Convexity of Symmetric Cone trace functions in Euclidean Jordan algebras

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Journal of Nonlinear and Convex Analysis, vol. 14, no. 1, pp. 53-61, 2013.

Convexity of Symmetric Cone trace functions in Euclidean Jordan algebras

Yu-Lin Chang

Department of Mathematics National Taiwan Normal University

Taipei 11677, Taiwan

E-mail: [email protected]

Jein-Shan Chen ¹ Department of Mathematics National Taiwan Normal University

Taipei 11677, Taiwan E-mail: [email protected]

February 23, 2012

(in honor of Prof. Mau-Hsiang Shih’s 66th birthday)

Abstract. In this paper, we establish convexity of some functions associated with symmetric cones, called SC trace functions. As illustrated in the paper, these functions play a key role in the development of penalty and barrier functions methods for symmtric cone programs.

Key words. Symmetric cone, L¨owner operator, convexity

AMS subject classifications. 26A27, 26B05, 26B35, 49J52, 90C33.

1 Introduction

The second-order cone (SOC) in Rⁿ, also called Lorentz cone, is the set defined as Kⁿ :=n

(x₁, x₂) ∈ R × Rⁿ⁻¹ | x₁ ≥ kx₂ko

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1Member of Mathematics Division, National Center for Theoretical Sciences, Taipei Office. The author’s work is supported by National Science Council of Taiwan.

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where k·k denotes the Euclidean norm. When n = 1, Kⁿreduces to the set of nonnegative real numbers R+. As shown in [13], Kⁿ is also a set composed of the squared elements from Jordan algebra (Rⁿ, ◦), where the Jordan product “◦” is a binary operation defined by

x ◦ y := (hx, yi, x₁y₂+ y₁x₂) (2) for any x = (x₁, x₂), y = (y₁, y₂) ∈ R × Rⁿ⁻¹. Here for any x ∈ Rⁿ, we use x₁ to denote the first component of x, and x2 to denote the vector consisting of the rest n − 1 compo- nents.

From [12, 13], we recall that each x ∈ Rⁿ admits a spectral decomposition associated with Kⁿ of the following form

x = λ₁(x)u⁽¹⁾_x + λ₂(x)u⁽²⁾_x , (3) where λ_i(x) and u⁽ⁱ⁾x for i = 1, 2 are the spectral values and the associated spectral vectors of x, respectively, defined by

λ_i(x) = x₁+ (−1)ⁱkx₂k, u⁽ⁱ⁾_x = 1 2

1, (−1)ⁱx¯₂

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with ¯x₂ = _kx^x²

2k if x₂ 6= 0, and otherwise ¯x₂ being any vector in Rⁿ⁻¹ such that k¯x₂k = 1.

When x2 6= 0, the spectral factorization is unique. The determinant and trace of x are defined as det(x) := λ₁(x)λ₂(x) and tr(x) := λ₁(x) + λ₂(x), respectively.

With the spectral decomposition above, for any given scalar function φ : J ⊆ R → R, we may define a vector-valued function φ^soc: S ⊆ Rⁿ→ Rⁿ by

φ^soc(x) := f (λ₁(x))u⁽¹⁾_x + f (λ₂(x))u⁽²⁾_x (5) where J is an interval (finite or infinite, open or closed) of R, and S is the domain of φ^soc determined by φ. Then, we can define the SOC trace function associated with φ

φ^tr(x) := φ(λ₁(x)) + φ(λ₂(x)) = tr(φ^soc(x)) ∀x ∈ S. (6) Chen, Liao and Pan [11] give the following relation between φ^tr and φ^soc

∇φ^tr(x) = (φ⁰)^soc(x) and ∇²φ^tr(x) = ∇(φ⁰)^soc(x) ∀x ∈ intS. (7) By using Schur Complement Theorem, they establish the convexity of SOC trace functions and the compounds of SOC trace functions. Some of these functions are the key of penalty and barrier function methods for second-order cone programs (SOCPs), as well as the establishment of some important inequalities associated with SOCs, for which the proof of convexity of these functions is a necessity.

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Some similar results associated with positive semidefinite cone are also investigated by Auslender in [1, 2]. Since both SOC and positive semidefinite cone are special cases of symmetric cone (SC for short). A natural question leads us to consider the more general case. To this end, we need to recall some concepts regarding Euclidean Jordan algebra.

