Figure 1. Relationship among various distances.
(ii) x◦ (x2◦ y) = x2◦ (x ◦ y) for all x, y ∈ V where x2:= x◦ x;
(iii) hx ◦ y, zi = hx, y ◦ zi for all x, y, z ∈ V.
Here x◦ y is called the Jordan product of x and y. If a Jordan product only satisfies the conditions (i) and (ii) in the above definition, the algebra V is said to be a Jordan algebra. Moreover, if there is an (unique) element e∈ V such that x ◦ e = x for all x∈ V, the element e is called the identity element in V. Note that a Jordan algebra does not necessarily have an identity element. Throughout this paper, we assume that V is a Euclidean Jordan algebra with an identity element e.
In a given Euclidean Jordan algebra V, the set of squares K := {x2| x ∈ V}
is a symmetric cone [10, Theorem III.2.1]. This means that K is a self-dual closed convex cone and, for any two elements x, y∈ int(K), there exists an invertible linear transformation Γ :V → V such that Γ(x) = y and Γ(K) = K. It is well known that second-order cone is a special symmetric cone, which is defined as follows in Rn:
Kn:=
x = (x0, ¯x)∈ R × Rn−1 | x0 ≥ k¯xk ,
and the corresponding Jordan product of x and y in Rn with x = (x0, ¯x), y = (y0, ¯y)∈ R × Rn−1 is given by
x◦ y :=
xTy x0y + y¯ 0x¯
.
In particular, in the setting of the second-order cone Kn, the identity element e = (1, 0)∈ R × Rn−1, where 0 denotes the zero vector in Rn−1.
For x∈ V, we denote m(x) the degree of the minimal polynomial of x, that is, m(x) := min
n
k > 0| {e, x, . . . , xk} is linearly dependento ,
and the rank of V is well-defined by r := max{m(x) | x ∈ V}. In Euclidean Jordan algebra V, an element ei ∈ V is an idempotent if (ei)2 = ei, and it is a primi- tive idempotent if it is nonzero and cannot be written as a sum of two nonzero idempotents. The idempotents ei and ej are said to be orthogonal if ei◦ ej = 0. In
addition, we say that a finite set {e1, e2, . . . , er} of primitive idempotents in V is a Jordan frame if
ei◦ ej = 0 for i6= j, and Xr i=1
ei = e.
Note that hei, eji = hei◦ ej, ei whenever i 6= j.
With the above, there have the spectral decomposition and Peirce decomposition of an element x in V.
Theorem 1.1 ((The Spectral Decomposition Theorem) [10, Theorem III.1.2]). Let V be a Euclidean Jordan algebra. Then, there is a number r such that, for every x ∈ V, there exists a Jordan frame {e1, . . . , er} and real numbers λ1(x), . . . , λr(x) with
x = λ1(x)e1+· · · + λr(x)er.
Here, the numbers λi(x) (i = 1, . . . , r) are the eigenvalues of x, the expression λ1(x)e1 + · · · + λr(x)er is the spectral decomposition of x. Moreover, tr(x) :=
Pr
i=1λi(x) is called the trace of x, and det(x) := λ1(x) . . . λr(x).
We point out that different elements x, y have their own Jordan frames in the spectral decomposition, which are not easy to handle when we need to do operations for x and y. Thus, we need another so-called Peirce decomposition to conquer such difficulty. In other words, in the Peirce decomposition, two different elements x, y share the same Jordan frame. We elaborate them more as below.
The Peirce decomposition: Fix a Jordan frame {e1, e2, . . . , er} in a Euclidean Jordan algebra V. For i, j ∈ {1, 2, . . . , r}, we define the following eigen-spaces
Vii:={x ∈ V | x ◦ ei = x} = Rei and
Vij :=
x∈ V x◦ ei= 1
2x = x◦ ej
for i6= j.
Theorem 1.2 ([10, Theorem IV.2.1]). The space V is the orthogonal direct sum of spaces Vij(i≤ j). Furthermore,
Vij ◦ Vij ⊂ Vii+Vjj, Vij ◦ Vjk ⊂ Vik, if i6= k,
Vij ◦ Vkl ={0}, if {i, j} ∩ {k, l} = ∅.
