1. Introduction
Optimization theory primarily explores the existence of an optimal solution for an objective function under specific conditions and the methods for finding it. The content includes studying the conditions for the existence of the optimal solution and some related criteria and designing the corresponding algorithm to find the optimal solution. We recall the formulation of a nonlinear symmetric cone programming:
where
is a nonlinear objective function,
is a Euclidean Jordan algebra, and
denotes the symmetric cone associated with
. A commonly employed approach for solving symmetric cone programming problems is the proximal point algorithm, which produces a sequence
according to the iterative scheme
Here, is a certain function that satisfies some desirable properties, and a positive sequence. The choice of is important, and several well-known examples are the distances induced by the Euclidean norm, the Bregman distance, the proximal distance, the quasi-distance, and the -divergence.
In previous research, it can be observed that when an algorithm is designed to solve symmetric cone programming problems and investigate its convergence, it is essential to consider inequalities on symmetric cones. Most of these inequalities differ from those in real numbers. Due to the special algebraic structure, deriving inequalities analogous to fundamental ones in real numbers is not always feasible, such as the most fundamental arithmetic–geometric inequality and the Cauchy–Schwarz inequality, among others. Historically, the development of inequalities associated with symmetric cones has been mainly centered on matrix inequalities, as detailed in [
1,
2,
3].
In fact, there are only a few known inequalities associated with second-order cones. Over the past several years, one of our main researches has been devoted to the study of inequalities associated with second-order cones, including defining the means and weighted means and establishing trace inequalities. So far, we have accumulated numerous studies on this topic; see [
4,
5,
6,
7,
8].
The primary aim of this paper is to develop a series of results comparable to classical inequalities in matrix analysis. In this paper, we investigate the eigenvalue, trace, and norm inequalities associated with second-order cones. Furthermore, we investigate the trace versions of an inverse Young inequality and employ them to deduce the trace version of an inverse Hölder inequality and inverse Minkowski inequality associated with second-order cones. These conclusions align with the results established for the positive semidefinite cone, which is also a symmetric cone; see [
9,
10].
The structure of this paper is as follows.
Section 2 reviews the fundamental concepts and properties of symmetric cones, with particular emphasis on second-order cones.
Section 3 begins with a study of Young inequalities associated with second-order cones. The latter part is devoted to the inverse Young inequalities and their applications.
Section 4 concludes the paper with a discussion of potential directions for future research.
2. Preliminaries
In this section, we review the fundamental concepts and properties of Jordan algebras, with reference to [
11] for symmetric cones and [
12,
13,
14] for second-order cones (also Lorentz cones). These concepts are essential for the developments in subsequent sections. Throughout this paper, we employ the following notation and conventions:
: the n-dimensional Euclidean space.
: canonical inner product in .
: Euclidean norm of x given by .
: interior of .
: boundary of .
.
A Euclidean Jordan algebra is a finite-dimensional real inner product space ( for short) equipped with a bilinear map satisfying
- (i)
for all ,
- (ii)
for all ,
- (iii)
for all ,
where , and is called the Jordan product of x and y. If there exists a (unique) element such that for all , then e is called the identity element. It is worth noting that a Jordan algebra does not necessarily possess an identity element.
Given a Euclidean Jordan algebra
, the set of squares
forms a
symmetric cone ([
11], Theorem III.2.1). Specifically,
is a closed convex cone that is self-dual and homogeneous. A cone is called
self-dual if its dual cone with respect to the canonical inner product is itself, and
homogeneous provided that for any
, there is an invertible linear transformation
such that
and
. These properties endow symmetric cones with symmetry and make them particularly tractable in both theoretical and practical settings. In particular, symmetric cones play a fundamental role in conic optimization, generalizing classical cones such as the nonnegative orthant and the cone of positive semidefinite matrices. Symmetric cones also appear in a variety of other areas, including interior-point methods, differential geometry, and the theory of Euclidean Jordan algebras.
Next, we introduce some fundamental notions related to idempotents in :
An element is called an idempotent if .
A nonzero idempotent is said to be primitive if it cannot be written as the sum of two nonzero idempotents.
Two idempotents and are said to be orthogonal if .
Moreover, a finite collection of primitive idempotents in is called a Jordan frame if it satisfies the following two conditions:
- (i)
for all ,
- (ii)
.
Based on the above notions, any element admits a spectral decomposition with respect to a Jordan frame.
Theorem 1 ([
11], Theorem III.1.2)
. (The Spectral Decomposition Theorem) Let be a Euclidean Jordan algebra. Then, there is a number r such that, for every , there exists a Jordan frame and real numbers with Expression (
1) is called the
spectral decomposition of
x, and numbers
(for
) are called the
eigenvalues of
x. Moreover,
is called the
trace of
x, and
is called the
determinant of
x.
