Pacific Journal of Optimization, vol. 14, no. 3, pp. 399-419, 2018

### From symmetric cone optimization to nonsymmetric cone optimization: Spectral decomposition, nonsmooth analysis, and

### projections onto nonsymmetric cones

^{1}

Xin-He Miao^{2}

Department of Mathematics Tianjin University, China

Tianjin 300072, China

Yue Lu ^{3}

School of Mathematical Sciences Tianjin Normal University

Tianjin 300387, China

Jein-Shan Chen ^{4}
Department of Mathematics
National Taiwan Normal University

Taipei 11677, Taiwan October 6, 2017

(revised on February 8, 2018)

Abstract. It is well known that Euclidean Jordan algebra is an unified framework for symmetric cone programs, including positive semidefinite programs and second-order cone programs. Unlike symmetric cone programs, there is no unified analysis technique to deal with nonsymmetric cone programs. Nonetheless, there are several common concepts

1This is an extended version of “From symmetric cone optimization to nonsymmetric cone optimiza- tion: Projections onto nonsymmetric cones”, Proceedings of the Twenty-Eighth RAMP Symposium, Niigata University, pp. 25-34, October, 2016.

2E-mail: xinhemiao@tju.edu.cn. The author’s work is supported by National Natural Science Foun- dation of China (No. 11471241).

3The research is supported by National Natural Science Foundation of China (Grant Number:

11601389), Doctoral Foundation of Tianjin Normal University (Grant Number: 52XB1513) and 2017- Outstanding Young Innovation Team Cultivation Program of Tianjin Normal University (Grant Number:

135202TD1703).

4Corresponding author. The author’s work is supported by Ministry of Science and Technology, Taiwan. E-mail: jschen@math.ntnu.edu.tw

when dealing with general conic optimization. More specifically, we believe that spectral decomposition associated with cones, nonsmooth analysis regarding cone-functions, pro- jections onto cones, and cone-convexity are the bridges between symmetric cone programs and nonsymmetric cone programs. Hence, this paper is devoted to looking into the first three items in the setting of nonsymmetric cones. The importance of cone-convexity is recognized in the literature so that it is not discussed here. All results presented in this paper are very crucial to subsequent study about the optimization problems associated with nonsymmetric cones.

Key words. Spectral decomposition, nonsmooth analysis, projection, symmetric cone, nonsymmetric cone.

Mathematics Subject Classification: 49M27, 90C25

### 1 Introduction

Symmetric cone optimization, including SDP (positive semidefinite programming) and SOCP (second-order cone programming) as special cases, has been a popular topic during the past two decades. In fact, for many years, there has been much attention on sym- metric cone optimization, see [10, 11, 14, 20, 27, 30, 32, 35, 38] and references therein.

Recently, some researchers have paid attention to nonsymmetric cones, for example, ho- mogeneous cone [9, 28, 40], matrix norm cone [18], p-order cone [1, 23, 41], hyperbolicity cone [24, 26, 36], circular cone [13, 15, 42] and copositive cone [16], etc.. In general, the structure of symmetric cone is quite different from the one of non-symmetric cone.

In particular, unlike the symmetric cone optimization in which the Euclidean Jordan algebra can unify the analysis, so far no unified algebra structure has been found for non-symmetric cone optimization. This motivates us to find the common bridge between them. Based on our earlier experience, we think the following four items are crucial:

• spectral decomposition associated with cones.

• smooth and nonsmooth analysis for cone-functions.

• projection onto cones.

• cone-convexity.

The role of cone-convexity had been recognized in the literature. In this paper, we focus on the other three items that are newly explored recently by the authors. Moreover, we look into several kinds of nonsymmetric cones, that is, the circular cone, the p-order cone, the geometric cone, the exponential cone and the power cone, respectively. The symmetric cone can be unified under Euclidean Jordan algebra, which will be introduced

later. Unlike the symmetric cone, there is no unified framework for dealing with non- symmetric cones. This is the main source where the difficulty arises from. Note that the homogeneous cone can be unified under so-called T -algebra [28, 39, 40].