Let A = (V, h·, ·i, ◦) be an n-dimensional Euclidean Jordan algebra (see Section 2) and K be the symmetric cone in V. For any given scalar function φ : J ⊆ R → R, we define the associated function

φ^sc

V(x) := φ(λ₁(x))c₁+ · · · + φ(λ_r(x))c_r, (8) and SC trace function

φ^tr

V(x) := φ(λ1(x)) + · · · + φ(λr(x)) = tr(φ^sc

V(x)) ∀x ∈ S, (9)

where x ∈ V has the spectral decomposition

x = λ₁(x)c₁+ · · · + λ_r(x)c_r.

In this paper we extend the aforementioned results to general symmetric cone set- ting where we establish the convexity of SC trace functions and the compounds of SC trace functions. Throughout this note, for any x, y ∈ V, we write x K y if x − y ∈ K;

and write x _K y if x − y ∈ intK. For a real symmetric matrix A, we write A 0 (respectively, A 0) if A is positive semidefinite (respectively, positive definite). For any φ : J → R, φ⁰(t) and φ⁰⁰(t) denote the first derivative and second-order derivative of φ at the differentiable point t ∈ J , respectively. Suppose F : S ⊆ V → R, ∇F (x) and ∇²F (x) denote the gradient and the Hessian matrix of F at the differentiable point x ∈ S, respectively.

2 Preliminaries

This section recalls some results on Euclidean Jordan algebras that will be used in sub- sequent analysis. More detailed expositions of Euclidean Jordan algebras can be found in Koecher’s lecture notes [16] and the monograph by Faraut and Kor´anyi [13].

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·, ·i

V such that

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(iii) hx ◦ y, zi

V = hy, x ◦ zi

V 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 .

From [13, Theorem III.2.1], K is a symmetric cone which means that K is a self-dual closed convex cone with nonempty interior and for any two elements x, y ∈ intK, 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, · · · , x^k} 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 c_i and c_j are said to be orthogonal if c_i ◦ c_j = 0.

One says that {c₁, c₂, . . . , c_k} is a complete system of orthogonal idempotents if c²_j = c_j, c_j◦ c_i = 0 if j 6= i for all j, i = 1, 2, · · · , k and Pk

j=1c_j = 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 [13, Theorem III.1.2] Suppose that A is a Euclidean Jordan algebra with rank r. Then for any x ∈ V, there exists a Jordan frame {c1, . . . , c_r} and real numbers λ₁(x), . . . , λ_r(x), arranged in the decreasing order λ₁(x) ≥ λ₂(x) ≥ · · · ≥ λ_r(x), such that

x = λ₁(x)c₁+ λ₂(x)c₂+ · · · + λ_r(x)c_r.

The numbers λ_j(x) (counting multiplicities), which are uniquely determined by x, are called the eigenvalues and tr(x) =Pr

j=1λj(x) the trace of x.

Since, by [13, Proposition 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 [13, Prop. II. 4.3], i.e., hx, y ◦ zi = hy, x ◦ zi for any x, y, z ∈ V. For any given x ∈ V, let L(x) be the linear operator of V defined by

L(x)y := x ◦ y ∀y ∈ V.

Then, L(x) is symmetric with respect to the inner product h·, ·i in the sense that hL(x)y, zi = hy, L(x)zi ∀y, z ∈ V.

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In the sequel, we let k · k be the norm on V induced by the inner product, namely, kxk := phx, xi =

Pr

j=1λ²_j(x)1/2

∀x ∈ V. (10)

A Euclidean Jordan algebra is called simple if it cannot be written as a direct sum of the other two Euclidean Jordan algebras. It is known that every Euclidean Jordan algebra is a direct sum of simple Euclidean Jordan algebras. Unless otherwise stated, in the rest of this paper, we assume that A = (V, ◦, h·, ·i) is a simple Euclidean Jordan algebra of rank r. Let {c₁, c₂, . . . , c_r} be a Jordan frame of A. From [13, Lemma IV. 1.3], we know that the operators L(c_j), j = 1, 2, . . . , r commute and admit a simultaneous diagonalization. For i, j ∈ {1, 2, . . . , r}, define the subspaces

Vii:= Rci and Vij :=

x ∈ V | ci◦ x = c_j ◦ x = 1 2x

when i 6= j.