The second-order cone (in short SOC) is a well-known example of symmetric cones in
, and it is defined by
In the case of
,
is defined as the set
. Since
is a pointed closed convex cone, it allows us to define partial orders on
as follows:
for any
. The
Jordan product of
x and
y is defined by
for all
. We notice that
serves as the Jordan identity
e. In general, the Jordan product is not associative. However, it is power associative, which means that for all
,
. Without ambiguity, we denote by
the Jordan product of
m copies of
x, and
for any positive integer
m and
n, with the convention that
. Moreover, we remark that
is
not closed under the Jordan product.
Given any
, there exists a unique vector in
, denoted by
, such that
. In fact,
is given explicitly by
where
. For any
, it always holds that
. Consequently, there is a unique vector
, denoted by
, such that
. It is straightforward to claim that for any
,
, and
. For further details, we refer the reader to [
4,
11,
15].
In the setting of a second-order cone in
, any vector
x admits the following decomposition:
where
and
are the eigenvalues (or spectral values) of
x, with the associated eigenvectors (or spectral vectors) denoted by
and
, respectively, as follows:
for
, where
is any vector with
. For the purposes of the subsequent analysis, it is important to note that
is the smaller eigenvalue of
x, and
is the larger one. The decomposition is unique provided that
. As a result, we have the following expressions in terms of eigenvalues associated with second-order cones:
The trace of x is .
The determinant of x is .
The Euclidean norm for x is .
For any real-valued function
, the corresponding vector-valued function associated with second-order cones is given by
for all
; see [
4,
12,
13]. The domain of
will correspond to the domain of
f. In fact, the eigenvalues of
x in the domain of
must belong to the domain of
f. The definition in (
5) remains unambiguous, regardless of whether
or
. Let
m be any real number and consider the real-valued function
. The corresponding vector-valued function allows us to define the
power of any
as follows:
With this definition in place, we are now able to investigate properties of the Young inequality associated with second-order cones.
Recent studies by Maldonado and López have demonstrated the effectiveness of second-order cone programming in statistical learning, especially for Support Vector Machines and Support Vector Regression; see [
16,
17,
18,
19] and references therein. This approach stands in contrast to the prevalent use of semidefinite programming, presenting a novel and potentially impactful research direction.
We conclude this section by summarizing some crucial properties and listing a notation table, see
Table 1. For detailed proofs, the reader is referred to [
4,
11,
12,
15].
Lemma 1. Suppose admit spectral decompositions of the form given in (2)–(4). Then, the following statements are true: - (a)
.
- (b)
, and .
- (c)
, and .
- (d)
If , then , for all .
- (e)
.
- (f)
.
Lemma 2. Suppose that . Then, the following inequality is satisfied. 3. Young Inequality and Inverse Young Inequality
We begin by briefly recalling the classical Young inequality in real numbers. Suppose that
with
. The classical Young inequality states that for all
,
In 1995, a singular value version of the Young inequality for positive definite matrices was established by Ando [
20], stating that for all
,
where
A and
B are positive definite matrices. Recently, Huang, Chen, and Hu [
7] proposed some trace versions of Young inequalities associated with second-order cones. The authors also conjectured the existence of an eigenvalue version of the Young inequality.
Conjecture 1. For any and , the following hold: Later, Huang et al. [
8] established that the Young inequality under the partial order
holds if
x and
y share the same Jordan frame. Furthermore, it would deduce the trace, determinant, and norm version of Young inequalities.
In this section, we may assume that for any , and . In fact, x and y will share the same Jordan frame if or . We first illustrate some inequalities of the eigenvalue associated with second-order cones.
Lemma 3. Suppose that . Then, the following inequalities for the eigenvalues are satisfied:
- (a)
.
- (b)
.
Proof. It is well known that for any
, the minimum and maximum of
a and
b can be expressed as
The eigenvalues of
are given by
(a) By the triangle inequality for norms, it is well known that
It follows from this inequality that
The other part of the inequality follows from
(b) Analogously, we find that
and the remaining inequality is derived from
The proof is thereby concluded. □
Proposition 1. Suppose that and with . Then, the following inequalities are satisfied:
- (a)
.
- (b)
Proof. According to the decomposition of
, it is clear that
since
p and
q are positive and
for
.