We begin with introducing Euclidean Jordan algebra [29] and symmetric cone [19].

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 ◦ (x^{2}◦ y) = x^{2}◦ (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

(iii) hx ◦ y, zi

V = hy, x ◦ zi

V for all x, y, z ∈ V.

Given a Euclidean Jordan algebra A = (V, ◦, h·, ·i_{V}), we denote the set of squares as
K := x^{2}| x ∈ V .

By [19, 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 ∈ intK, there exists an invertible linear transformation T : V → V such that T (K) = K and T (x) = y.

Below are three well-known examples of Euclidean Jordan algebras.

Example 1.1. Consider R^{n} with the (usual) inner product and Jordan product defined
respectively as

hx, yi =

n

X

i=1

x_{i}y_{i} and x ◦ y = x ∗ y ∀x, y ∈ R^{n}

where x_{i} denotes the ith component of x, etc., and x ∗ y denotes the componentwise
product of vectors x and y. Then, R^{n}is a Euclidean Jordan algebra with the nonnegative
orthant R^{n}+ as its cone of squares.

Example 1.2. Let S^{n} be the space of all n × n real symmetric matrices with the trace
inner product and Jordan product, respectively, defined by

hX, Y i_{T}:= Tr(XY ) and X ◦ Y := 1

2(XY + Y X) ∀X, Y ∈ S^{n}.

Then, (S^{n}, ◦, h·, ·i_{T}) is a Euclidean Jordan algebra, and we write it as Sn. The cone of
squares S^{n}+ in Sn is the set of all positive semidefinite matrices.

Example 1.3. The Jordan spin algebra Ln. Consider R^{n} (n > 1) with the inner product
h·, ·i and Jordan product

x ◦ y :=

hx, yi
x_{0}y + y¯ _{0}x¯

for any x = (x_{0}, ¯x), y = (y_{0}, ¯y) ∈ R × R^{n−1}. We denote the Euclidean Jordan algebra
(R^{n}, ◦, h·, ·i) by Ln. The cone of squares, called the Lorentz cone (or second-order cone),
is given by

L^{+}n :=(x_{0}; ¯x) ∈ R × R^{n−1}| x_{0} ≥ k¯xk .

For any given x ∈ A, let ζ(x) be the degree of the minimal polynomial of x, i.e.,
ζ(x) := mink : {e, x, x^{2}, · · · , 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 c^{2} = c. Two idempotents c_{i} and c_{j} are said to be orthogonal if c_{i} ◦ c_{j} = 0.

One says that {c_{1}, c_{2}, . . . , c_{k}} is a complete system of orthogonal idempotents if

c^{2}_{j} = c_{j}, c_{j} ◦ c_{i} = 0 if j 6= i for all j, i = 1, 2, · · · , k, and

k

X

j=1

c_{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 1.1. [19, 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, . . . , c_{r}} and real
numbers λ1(x), . . . , λr(x), arranged in the decreasing order λ1(x) ≥ λ2(x) ≥ · · · ≥ λr(x),
such that

x = λ_{1}(x)c_{1}+ λ_{2}(x)c_{2}+ · · · + λ_{r}(x)c_{r}.

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

j=1λ_{j}(x) the trace of x.

From [19, 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. Then, 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 [19, 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 1.2. [19, Chapter V.3.7] Every simple Euclidean Jordan algebra is isomorphic to one of the following.

(i) The Jordan spin algebra L^{n}.

(ii) The algebra S^{n} of n × n real symmetric matrices.

(iii) The algebra H^{n} of all n × n complex Hermitian matrices.

(iv) The algebra Q^{n} of all n × n quaternion Hermitian matrices.

(v) The algebra O^{3} of all 3 × 3 octonion Hermitian matrices.