Hence, given any Jordan frame {e1, e2, . . . , er}, we can write any element x ∈ V as x =
Xr i=1
xiei+X
i<j
xij,
where xi ∈ R and xij ∈ Vij. The expression Pr
i=1xiei +P
i<jxij is called the Peirce decomposition of x.
Theorem 1.3 ([22, Theorem 4.6]). Suppose that V is simple and {e1, . . . , er} is any fixed Jordan frame in V. Let z =Pr
i=1ziei+P
i<jzij ∈ K. Then, we have Xr
i=1
zip≤ tr(zp) for p > 1 and Xr i=1
zpi ≥ tr(zp) for 0 < p < 1, where the equalities hold if and only if z =Pr
i=1ziei.
In a Euclidean Jordan algebrasV, for any x ∈ V, a linear transformation L(x) : V → V is called Lyapunov transformation, which is defined as L(x)(y) := x ◦ y for all y ∈ V. The so-called quadratic representation P (x) is define by P (x) :=
2L2(x)− L(x2). For any x∈ V, the endomorphisms L(x) and P (x) are self-adjoint.
We say that two elements x and y of a Euclidean Jordan algebraV operator commute if x◦(y ◦z) = y ◦(x◦z) for all z ∈ V, which is equivalent to stating that L(x)L(y) = L(y)L(x). For the quadratic representation P (x), if x is invertible, then we have
P (x)K = K and P (x)int(K) = int(K).
Below is a useful property regarding the quadratic representation P (x), which is needed for our subsequent analysis.
Theorem 1.4 ([12, Proposition 2.5]). Suppose that{e1, e2, . . . , er} is Jordan frame in V and the spectral decomposition of x can be expressed as x = λ1(x)e1 +· · · + λr(x)er. For any z ∈ V, if the Peirce decomposition of z is z =Pr
i=1ziei+P
i<jzij, we have
P (x)z = Xr
i=1
λi(x)2ziei+X
i<j
λi(x)λj(x)zij.
In light of the trace function tr(·), a semi-distance function associated with sym- metric cone was proposed in [16]:
(1.3) d(x, y) := tr(x + y)− 2 tr
P (x12)y
1
2 , for x, y∈ K.
When the symmetric cone reduces to the second-order coneKn, the function d(x, y) was modified a bit as below distance function (1.4) and is further proved a proximal distance in [16]. When the symmetric cone reduces to the semi-definite positive matrix cone, the function d(x, y) corresponds to the matrix distance proposed by Givens and Shortt [11]:
d(A, B) := tr(A + B)− 2 tr
A12BA12
1
2 .
Theorem 1.5 ( [16, Theorem 2.3]). Let d(·, ·) be defined as in (1.3). For any x, y ∈ K, assume that x and y operator commute. Then, the function d(·, ·) is a semi-distance, i.e.,
(a) d(x, y)≥ 0;
(b) d(x, y) = 0 if and only if x = y;
(c) d(x, y) = d(y, x).
As mentioned earlier, the function d(x, y) was modified a bit in the setting of second-order cone in [16], which could become a proximal distance. In particular, for any x, y∈ Rn, there defines d :Rn× Rn→ R+∪ {+∞} by
(1.4) d(x, y) :=
tr(x + y)− 2 tr
P (x12)y
1
2 ∀x ∈ int(Kn), y∈ Kn,
+∞ otherwise.
This modified function d(x, y) is a proximal distance on int(Kn), see [16, Theorem 3.7].
Theorem 1.6 ( [16, Theorem 3.7]). Let the function d(·, ·) be defined by (1.4).
Then, the function d(·, ·) is a proximal distance with respect to int(Kn), i.e., (a) d(·, y) is proper, l.s.c., convex, continuously differentiable on int(Kn);
(b) dom d(·, y) ⊂ int(Kn) and dom ∂1d(·, y) = int(Kn), where the symbol ∂1d(·, y) denotes the classical subgradient map of the function d(·, y) with respect to the first variable;
(c) d(·, y) is level bounded on Rn i.e., lim∥u∥→∞d(u, y) = +∞;
(d) d(y, y) = 0.
In this short paper, we improve these two results by showing that without as- suming operator commute, the function d(·, ·) is a semi-distance, and the function d(·, ·) is a proximal distance in the setting of symmetric cone. These generaliza- tions enable them applicable to proximal-like algorithm for nonlinear symmetric cone programming.
2. Main results
In this section, without assuming operator commute, we show our main results.