Then, [13, Corollary IV.2.6] says

dim(Vij) = dim(Vst) for any i 6= j ∈ {1, 2, . . . , r} and s 6= t ∈ {1, 2, . . . , r}, and n = r + ^d₂r(r − 1) where d denotes this common dimension. Moreover, from [13, Theorem IV.2.1], we have the following conclusion.

Theorem 2.2 The space V is the orthogonal direct sum of subspaces Vij (1 ≤ i ≤ j ≤ r), i.e., V = ⊕i≤jV^ij. Furthermore,

Vij ◦ Vij ⊂ Vii+ Vij, Vij ◦ Vjk ⊂ Vik, if i 6= k,

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

Let x ∈ V have the spectral decomposition x = Pr

j=1λ_j(x)c_j, where λ₁(x) ≥ λ₂(x) ≥

· · · ≥ λ_r(x) are the eigenvalues of x and {c₁, c₂, . . . , c_r} is the corresponding Jordan frame.

For i, j ∈ {1, 2, . . . , r}, let Cij(x) be the orthogonal projection operator onto V^ij. Then, from Theorem IV 2.1 of [13], it follows that for all i, j = 1, 2, . . . , r,

Cjj(x) = 2L²(cj) − L(cj) and Cij(x) = 4L(ci)L(cj) = 4L(cj)L(ci) = Cji(x). (11) Moreover, the orthogonal projection operators {C_ij(x) : i, j = 1, 2, . . . , r} satisfy

C_ij(x) = C_ij^∗(x), C_ij²(x) = C_ij(x), C_ij(x)C_kl(x) = 0 if {i, j} 6= {k, l} (12) and

X

1≤i≤j≤r

C_ij(x) = I. (13)

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Suppose φ : R → R be a scalar valued function and we define the L¨owner operator associated with φ as

φ^sc

V(x) =:

r

X

j=1

φ(λ_j(x))c_j, where x ∈ V has the spectral decomposition x =Pr

j=1λ_j(x)c_j. Kor´anyi [15] (or see [19]) proves the following result, which generalizes L¨owner result on symmetric matrices to Euclidean Jordan algebras.

Theorem 2.3 Let x = Pr

j=1λ_j(x)c_j and (a, b) be an open interval in R that contains λj(x), j = 1, 2, . . . , r. If φ is continuously differentiable on (a, b), then φ^sc_V is differentiable at x and its derivative, for any h ∈ V, is given by

∇φ^sc

V (x)(h) =

r

X

j=1

φ^[1](λ(x))

jjC_jj(x)h + X

1≤j<l≤r

φ^[1](λ(x))

jlC_jl(x)h (14) where the coefficient is defined as

φ^[1](λ(x))_jl :=

( φ⁰(λ_j) if λ_j = λ_l,

φ(λj)−φ(λl)

λj−λ_l if λ_j 6= λ_l. (15)

Moreover, based on this theorem, Sun and Sun [19] show that φ^sc

V is continuously differentiable at x if and only if φ is continuously differentiable at λ_j(x), j = 1, 2, · · · , r.

We will exploit such property to achieve Lemma 3.1 which paves a way to our main result.

3 Main results

In this section, we present how we achieve the convexity of symmetric cone trace functions. We start with a technical lemma.

Lemma 3.1 For any given scalar function φ : J ⊆ R → R, let φ^sc_V : S → V and φ^tr

V : S → R be given by (8) and (9), respectively. Assume that J is an open interval in R. Then, the following results hold.

(a) The domain S of φ^sc

V and φ^tr

V is open and convex.

(b) If φ is (continuously) differentiable, then φ^tr

V is (continuously) differentiable on S with ∇φ^tr

V(x)(h) = hh, (φ⁰)^sc

V(x)i for all h ∈ V.

(c) If φ is twice (continuously) differentiable, then φ^tr

V is twice (continuously) differentiable on S with ∇²φ^tr

V(x)(h, k) = hh, ∇(φ⁰)^sc

V(x)ki for all h, k ∈ V.