Given an n-dimensional Euclidean Jordan algebra A = (V, h·, ·i, ◦) with K being its
corresponding symmetric cone in V. For any scalar function f : R → R, we define a
vector-valued function f^{sc}(x) (called L¨owner function) on V as

f^{sc}(x) = f (λ_{1}(x))c_{1}+ f (λ_{2}(x))c_{2}+ · · · + f (λ_{r}(x))c_{r} (1)
where x ∈ V has the spectral decomposition

x = λ_{1}(x)c_{1}+ λ_{2}(x)c_{2}+ · · · + λ_{r}(x)c_{r}.

When V is the space S^{n} which means n × n real symmetric matrices. The spectral
decomposition reduces to the following: for any X ∈ S^{n},

X = P

λ_{1}

. ..

λ_{n}

P^{T},

where λ_{1}, λ_{2}, · · · , λ_{n} are eigenvalues of X and P is orthogonal (i.e., P^{T} = P^{−1}). Under
this setting, for any function f : R → R, we define a corresponding matrix valued
function associated with the Euclidean Jordan algebra S^{n} := Sym(n, R), denoted by
f^{mat}: S^{n} → S^{n}, as

f^{mat}(X) = P

f (λ_{1})

. ..

f (λn)

P^{T}.

For this case, Chen, Qi and Tseng in [12] show that the function f^{mat} inherits from
f the properties of continuity, Lipschitz continuity, directional differentiability, Fr´echet
differentiability, continuous differentiability, as well as semismoothness. We state them
as below.

Theorem 1.3. (a) f^{mat} is continuous if and only if f is continuous.

(b) f^{mat} is directionally differentiable if and only if f is directionally differentiable.

(c) f^{mat} is Fr´echet-differentiable if and only if f is Fr´echet-differentiable.

(d) f^{mat} is continuously differentiable if and only if f is continuously differentiable.

(e) f^{mat} is locally Lipschitz continuous if and only if f is locally Lipschitz continuous.

(f ) f^{mat} is globally Lipschitz continuous with constant κ if and only if f is globally Lip-
schitz continuous with constant κ.

(g) f^{mat} is semismooth if and only if f is semismooth.

When V is the Jordan spin algebra L^{n} in which K corresponds to the second-order
cone (SOC), which is defined as

K^{n} := {(x_{1}, ¯x) ∈ R × R^{n−1}| k¯xk ≤ x_{1}},

the function f^{sc} reduces to so-called SOC-function f^{soc} studied in [4, 6, 7, 8]. More
specifically, under such case, the spectral decomposition for any x = (x_{1}, ¯x) ∈ R × R^{n−1}
becomes

x = λ_{1}(x)u^{(1)}_{x} + λ_{2}(x)u^{(2)}_{x} , (2)
where λ_{1}(x), λ_{2}(x), u^{(1)}x and u^{(2)}x with respect to K^{n} are given by

λi(x) = x1+ (−1)^{i}k¯xk,
u^{(i)}_{x} =

1 2

1, (−1)^{i} x¯
k¯xk

if ¯x 6= 0,

1 2

1, (−1)^{i}w

if ¯x = 0,

for i = 1, 2, with w being any vector in R^{n−1} satisfying kwk = 1. If ¯x 6= 0, the decompo-
sition (2) is unique. With this spectral decomposition, for any function f : R → R, the
L¨owner function f^{sc} associated with K^{n} reduces to f^{soc} as below:

f^{soc}(x) = f (λ_{1}(x))u^{(1)}_{x} + f (λ_{2}(x))u^{(2)}_{x} ∀x = (x_{1}, ¯x) ∈ R × R^{n−1}. (3)
The picture of second-order cone K^{n} in R^{3} is depicted in Figure 1.

For general symmetric cone case, Baes [2] consider the convexity and differentiability
properties of spectral functions. For this SOC setting, Chen, Chen and Tseng in [8] show
that the function f^{soc} inherits from f the properties of continuity, Lipschitz continuity,
directional differentiability, Fr´echet differentiability, continuous differentiability, as well
as semismoothness. In other words, the following hold.

Theorem 1.4. (a) f^{soc} is continuous if and only if f is continuous.

(b) f^{soc} is directionally differentiable if and only if f is directionally differentiable.