Indeed, there exists a difficulty that the same Jordan frame is not available for any two elements x and y in V, when there is no condition of operator commute. Our novel idea to tackle with it is using the spectral decomposition of x, whereas employ- ing the Peirce decomposition of y. These together with the quadratic representation P (x) paves a way to do the analysis.
Theorem 2.1. Let d(·, ·) be defined as in (1.3). For any x, y ∈ K, the function d(·, ·) is a semi-distance, i.e., there hold
(a) d(x, y)≥ 0;
(b) d(x, y) = 0 if and only if x = y;
(c) d(x, y) = d(y, x).
Proof. (a) Suppose that{e1, e2, . . . , er} is a Jordan frame in V. With this, we write out the spectral decomposition of x and the Peirce decomposition of y, respectively, as below:
x = λ1(x) e1+· · · + λr(x) er, y = y1e1+· · · + yrer+X
i<j
yij.
Based on the spectral decomposition of x, it follows that x12 =p
λ1(x) e1+· · · +p
λr(x) er. Combining with Theorem 1.5, this implies that
P (x12)y = λ1(x)y1e1+· · · + λr(x)yrer+X
i<j
q
λi(x)λj(x) yij.
Then, applying Theorem 1.3, we have tr
P (x12)y
1
2 ≤
Xr i=1
pλi(x)yi.
According to this, for any x, y∈ K, we achieve d(x, y) = tr(x + y)− 2 tr
P (x12)y
1
2
≥ tr(x) + tr(y) − 2 Xr i=1
pλi(x)yi
≥ Xr i=1
λi(x) + Xr i=1
yi− 2 Xr
i=1
pλi(x)yi
= Xr i=1
pλi(x)−√ yi
2
≥ 0,
where the second inequality follows from [12, Corollary 4.6]. Hence, we prove that d(x, y)≥ 0.
(b) From the proof of part (a), we know that d(x, y) = tr(x + y)− 2 tr
P (x12)y
1
2 ≥ tr(x) + tr(y) − 2 Xr
i=1
pλi(x)yi
≥ Xr
i=1
pλi(x)−√ yi
2
≥ 0.
Hence, it follows from d(x, y) = 0 that tr(x + y)− 2 tr
P (x12)y
1
2 = tr(x) + tr(y)− 2 Xr i=1
pλi(x)yi
and Xr
i=1
pλi(x)−√ yi
2
= 0.
These lead that tr
P (x12)y
1
2 =
Xr i=1
pλi(x)yi and p
λi(x) =√
yi ∀i = 1, . . . , r.
In addition, applying Theorem 1.3 yields x =
Xr i=1
λi(x)ei = Xr
i=1
yiei = y.
Therefore, it is clear to see that d(x, y) = 0 if and only if x = y.
(c) First, from [14, Proposition 3.2], for any x, y∈ K, we have λi
P (x12)y
= λi
P (y12)x
for i = 1, . . . , r. This leads to λi
P (x12)y
1
2 = λi
P (y12)x
1
2 ∀i = 1, . . . , r.
Hence, it follows that tr
P (x12)y
1
2 = tr
P (y12)x
1
2, which implies that d(x, y) = tr(x + y)− 2 tr
P (x12)y
1
2
= tr(y + x)− 2 tr
P (y12)x
1
2 = d(y, x).
Then, the proof is complete. □
Theorem 2.2. Let d(·, ·) be defined as in (1.3). Then, the function d(x, y) is convex, for any a fixed x∈ K or y ∈ K.
Proof. The proof is the similar to [16, Theorem 2.4]. Hence, we omit it. □ From Theorem 2.1 and Theorem 2.2, we have shown that the function d(·, ·) defined as in (1.3) is a convex semi-distance associated with symmetric cone. How- ever, as indicated in [16], it can be verified by using the convexity of d(·, ·) that the triangle inequality fails. To see this, for given any x, y ∈ K, taking z = λx+(1−λ)y and 0 < λ < 1, there have
d(x, z) = d (x, λx + (1− λ)y) (2.1)
≤ λd(x, x) + (1 − λ)d(x, y) = (1 − λ)d(x, y), d(z, y) = d (λx + (1− λ)y, y)
(2.2)
≤ λd(x, y) + (1 − λ)d(y, y) = λd(x, y).
