to appear in Journal of Nonlinear and Convex Analysis, 2012
Strong semismoothness of Fischer-Burmeister complementarity function associated with symmetric cones
Yu-Lin Chang Department of Mathematics National Taiwan Normal University
Taipei 11677, Taiwan E-mail: [email protected]
Jein-Shan Chen1 Department of Mathematics National Taiwan Normal University
Taipei, Taiwan 11677
Shaohua Pan2
School of Mathematical Sciences South China University of Technology
Guangzhou 510640, China
July 26, 2011
(revised on November 22, 2011)
Abstract. We provide an affirmative answer to an question that the Fischer-Burmeister complementar- ity function associated with symmetric cones, named the FB SC complementarity function, is globally Lipschitz continuous and strongly semismooth everywhere for Hn and Qn. This is achieved with the help of embedding Hn and Qn into certain Sm.
Key words. Fischer-Burmeister function, symmetric cones, strong semismoothness.
1 Introduction
Let A = (V, h·, ·i, ◦) be an n-dimensional Euclidean Jordan algebra (see Section 2) and K be the sym- metric cone in V. We call φ : V × V → V a complementarity function associated with symmetric cone (SC complementarity function for short) if
φ(x, y) = 0 ⇐⇒ x ∈ K, y ∈ K, x ◦ y = 0. (1)
1Member of Mathematics Division, National Center for Theoretical Sciences, Taipei Office.
The author’s work is partially supported by National Science Council of Taiwan. E-mail:
2The author’s work is supported by National Young Natural Science Foundation (No. 10901058) and Guangdong Natural Science Foundation (No. 9251802902000001). E-mail: [email protected].
This class of functions and the merit functions induced play an important role in the development of algorithms for the symmetric cone complementarity problem (SCCP) and the symmetric cone program- ming (SCP), and attract much attention in current optimization field; see [6, 7, 8, 11, 12, 14].
A popular choice for φ is the natural residual (NR) SC complementarity function [5]
φscNR(x, y) := x − (x − y)+ ∀x, y ∈ V, (2) where (z)+ denotes the minimum metric projection of z onto K. This function is shown to be strongly semismooth in [14]. Recently, the nonsingularity of Clarke’s generalized Jacobian of the KKT nons- mooth system based on φscNR for linear SCP is studied in [9, 15]. These works lay the foundations for the corresponding nonsmooth Newton methods and smoothing Newton methods of the SCCPs and the SCPs.
Another popular choice of φ is the Fischer-Burmeister(FB) SC complementarity function [5] defined as
φscFB(x, y) := (x2+ y2)1/2− (x + y) ∀x, y ∈ V, (3) where x2 = x ◦ x, and x1/2 denotes 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 φsc
FB.
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 IRn space 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 φsc
FB 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, h·, ·iV, such that
(iii) hx ◦ y, ziV = hy, x ◦ ziV for all x, y, z ∈ V.
Given a Euclidean Jordan algebra A = (V, ◦, h·, ·iV), 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 transformation T : 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) := mink : {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 ciand 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 j 6= i for all j, i = 1, 2, · · · , k, and Pk
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 eigen- values and tr(x) =Pr
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 hx, yi := tr(x ◦ y) for any x, y ∈ V. The inner product h·, ·i is associative by [2, Prop. II. 4.3], i.e., hx, y ◦ zi = hy, x ◦ zi 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 isomorphic to one of the following.
(i) The Jordan spin algebra Ln.
(ii) The algebra Sn of n × n real symmetric matrices.
(iii) The algebra Hn of all n × n complex Hermitian matrices.
(iv) The algebra Qn of all n × n quaternion Hermitian matrices.
(v) The algebra O3 of all 3 × 3 octonion Hermitian matrices.
Next we provide the definition of semismoothness [14] for readers’ convenience. Let X 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 theorem, Φ is almost everywhere differentiable. We denote DΦ the set of points where Φ is differentiable and denote Φ0(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→∞Φ0(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 2.1 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(ky − xk).
3 Main results
As mentioned earlier, the Fischer-Burmeister SC complementarity function φscFB defined as in (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 hX, Y i := trace(XY ). Then Hn is 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 in S4:
H23
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
Hn3
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 over R, 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. Let Qn 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 hX, Y i := 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 for Q2is
Q23
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 3
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 between Hn or Qn and some Jordan sub-algebra of Sm respectively for certain m. Hence the Fischer-Burmeister SC complementarity function φscFB defined on Hn or Qn can be viewed as φscFB defined on Smrestricted on certain Jordan sub-algebra of Sm. Note here in defining φscFB we only use square operation, square root operation, addition and subtraction operations which are preserved under the algebra isomorphism that we just constructed. Moreover the strong semismoothness of φsc
FB defined on ambient Smis shown in [13]. Hence the strong semismoothness of φsc
FB holds when restricted on those Jordan sub-algebra of Sm(see Definition 2.1). Thus, we conclude the following theorem.
Theorem 3.1 The Fischer-Burmeister SC complementarity function φscFB defined as in (3) is strongly semismooth for each one of the following.
(i) The Jordan spin algebra Ln.
(ii) The algebra Sn of n × n real symmetric matrices.
(iii) The algebra Hn 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 semismooth. The exceptional case where we cannot draw a conclusion is O3which is also called Albert algebra, a 27-dimensional Jordan algebra. Since O is not an associative 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 proce- dure. 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].
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