Figure 1: The second-order cone in R^{3}

(c) f^{soc} is Fr´echet-differentiable if and only if f is Fr´echet-differentiable.

(d) f^{soc} is continuously differentiable if and only if f is continuously differentiable.

(e) f^{soc} is locally Lipschitz continuous if and only if f is locally Lipschitz continuous.

(f ) f^{soc} is globally Lipschitz continuous with constant κ if and only if f is globally Lips-
chitz continuous with constant κ.

(g) f^{soc} is semismooth if and only if f is semismooth.

As for general symmetric cone case, Sun and Sun [38] uses φ

V to denote f^{sc} defined
as in (1). More specifically, for any function φ : R → R, they define a corresponding
function associated with the Euclidean Jordan algebra V by

φV(x) = φ(λ_{1}(x))c_{1}+ φ(λ_{2}(x))c_{2}+ · · · + φ(λ_{r}(x))c_{r},

where λ_{1}(x), λ_{2}(x), · · · , λ_{r}(x) and c_{1}, c_{2}, · · · , c_{r} are the spectral values and spectral vec-
tors of x, respectively. In addition, Sun and Sun [38] extend some of the aforementioned
results to more general symmetric cone case regarding f^{sc} (i.e., φ

V).

Theorem 1.5. Assume that the symmetric cone is simple in the Euclidean Jordan algebra V.

(a) φ

V is continuous if and only if φ is continuous.

(b) φ_{V} is directionally differentiable if and only if φ is directionally differentiable.

(c) φ

V is Fr´echet-differentiable if and only if φ is Fr´echet-differentiable.

(d) φ

V is continuously differentiable if and only if φ is continuously differentiable.

(e) φ

V is semismooth if and only if φ is semismooth.

With respect to matrix cones, Ding et al. [17] recently introduce a class of matrix- valued functions, which is called spectral operator of matrices. This class of functions generalizes the well known L¨owner operator and has been used in many important appli- cations related to structured low rank matrices and other matrix optimization problems in machine learning and statistics. Similar to Theorem 1.4 and Theorem 1.5, the con- tinuity, directional differentiability and Frechet-differentiability of spectral operator are also obtained. See [17, Theorem 3, 4 and 5] for more details.

For subsequent needs, for a closed convex cone K ⊆ R^{n}, we also recall its dual cone,
polar cone, and the projection onto itself. For any a given closed convex cone K ⊆ R^{n},
its dual cone is defined by

K^{∗} := {y ∈ R^{n}| hy, xi ≥ 0, ∀x ∈ K},

and its polar cone is K^{◦} := −K^{∗}. Let ΠK(z) denote the Euclidean projection of z ∈ R^{n}
onto the closed convex cone K. Then, it follows that z = ΠK(z) − ΠK^{∗}(−z) and

ΠK(z) = argmin_{x∈K} 1

2kx − zk^{2}.

### 2 Circular cone

The definition of the circular cone L_{θ} is defined as [42]:

L_{θ} := x = (x1, ¯x) ∈ R × R^{n−1}| kxk cos θ ≤ x_{1}

= x = (x_{1}, ¯x) ∈ R × R^{n−1}| k¯xk ≤ x_{1}tan θ .

From the concept of the circular cone L_{θ}, we know that when θ = ^{π}_{4}, the circular cone
is exactly the second-order cone K^{n}. In addition, we also see that L_{θ} is solid (i.e.,
int L_{θ} 6= ∅), pointed (i.e., L_{θ}∩ −L_{θ} = 0), closed convex cone, and has a revolution axis
which is the ray generated by the canonical vector e_{1} := (1, 0, · · · , 0)^{T} ∈ R^{n}. Moreover,
its dual cone is given by

L^{∗}_{θ} := {y = (y_{1}, ¯y) ∈ R × R^{n−1}| kyk sin θ ≤ y_{1}}

= {y = (y_{1}, ¯y) ∈ R × R^{n−1}| k¯yk ≤ y_{1}cot θ}

= L^{π}

2−θ.

