) for any x→ y and A ∈ ∂Φ(x), there have
Φ(x)− Φ(y) − A(y − x) = o(∥y − x∥).
3. Main results
As mentioned earlier, the Fischer-Burmeister SC complementarity function ϕsc
FB
defined as in (1.3) was shown to be strongly semismooth in [13] for classes (i) and (ii) of Theorem 2.2. Thus, we only need to check the remainder classes (iii)-(v). Let us start with class (iii).
Class(iii): The algebra Hn of n× n complex Hermitian matrices.
A square matrix A of complex entries is said to be Hermitian if A∗ := ¯AT = A, where ‘bar’ denotes the complex conjugate, and the superscript ‘T’ means the transpose. Let Hn be the set of all n× n complex Hermitian matrices. On Hn, we define the Jordan product and inner product by X ◦ Y := 12(XY + Y X) and
⟨X, Y ⟩ := trace(XY ). Then Hnis a Euclidean Jordan algebra of rank n with e being the n× n identity matrix I.
For example,H2 is the set which contains all [ α1 β
β¯ α2 ]
, α1, α2∈ R and β ∈ C.
We also know that each complex number a + bi can be represented as a 2× 2 real matrix:
a [ 1 0
0 1 ]
+ b
[ 0 1
−1 0 ]
,
where
[ 0 1
−1 0 ]
satisfies
[ 0 1
−1 0 ]2
=− [ 1 0
0 1 ]
. Hence, we can embed
[ α1 β β¯ α2
]
into an element inS4:
H2∋
[ α1 β β¯ α2
] 7−→
[ α1 0 0 α1
] [ a b
−b a ] [ a −b
b a
] [ α2 0 0 α2
]
∈ S4
where β = a + ib.
It is easy to check that this embedding is one-to-one and onto, and also preserves the Jordan algebra structures on the both sides by matrix block multiplication.
Therefore, we can view H2 as a Jordan sub-algebra of S4. For general n it is also true that Hn is a Jordan sub-algebra of S2n. In fact, the general embedding map is
given by
Hn ∋
α1 β · · · γ β¯ α2 · · · δ ... ... . .. ...
¯
γ ¯δ · · · αn
7−→
[ α1 0 0 α1
] [ a b
−b a ]
· · ·
[ c d
−d c ] [ a −b
b a
] [ α2 0 0 α2
]
· · ·
[ e f
−f e ]
... ... . .. ...
[ c −d d c
] [ e −f
f e
] . . .
[ αn 0 0 αn
]
∈ S2n
where β = a + ib, γ = c + id, δ = e + if .
Class(iv): The algebra Qn of n× n quaternion Hermitian matrices.
The linear space of quaternions overR, denoted by Q, is 4-dimensional vector space [16] with a basis {1, i, j, k}. This space becomes an associated algebra via the following multiplication table.
1 i j k
1 1 i j k
i i −1 k −j
j j −k −1 i
k k j −i −1
For any x = x01 + x1i + x2j + x3k ∈ Q, we define its real part by IR(x) := x0, its conjugate by ¯x := x01− x1i− x2j − x3k, and its norm by |x| = √
x¯x. A square matrix A with quaternion entries is called Hermitian if A coincides with its conjugate transpose. LetQn be the set of all n× n quaternion Hermitian matrices.
For any X, Y ∈ Qn, we define X◦ Y := 1
2(XY + Y X) and ⟨X, Y ⟩ := IR(trace(XY )).
Then Qn is a Euclidean Jordan algebra of rank n with e being the n× n identity matrix I. Analogous to complex number, each quaternion x = a1 + bi + cj + dk∈ Q can be represented as a 4 × 4 real matrix
a b c d
−b a −d c
−c d a −b
−d −c b a
which is also equivalent to
a
1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
+b
0 1 0 0
−1 0 0 0 0 0 0 −1
0 0 1 0
+c
0 0 1 0
0 0 0 1
−1 0 0 0
0 −1 0 0
+d
0 0 0 1
0 0 −1 0
0 1 0 0
−1 0 0 0
.
Following the same lines for Hn, we can embed Qn into S4n such that Qn can be viewed as a Jordan sub-algebra of S4n. Again, the embedding map under the case
forQ2 is
Q2∋
[ α1 x
¯ x α2
] 7−→
α1 0 0 0
0 α1 0 0
0 0 α1 0
0 0 0 α1
a b c d
−b a −d c
−c d a −b
−d −c b a
a −b −c −d
b a d −c
c −d a b
d c −b a
α2 0 0 0
0 α2 0 0
0 0 α2 0
0 0 0 α2
∈ S8
where x = a1 + bi + cj + dk.
Moreover, the general embedding map under this case is given by
Qn∋
α1 x · · · y
¯
x α2 · · · z ... ... . .. ...
¯
y z¯ · · · αn
7−→
α1 0 0 0
0 α1 0 0
0 0 α1 0
0 0 0 α1
a b c d
−b a −d c
−c d a −b
−d −c b a
· · ·
e f g h
−f e −h g
−g h e −f
−h −g f e
a −b −c −d
b a d −c
c −d a b
d c −b a
α2 0 0 0
0 α2 0 0
0 0 α2 0
0 0 0 α2
· · ·
p q r s
−q p −s r
−r s p −q
−s −r q p
... ... . .. ...
e −f −g −h
f e h −g
g −h e f
h g −f e
p −q −r −s
q p s −r
r −s p q
s r −q p
· · ·
αn 0 0 0
0 αn 0 0
0 0 αn 0
0 0 0 αn
∈ S4n
where x = a1 + bi + cj + dk, y = e1 + f i + gj + hk and z = p1 + qi + rj + sk.