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Proof. (a) Suppose J = (a, b). Then the domain S is open because it is the intersection of two open sets S₁ and S_r, where S₁ and S_r are defined as

S₁ = {x ∈ V : λ1(x) < b} and S_r = {x ∈ V : λr(x) > a}.

We note here that the eigenvalue functions λj(x) are continuous, see [19]. For convexity of S, we suppose x, y ∈ S and 0 ≤ λ ≤ 1. We want to verify that λx + (1 − λ)y ∈ S.

First, we know that the largest eigenvalue function λ₁(x) is a convex function [8] which implies

λ1(λx + (1 − λ)y) ≤ λλ1(x) + (1 − λ)λ1(y) < λa + (1 − λ)a = a.

This means λx + (1 − λ)y ∈ S₁. Analogously, we know that the smallest eigenvalue function λ_r(x) is concave which leads to λ_r(λx + (1 − λ)y) > b, i.e. λx + (1 − λ)y ∈ S_r. (b) As mentioned earlier, the (continuous) differentiability is known. From the following formula

φ^tr

V(x) :=

r

X

j=1

φ(λ_j(x)) =

* _r X

j=1

φ(λ_j(x))c_j, e +

= hφ^sc

V(x), ei, we have that, for any h ∈ V,

∇φ^tr

V(x)(h) =∇φ^sc

V(x)h, e = h, ∇φ^sc

V(x)e where we use symmetry property of ∇φ^sc

V(x) in the second equation. By applying equa- tions (14) and (15), we obtain

∇φ^sc

V(x)e =

r

X

j=1

φ^[1](λ(x))

jjC_jj(x)e + X

1≤j<l≤r

(φ^[1](λ(x)))_jlC_jl(x)e

=

r

X

j=1

(φ^[1](λ(x)))jjcj

=

r

X

j=1

φ⁰(λ_j(x))c_j = (φ⁰)^sc

V (x) (16)

Note that e = c1+ · · · + cr. Hence Cjj(x)e = cj and Cjl(x)e = 0 for j 6= l.

(c) Suppose now that φ is twice (continuously) differentiable. It is not hard to see that φ^tr

V is twice (continuously) differentiable on S with ∇²φ^tr

V(x)(h, k) = hh, ∇(φ⁰)^sc

V (x)ki by the expression ∇φ^tr

V(x)(h) = hh, (φ⁰)^sc

V(x)i. 2

Theorem 3.1 For any given φ : J → R, let φ^sc_V: S → Rⁿ and φ^tr

V : S → R be given by (5) and (6), respectively. Assume that J is an open interval in R. If φ is twice differentiable on J , then

(a) φ⁰⁰(t) ≥ 0 for any t ∈ J ⇐⇒ ∇²φ^tr

V(x) 0 for any x ∈ S ⇐⇒ φ^tr

V is convex in S.

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(b) φ⁰⁰(t) > 0 for any t ∈ J ⇐⇒ ∇²φ^tr

V(x) 0 ∀x ∈ S =⇒ φ^tr

V is strictly convex in S.

Proof. (a) We substitute φ by φ⁰, then the coefficient equation (15) becomes

φ^0[1](λ(x))_jl:=

( φ⁰⁰(λ_j) if λ_j = λ_l;

φ⁰(λj)−φ⁰(λl)

λj−λ_l if λ_j 6= λ_l.

Hence the the coefficients are all nonnegative because of the assumption φ⁰⁰(t) ≥ 0.

Observing that V is a direct sum of orthogonal spaces V = ⊕^i≤jV^ij, we can give an orthonormal basis B = {c₁. . . , c_r, c⁽¹⁾₁₂, . . . , c^(d)₁₂, c⁽¹⁾₁₃, . . . , c^(d)₁₃, . . . , c⁽¹⁾_r−1,r, . . . , c^(d)_r−1,r} for V and {c⁽¹⁾_jl , . . . , c^(d)_jl } spans the space V^jl, where d is the common dimension of V^jl, j < l.