The pictures of circular cone L_{θ} in R^{3} are depicted in Figure 2.

In view of the expression of the dual cone L^{∗}_{θ}, we see that the dual cone L^{∗}_{θ} is also a
solid, pointed, closed convex cone. By the reference [42], the explicit formula of projection
onto the circular cone L_{θ} can be expressed by in the following theorem.

Figure 2: Three different circular cones in R^{3}.

Theorem 2.1. ([42]) Let x = (x_{1}, ¯x) ∈ R × R^{n−1} and x_{+} denote the projection of x onto
the circular cone L_{θ}. Then x_{+} is given below:

x_{+}=

x if x ∈ L_{θ},
0 if x ∈ −L^{∗}_{θ},
u otherwise,
where

u =

x_{1}+ k¯xk tan θ
1 + tan^{2}θ

x_{1}+ k¯xk tan θ
1 + tan^{2}θ tan θ

x¯ k¯xk

.

Zhou and Chen [42] also present the decomposition of x, which is similar to the one in the setting of second-order cone.

Theorem 2.2. ([42, Theorem 3.1]) For any x = (x_{1}, ¯x) ∈ R × R^{n−1}, one has
x = λ_{1}(x)u^{(1)}_{x} + λ_{2}(x)u^{(2)}_{x} ,

where

λ1(x) = x1 − k¯xk cot θ
λ_{2}(x) = x_{1} + k¯xk tan θ
and

u^{(1)}_{x} = 1
1 + cot^{2}θ

1 0

0 cot θ

1

−w

u^{(2)}_{x} = 1
1 + tan^{2}θ

1 0

0 tan θ

1 w

with w = _{k¯}^{x}_{xk}^{¯} if ¯x 6= 0, and any vector in R^{n−1} satisfying kwk = 1 if ¯x = 0.

Theorem 2.3. ([42, Theorem 3.2]) For any x = (x_{1}, ¯x) ∈ R^{n}× R, we have
x+= (λ1(x))+u^{(1)}_{x} + (λ2(x))+u^{(2)}_{x} ,

where (a)_{+} := max{0, a}, λ_{i}(x) and u^{(i)}x for i = 1, 2 are given as in Theorem 2.2.

With this spectral decomposition of x, for any function f : R → R, the L¨owner
function f^{circ} associated with L_{θ} is defined as below:

f^{circ}(x) = f (λ_{1}(x))u^{(1)}_{x} + f (λ_{2}(x))u^{(2)}_{x} ∀x = (x_{1}, ¯x) ∈ R × R^{n−1}. (4)
In [15], Chang, Yang and Chen have obtained that many properties of the function f^{circ}
are inherited from the function f , which is represented in the following theorem.

Theorem 2.4. ([15]) For any the function f : R → R, the vector-valued function f^{circ}
is defined by (4). Then, the following results hold.

(a) f^{circ} is continuous at x ∈ R^{n} with spectral values λ_{1}(x), λ_{2}(x) if and only if f is
continuous at λ1(x), λ2(x).

(b) f^{circ} is directionally differentiable at x ∈ R^{n} with spectral values λ_{1}(x), λ_{2}(x) if and
only if f is directionally differentiable at λ1(x), λ2(x).

(c) f^{circ} is differentiable at x ∈ R^{n} with spectral values λ_{1}(x), λ_{2}(x) if and only if f is
differentiable at λ1(x), λ2(x).

(d) f^{circ} is strictly continuous at x ∈ R^{n} with spectral values λ_{1}(x), λ_{2}(x) if and only if
f is strictly continuous at λ1(x), λ2(x).

(e) f^{circ} is semismooth at x ∈ R^{n} with spectral values λ_{1}(x), λ_{2}(x) if and only if f is
semismooth at λ_{1}(x), λ_{2}(x).

(f ) f^{circ} is continuously differentiable at x ∈ R^{n} with spectral values λ_{1}(x), λ_{2}(x) if and
only if f is continuously differentiable at λ_{1}(x), λ_{2}(x).