In summary, we construct algebra isomorphism betweenHn orQnand some Jor- dan sub-algebra ofSm respectively for certain m. Hence the Fischer-Burmeister SC complementarity function ϕsc
FB defined on Hn or Qn can be viewed as ϕsc
FB defined onSm restricted on certain Jordan sub-algebra ofSm. Note here in defining ϕscFB we only use square operation, square root operation, addition and subtraction opera- tions which are preserved under the algebra isomorphism that we just constructed.
Moreover the strong semismoothness of ϕscFB defined on ambientSmis shown in [13].
Hence the strong semismoothness of ϕscFB holds when restricted on those Jordan sub-algebra ofSm(see Definition 2.3). Thus, we conclude the following theorem.
Theorem 3.1. The Fischer-Burmeister SC complementarity function ϕscFB defined as in (1.3) is strongly semismooth for each one of the following.
(i) The Jordan spin algebraLn.
(ii) The algebra Sn of n× n real symmetric matrices.
(iii) The algebraHn of all n× n complex Hermitian matrices.
(iv) The algebra Qn of all n× n quaternion Hermitian matrices.
Suppose that A is a Euclidean Jordan algebra which is a direct sum of ones taken only from classes (i)-(iv) of Theorem 3.1. Theorem 3.1 says that the Fischer- Burmeister SC complementarity function ϕscFB defined on such A is strongly semis- mooth. The exceptional case where we cannot draw a conclusion isO3 which is also called Albert algebra, a 27-dimensional Jordan algebra. SinceO is not an associa- tive algebra, there is no way (to our best knowledge) to represent an element in O as a real matrix. Hence we can not embed O3 into Sm as what we do for classes (iii)-(iv). This is the big hurdle which causes the uncertainty of the function ϕscFB being strongly semismooth under this case of O3.
In fact, the aforementioned result could be obtained by using analysis associated with Euclidean Jordan algebra. However, in that approach there comes up similar barrier during some analysis procedure. Moreover, the arguments by employing direct analysis associated with Jordan algebra are harder to follow. Therefore, we decide to use the current way to present this result. Even though the outcome is not perfect because there is one case not concluded, we still think the update result should be known in public so that subsequent research can be continued. We leave this unsolved case for future study. For readers who are interested in knowing more details about the structure of O (so that they can understand why it is a difficult case), please refer to [4].
References
[1] R. Bhatia, Matrix Analysis, Springer-Verlag, New York, 1997.
[2] J. Faraut, and A. Kor´anyi, Analysis on Symmetric Cones, Oxford Mathematical Monographs, Oxford University Press, New York, 1994.
[3] M. Koecher, The Minnesota Notes on Jordan Algebras and Their Applications, edited and annotated by A. Brieg and S. Walcher, Springer, Berlin, 1999.
[4] R. L. GRIESS, A Moufang loop, the exceptional Jordan algebra, and a cubic form in 27 variables, Journal of Algebra 131 (1990), 281–293.
[5] M. S. Gowda, R. Sznajder and J. Tao, Some P-properties for linear transformations on Eu- clidean Jordan algebras, Linear Algebra and Its Applications 393 (2004), 203–232.
[6] Z.-H. Huang and T. Ni, Smoothing algorithms for complementarity problems over symmetric cones, Computational Optimization and Applications 45 (2010), 557–579.
[7] L. C. Kong, J. Sun and N. H. Xiu, A regularized smoothing Newton method for symmetric cone complementarity problems, SIAM Journal on Optimization 19 (2008), 1028–1047.
[8] L. C. Kong, L. Tuncel and N. H. Xiu, Vector-valued implicit Lagrangian for symmetric cone complementarity problems, Asia-Pacific Journal of Operational Research 26 (2009), 199–233.
[9] L. C. Kong, L. Tuncel and N. H. Xiu, Equivalent conditions for Jacobian nonsingularity in linear symmetric cone programming, Journal of Optimization Theory and Applications 148 (2011), 364–389.
[10] L. C. Kong and N. H. Xiu, The Fischer-Burmeister complementarity function on Euclidean Jordan algebras, Pacific Journal of Optimization 6 (2010), 423–440.
[11] Y. Liu, L. Zhang and Y. Wang, Some propeties of a class of merit functions for symmetric cone complementarity problems, Asia-Pacific Journal of Operational Research 23 (2006), 473–496.
[12] S.-H. Pan and J.-S. Chen, A one-parametric class of merit functions for the symmetric cone complementarity problem, Journal of Mathematical Analysis and Applications 355 (2009), 195–215.
[13] D. Sun and J. Sun, Strong semismoothness of the Fischer-Burmeister SDC and SOC comple- mentarity functions, Mathematical Programming 103 (2005), 575–581.
[14] D. Sun and J. Sun, L¨owner’s operator and spectral functions in Euclidean Jordan algebras, Mathematics of Operations Research 33 (2008), 421–445.
[15] Y. Wang, Perturbation Analysis of Optimization Problems over Symmetric Cones. Ph.D. the- sis, Dalian University of Technology, September, 2008.
[16] F. Zhang, Quaternions and matrices of quaternions, Linear Algebra and its Applications 251 (1997), 21–57.
Manuscript received July 26, 2011 revised November 22, 2011
Yu-Lin Chang
Department of Mathematics, National Taiwan Normal University, Taipei 11677, Taiwan E-mail address: ylchang@math.ntnu.edu.tw
Jein-Shan Chen
Department of Mathematics, National Taiwan Normal University, Taipei, 11677, Taiwan E-mail address: jschen@math.ntnu.edu.tw
Shaohua Pan
School of Mathematical Sciences, South China University of Technology, Guangzhou 510640, China E-mail address: shhpan@scut.edu.cn