Then, adding (2.1) and (2.2) together yields
d(x, z) + d(z, y)≤ d(x, y).
In other words, the semi-distance d(·, ·) defined as in (1.3) could not become a
“distance function” (metric function). Thus, we turn our attention to the possibility of d(·, ·) becoming a proximal distance.
In order to prove d(·, ·) could become a proximal distance, we need to modify it a bit. For any x, y ∈ Rn, we define d :Rn× Rn→ R+∪ {+∞} by
(2.3) d(x, y) :=
tr(x + y)− 2 tr
P (x12)y
1
2 ∀x ∈ intK, y ∈ K,
+∞ otherwise.
The above function d(·, ·) is different from the ones given in [1]. To our best knowl- edge, it may be the only proximal distance which is not induced from Bregman distance or φ-divergence. This function, as will be shown below, is a proximal distance on intK.
Theorem 2.3. Let d(·, ·) be defined as in (2.3) in the setting of symmetric cone.
Then, the function d(·, ·) is a proximal distance, i.e., it satisfies
(a) d(·, y) is proper, l.s.c., convex, continuously differentiable on intK;
(b) dom d(·, y) ⊂ intK and dom ∂1d(·, y) = intK, where ∂1d(·, y) denotes the classical subgradient map of the function d(·, y) with respect to the first vari- able;
(c) d(·, y) is level bounded on Rn i.e., lim∥u∥→∞d(u, y) = +∞;
(d) d(y, y) = 0.
Proof. (a) The proof is similar to [5, Lemma 3.1], we omit the details.
(b) The arguments are similar to [16, Proposition 3.5], due to only the general cone structure is used. We also omit them.
(c) Suppose y∈ intK. For any x ∈ intK, as what we do in Theorem 2.1, we write out the spectral decomposition of x and the Peirce decomposition of y, respectively, i.e.,
x = λ1(x) e1+· · · + λr(x) er and y = y1e1+· · · + yrer+X
i<j
yij.
Note that kxk2 = λ1(x)2ke1k2 + · · · + λr(x)2kerk2 ≤ rλ1(x)2ke1k2, where the inequality holds because keik is a constant on V for any primitive idempotent ei (i = 1, . . . , r) and λ1(x) ≥ · · · ≥ λr(x) ≥ 0. Hence, it is easy to check that λ1(x)→ ∞ as kxk → ∞. From this and the proof of part (a) in Theorem 2.1, we have
d(x, y)≥ Xr
i=1
pλi(x)−√ µi
2
≥p
λ1(x)−√ µ1
2
→ ∞.
It follows that d(x, y)→ ∞ as kxk → ∞ for any x ∈ intK. Moreover, d(x, y) = ∞ when x /∈ intK. Then, we prove that d(x, y) → ∞ as kxk → ∞ for any x ∈ Rn. Thus, we conclude that d(·, y) is level bounded on Rn.
(d) This property is trivial.
To sum up, the function d(·, ·) defined as in (2.3) is a proximal distance in the
setting of symmetric cone. □
Remark 2.4. We say a few words about Theorem 2.1 and Theorem 2.3. In fact, when the symmetric cone K reduces to the second-order cone Kn, the conclusions of Theorem 2.1 and Theorem 2.3 correspond to the contents of Theorem 2.5 and theorem 3.7 in [16], respectively. In other words, our results are generalizations of Theorem 2.5 and theorem 3.7 in [16] in a broader framework.
3. Concluding remarks
In this paper, we study a semi-distance associated with symmetric cone K. Fur- thermore, based on it, we construct a proximal distance on intK, which also answers a question raised in [16]. Again, we would like to point out some possible future directions as mentioned in [16].
• Can the function d(·, ·) further become a Bregman distance or φ-divergence?
• Can the function d(·, ·) be extended to nonsymmetric cone setting? In particular, for circular coneLθ, we have already known one type of spectral decomposition of x and some differentiabilities of λi(x), see [24]. By using these facts, we may consider to construct an analogous distance function d(·, ·) in the setting of circular cone.
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Manuscript received December 25, 2020 revised December 19, 2021
X. Miao
School of Mathematics, Tianjin University, Tianjin 300072, P.R. China E-mail address: [email protected]
J.-S. Chen
Department of Mathematics, National Taiwan Normal University, Taipei 11677, Taiwan E-mail address: [email protected]