We point out that there is a close relation between L_{θ} and K^{n} (see [34, 42]) as below
K^{n} = AL_{θ} where A :=tan θ 0

0 I

.

We point out a few points regarding circular cones. First, as mentioned in [43], it is possible to construct a new inner product which ensures the circular cone is self-dual.

However, it is not possible to make both L_{θ} and K^{n} are self-dual under a certain inner
product. Secondly, as shown in [43], the relation K^{n}= ALθ does not guarantee that there
exists a similar close relation between f^{circ} and f^{soc}. The third point is that the structure
of circular cone helps on constructing complementarity functions for the circular cone
complementarity problem as indicated in [34].

### 3 The p-order cone

The p-order cone in R^{n}, which is a generalization of the second-order cone K^{n}[14], is
defined as

K_{p} :=

x ∈ R^{n}

x_{1} ≥

n

X

i=2

|x_{i}|^{p}

!_{p}^{1}

(p ≥ 1). (5)

In fact, the p-order cone K_{p} can be equivalently expressed as

K_{p} =x = (x_{1}, ¯x) ∈ R × R^{n−1}| x_{1} ≥ k¯xk_{p} , (p ≥ 1),

where ¯x := (x_{2}, · · · , x_{n})^{T} ∈ R^{n−1}. From (5), it is clear to see that when p = 2, K_{2} is
exactly the second-order cone K^{n}. That means that the second-order cone is a special
case of p-order cone. Moreover, it is known that K_{p} is a convex cone and its dual cone is
given by

K^{∗}_{p} =

y ∈ R^{n}

y_{1} ≥

n

X

i=2

|y_{i}|^{q}

!^{1}_{q}

or equivalently

K_{p}^{∗} =y = (y1, ¯y) ∈ R × R^{n−1}| y_{1} ≥ k¯yk_{q} = Kq

with ¯y := (y_{2}, · · · , y_{n})^{T} ∈ R^{n−1}, where q ≥ 1 and satisfies ^{1}_{p}+^{1}_{q} = 1. From the expression
of the dual cone K^{∗}_{p}, we see that the cone K^{∗}_{p} is also a convex cone. For an application of
p-order cone programming, we refer the readers to [41], in which a primal-dual potential
reduction algorithm for p-order cone constrained optimization problems is studied. Be-
sides, in [41], a special optimization problem called sum of p-norms is transformed into
an p-order cone constrained optimization problems. The pictures of three different cones
K_{p} in R^{3} are depicted in Figure 3.

Figure 3: Three different p-order cones in R^{3}

In [33], Miao, Qi and Chen explore the expression of the projection onto p-order cone and the spectral decomposition associated with p-order cone, which are shown the following theorems.

Theorem 3.1. ([33, Theorem 2.1]) For any z = (z_{1}, ¯z) ∈ R × R^{n−1}, then the projection
of z onto K_{p} is given by

ΠKp(z) =

z, z ∈ K_{p}

0, z ∈ −K_{p}^{∗} = −K_{q}

u, otherwise (i.e., −k¯zk_{q} < z_{1} < k¯zk_{p})
where u = (u1, ¯u) with ¯u = (u2, u3, · · · , un)^{T} ∈ R^{n−1} satisfying

u1 = k¯ukp = (|u2|^{p}+ |u3|^{p}+ · · · + |un|^{p})^{1}^{p}
and

u_{i}− z_{i}+ u_{1}− z_{1}

u^{p−1}_{1} |u_{i}|^{p−2}u_{i} = 0, ∀i = 2, · · · , n.

Theorem 3.2. ([33, Theorem 2.2]) Let z = (z_{1}, ¯z) ∈ R × R^{n−1}. Then, z can be decom-
posed as

z = α_{1}(z) · v^{(1)}(z) + α_{2}(z) · v^{(2)}(z),
where

α_{1}(z) = z1+ k¯zkp

2 α2(z) = z1− k¯zkp

2

and

v^{(1)}(z) =

1

¯ w

v^{(2)}(z) =

1

− ¯w

with ¯w = _{k¯}_{zk}^{z}^{¯}

p if ¯z 6= 0; while ¯w being an arbitrary element satisfying k ¯wk_{p} = 1 if ¯z = 0.