(a) It follows by Lemma 3 that
(b) Similarly, the desired inequality follows by
This completes the proof. □
Proposition 2. Suppose that . Then, the following inequality holds: Proof. We note that
, and hence,
Consequently, the result follows from
where the inequalities hold by the triangle inequality for norms and the Cauchy–Schwarz inequality. □
Remark 1. Based on Propositions 1–2, for with , we can establish a picture of the ordered relationship between the eigenvalues of x, y, , as depicted in Figure 1. However, we have no results regarding the relationship between and . In fact, does not always belong to even if . That is, it is possible that . Proposition 3. Suppose that , with . Then, the following inequality holds: Proof. First, we observe by a straightforward calculation that
where
It follows from the triangle inequality for norms that
and similarly,
since
and
. In combining the above inequalities with the definition of a determinant, the first part of the desired inequality follows from
Likewise, the second inequality is valid since
Therefore, we conclude the desired inequalities. □
Remark 2. According to Lemma 2 and the classical Young inequality for real numbers, we obtain a determinant version of Young’s inequality associated with second-order cones. Specifically, for and with , we conclude that In fact, Huang et al. [5] established the determinant version of Young’s inequality based on the SOC weighted mean inequality. However, we obtain a refined inequality by direct computation. Proposition 4. Suppose that , with . Then the following inequality holds. Proof. Let
be expressed as in (
6). Following the similar argument in Proposition 3, we deduce that
Similarly, the other inequality holds by
Thus, the desired result follows. □
Proposition 5. Suppose that . Then, the following inequality holds Proof. It is evident that the inequalities hold if
or
. In fact, the equality will hold if
or
. Assume that
and
, which implies
and
. Then,
where
is the angle between
and
in
. We notice that the value of
is determined by
if
,
,
, and
are fixed. Let
be defined by
Then, it is clear that 0 and
are the only two critical points of
f since
Therefore, the extreme values of
occur at
. For
, we have
On the other hand, for
, we obtain
Thus, the norm achieves its maximum and minimum at and , respectively. The proof is complete. □
Remark 3. According to the proof of Proposition 4, we remark that the maximum and minimum of the norm also occur at and , respectively. In addition, for with , we could obtain the relationship between these two maxima and minima by applying the classical Young inequality; see Figure 2. However, we have not reached a conclusion on whether the inequality is true or not. In the following, we investigate the inverse Young inequality associated with second-order cones and its applications. We first review the classic inverse Young inequality in
, namely
for
and
. In the context of matrix analysis, Manjegani and Norouzi [
21] established an inverse Young inequality for eigenvalues. Specifically, for all
and
,
where
A and
B are positive definite matrices. However, Drury [
22] provided counterexamples to inequality (
7) for
, and then modified it slightly. He proved that inequality (
7) holds only for
. In the following, we first discuss the trace version of the inverse Young inequality associated with second-order cones.
Theorem 2. (Inverse Young inequality—Type I) Suppose that and . Then, the following inequality holds: Proof. According to Lemma 1(e) and (f), the desired result follows from the following computation:
where the second inequality follows from the inverse Young inequality in
. □
Corollary 1. (Inverse Young inequality—Type II) Suppose that and . Then, the following inequality holds: Proof. Since , the results follow immediately from Lemma 1(d) and Theorem 2. □
Corollary 2. (Inverse Young inequality—Type III) Suppose that and . If x and y are not in , then the following inequality is satisfied: Proof. Since x and y are not in , it follows from Lemma 1(a)–(c) that and are both in . Therefore, the desired inequality is obtained by applying Theorem 2 to and . □
Analogously to the classical setting in real analysis, we derive the trace versions of the inverse Hölder inequality by applying the trace versions of the inverse Young inequality through a similar analytical approach.
Theorem 3. (Inverse Hölder inequality—Type I) Suppose that and . Then, the following inequality is satisfied: Proof. Let
,
. Since
, this implies that
and
. Applying Theorem 2 to
and
, we obtain
Multiplying both sides by
, we obtain
since
. □
Corollary 3. (Inverse Hölder inequality—Type II) Suppose that and . If x and y are not in , then the following inequality is satisfied: Proof. Following the similar argument in Corollary 2, the desired inequality follows by applying Theorem 3 to and . □
Next, we apply the trace version of the inverse Hölder inequality to establish the trace version of the inverse Minkowski inequality.
Theorem 4. (Inverse Minkowski inequality) Suppose that and . Then, the following inequality is satisfied: Proof. By Lemma 1(e) and the distributive property of the Jordan product, we can write
Since
is also in
, we apply Theorem 3 to each term in the above expression, and thus obtain
Dividing both sides by , which is positive since , we obtain as required. □
Remark 4. We provide a more detailed discussion of Theorem 4. Let . Then, the trace version of the inverse Minkowski inequality can be equivalently rewritten aswhere . Huang et al. [6] showed that the quantitydefines a quasinorm on , known as the Schatten p-quasinorm. More recently, Jeong [23] established that this expression also defines a quasinorm on Euclidean Jordan algebras. In fact, it is analogous to the Schatten p-norm (or quasi-norm), which is defined via the p-norm of the singular values of a matrix; see [3] for further discussion. Combining ([6], Theorem 3.7) with Theorem 4, we can conclude that for all ,where . At the end of this section, we provide counterexamples to illustrate that the eigenvalue version of the inverse Young inequality associated with second-order cones does not always hold. That is, for all
, the inequality
fails to hold for all
when
.
Example 1. Let , . Then, we computeand thus, . Consequently, we obtain Example 2. Let , . Then, we computewhich yields . Hence, the eigenvalues of and are derived bywhich implies . In Example 2, we observe that it also serves as a counterexample to the determinant version of the inverse Young inequality, namely,
which fails to hold when
. However, for an alternative determinant version of the inverse Young inequality involving the absolute value, namely,
no conclusive result has been obtained so far.