Let h, k ∈ B. Plug in Lemma 3.1 (c), then the Hessian ∇²φ^tr

V(x) can be presented as a diagonal matrix under the basis B

A = diag(φ^0[1](λ(x))₁₁, . . . , φ^0[1](λ(x))_rr,

d⁰s

z }| {

φ^0[1](λ(x))₁₂, . . . , φ^0[1](λ(x))₁₂, , . . . ,

d⁰s

z }| {

φ^0[1](λ(x))_r−1,r, . . . , φ^0[1](λ(x))_r−1,r).

Then, the first part equivalence follows clearly from Lemma 3.1 whereas the second part is a well-known result in analysis.

(b) The arguments are similar to those in part(a), we omit them here. 2

Indeed, the fact that the strict convexity of φ implies the strict convexity of φ^tr

V was

proved in [2, 8] via checking the definition of convex function. But, here our analysis is much simpler and we also give the relation between ∇(φ⁰)^sc

V and ∇²φ^tr

V to achieve the convexity of SC trace functions. In addition , we note that the necessity involved in the first equivalence of Theorem 3.1(a) was given in [12] for SOC case via a different way.

Next, we will illustrate the application of Theorem 3.1 with some SC trace functions.

Theorem 3.2 The following functions associated with K are all strictly convex.

(a) F₁(x) = − ln(det(x)) for x ∈ intK.

(b) F₂(x) = tr(x⁻¹) for x ∈ intK.

(c) F₃(x) = tr(h(x)) for x ∈ intK, where

h(x) =

( _xp+1−e

p+1 + ^x^1−q_q−1^−e if p ∈ [0, 1], q > 1;

x^p+1−e

p+1 − ln x if p ∈ [0, 1], q = 1.

(d) F₄(x) = − ln(det(e − x)) for x ≺_K e.

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(e) F₅(x) = tr((e − x)⁻¹◦ x) for x ≺_K e.

(f ) F6(x) = tr(exp(x)) for x ∈ V.

(g) F₇(x) = ln(det(e + exp(x))) for x ∈ V.

(h) F₈(x) = tr x + (x² + 4e)^1/2 2

for x ∈ V.

Proof. Note that F₁(x), F₂(x) and F₃(x) are the SC trace functions associated with φ₁(t) = − ln t (t > 0), φ₂(t) = t⁻¹ (t > 0) and φ₃(t) (t > 0), respectively, where

φ₃(t) =

( _tp+1−1

p+1 +^t^1−q_q−1⁻¹ if p ∈ [0, 1], q > 1,

t^p+1−1

p+1 − ln t if p ∈ [0, 1], q = 1,

F4(x) is the SC trace function associated with φ4(t) = − ln(1 − t) (t < 1), F5(x) is the SC trace function associated with φ₅(t) = _1−t^t (t < 1) by noting that

(e − x)⁻¹◦ x = λ₁(x)

1 − λ₁(x)c1(x) + · · · + λ_r(x)

1 − λ_r(x)cr(x);

F₆(x) and F₇(x) are the SC trace functions associated with φ₆(t) = exp(t) (t ∈ R) and φ₇(t) = ln(1 + exp(t)) (t ∈ R), respectively, and F8(x) is the SC trace function associated with φ8(t) = 2⁻¹ t +√

t²+ 4

(t ∈ R). It is easy to verify that the functions φ¹-φ8

have positive second-order derivatives in their respective domain, and therefore F₁-F₈ are strictly convex functions by Theorem 3.1(b). 2

Analogous to SOC case, e.g., [6, 7, 17, 18, 20], the functions F1, F2 and F3 can be served as barrier functions for symmetric cone programming (SCP) which also play a key role in the development of interior point methods for SCPs. The function F₃ covers a wide range of barrier functions for SCPs, including the classical logarithmic barrier function, the self-regular functions and the non-self-regular functions; see [7] for details.

The functions F₄ and F₅ are called shifted barrier functions [1, 2, 3] for SOCPs, and F₆-F₈ can be used as penalty functions for SCPs.

Besides the application in establishing convexity for SC trace functions, our establishment of convexity of some compound functions of SC trace functions and scalar-valued functions is much simpler, which is usually difficult to achieve by the definition of convex function.

4 Conclusions

We establish convexity of SC-functions, especially for SC trace functions, which are the key of penalty and barrier function methods for symmetric cone programming and some

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important inequalities associated with symmetric cones. We believe that the results in this paper will be helpful towards establishing further properties of other SC functions.

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