For the projection onto p-order cone, we notice that this projection is not an explicit formula because it is hard to solve the equations which in Theorem 3.1. Moreover, the decomposition for z is not an orthogonal decomposition, which is different from the case in the second-order cone and circular cone setting. Because the decomposition for z is not an orthogonal decomposition, the corresponding nonsmooth analysis for its cone- functions is not established.

### 4 Geometric cone

The geometric cone is defined as bellow [22]:

G^{n}:=

(

(x, θ) ∈ R^{n}+× R+

n

X

i=1

e^{−}^{xi}^{θ} ≤ 1
)

where x = (x_{1}, · · · , x_{n})^{T} ∈ R^{n}+ and we also use the convention e^{−}^{xi}^{0} = 0. From the
definition of the geometric cone G^{n}, we know that G^{n} is solid (i.e., int G^{n} 6= ∅), pointed
(i.e., G^{n}∩ −G^{n}= 0), closed convex cone, and its dual cone is given by

(G^{n})^{∗} =
(

(y, µ) ∈ R^{n}+× R+

µ ≥ X

yi>0

y_{i}ln y_{i}
Pn

i=1y_{i}
)

where µ ∈ R^{+} and y = (y1, · · · , yn)^{T} ∈ R^{n}+. In view of the expression of the dual cone
(G^{n})^{∗}, we see that the dual cone (G^{n})^{∗} is also a solid, pointed, closed convex cone, and
((G^{n})^{∗})^{∗} = G^{n}. When n = 1, we note that the geometric cone G^{1} is just nonnegative
octant cone R^{2}+. In addition, by the expression of the geometric cone G^{n} and its dual
cone (G^{n})^{∗}, it is not hard to verify that the boundary of the geometric cone G^{n} and its
dual cone (G^{n})^{∗} can be respectively expressed as follows:

bd G^{n}=
(

(x, θ) ∈ R^{n}+× R+

n

X

i=1

e^{−}^{xi}^{θ} = 1
)

and

bd (G^{n})^{∗} =
(

(y, µ) ∈ R^{n}+× R+

µ = X

yi>0

y_{i}ln y_{i}
Pn

i=1y_{i}
)

.

For an application of geometric cone programming, we refer the readers to [21], in which
the author shows how to transform a prime-dual pair of geometric optimization problem
into a constrained optimization problem related with G^{n} and (G^{n})^{∗}. The pictures of G^{n}
and its dual cone (G^{n})^{∗} in R^{3} are depicted in Figure 4.

Figure 4: The geometric cone (left) and its dual cone (right) in R^{3}
Next, we present the projection of (x, θ) ∈ R^{n}× R onto the geometric cone G^{n}.

Theorem 4.1. Let x = (x, θ) ∈ R^{n} × R. Then the projection of x onto the geometric
cone G^{n} is given by

ΠG^{n}(x) =

x, if x ∈ G^{n},
0, if x ∈ (G^{n})^{◦},
u, otherwise,

(6)

where u = (u, λ) ∈ R^{n}+× R^{+} with u = (u1, u2, · · · , un)^{T} ∈ R^{n}+ satisfying
u_{i}− x_{i}+ λ(λ − θ)

Pn

j=1e^{−}^{uj}^{λ}u_{j}

e^{−}^{ui}^{λ} = 0, i = 1, 2, · · · , n (7)

and n

X

i=1

e^{−}^{ui}^{λ} = 1. (8)

Proof. From Projection Theorem [3, Prop. 2.2.1], we know that, for every x = (x, θ) ∈
R^{n}× R, a vector u ∈ G^{n} is equal to the projection point ΠG^{n}(x) if and only if

u ∈ G^{n}, x − u ∈ (G^{n})^{◦}, and hx − u, ui = 0.

With this, the first two cases of (6) are obvious. Hence, we only need to consider the third
case. Based on (8) and the definition of G^{n}, it is obvious that u ∈ G^{n}. In addition, from
(7), we obtain that Pn

i=1ui(ui− xi) + λ(λ − θ) = 0, which explains that hx − u, ui = 0.

Next, we argue that x − u ∈ (G^{n})^{◦}. To see this, by (7) and (8), we have

n

X

i=1

(ui− xi) = − λ(λ − θ) Pn

j=1e^{−}^{uj}^{λ}u_{j}.
Together with (7) again, it follows that ^{P}n^{u}^{i}^{−x}^{i}

j=1(uj−x_{j}) = e^{−}^{ui}^{λ}, which leads to ln^{P}n^{u}^{i}^{−x}^{i}
j=1(uj−x_{j}) =

−^{u}_{λ}^{i}. Hence, we have

X

ui−xi>0

(u_{i}− x_{i}) ln u_{i}− x_{i}
Pn

j=1(u_{j} − x_{j})

= − X

ui−xi>0

(ui− xi)u_{i}
λ

= −1 λ

X

ui−xi>0

(u_{i}− x_{i})u_{i}

≤ 1

λ · λ(λ − θ) = λ − θ, where the inequality holds since Pn

i=1u_{i}(u_{i} − x_{i}) + λ(λ − θ) = 0. This explains that
u − x ∈ (G^{n})^{∗}, i.e, x − u ∈ (G^{n})^{◦}. Then, the proof is complete. 2

For the projection onto geometric cone G^{n}, we notice again that this projection is not
an explicit formula since the equations (7) and 8 cannot be easily solved. Moreover, the
decomposition associated with the geometric cone G^{n} and the corresponding nonsmooth
analysis for its cone-functions are not established.

### 5 The exponential cone

The exponential cone is defined as bellow [5, 37]:

Ke:= cl n

(x1, x2, x3)^{T} ∈ R^{3}

x2e^{x1}^{x2} ≤ x3, x2 > 0
o

. In fact, the exponential cone can be expressed as the union of two sets, i.e.,

Ke :=

n

(x1, x2, x3)^{T} ∈ R^{3}

x2e^{x1}^{x2} ≤ x3, x2 > 0
o

∪(x1, 0, x3)^{T}

x1 ≤ 0, x3 ≥ 0 .
As shown in [5], the dual cone K^{∗}_{e} of the exponential cone K_{e} is given by

K_{e}^{∗} = cln

(y_{1}, y_{2}, y_{3})^{T} ∈ R^{3}

− y_{1}e^{y2}^{y1} ≤ ey_{3}, y_{1} < 0o
.

In addition, the dual cone K^{∗}_{e} is expressed as the union of the two following sets:

K^{∗}_{e} =n

(y_{1}, y_{2}, y_{3})^{T} ∈ R^{3}

− y_{1}e^{y2}^{y1} ≤ ey_{3}, y_{1} < 0o

∪(0, y2, y_{3})^{T}

y_{2} ≥ 0, y_{3} ≥ 0 .
From the expression of the exponential cone K_{e} and its dual cone K^{∗}_{e}, it is known that
the exponential cone Ke and its dual cone K^{∗}_{e} are closed convex cone in R^{3}. Moreover,
based on the expression of K_{e} and K^{∗}_{e}, it is easy to verify that their boundary can be
respectively expressed as follows:

bd Ke :=

n

(x1, x2, x3)^{T} ∈ R^{3}

x2e^{x1}^{x2} = x3, x2 > 0
o

∪(x1, 0, x3)^{T}

x1 ≤ 0, x3 ≥ 0 . and

bd K^{∗}_{e} :=

n

(y1, y2, y3)^{T} ∈ R^{3}

− y1e^{y2}^{y1} = ey3, y1 < 0
o

∪(0, y2, y3)^{T}

y2 ≥ 0, y3 ≥ 0 .
For an application of exponential cone programming, we refer the readers to [5], in which
interior-point algorithms for structured convex optimization involving exponential have
been investigated. The pictures of the exponential cone Ke and its dual cone K^{∗}_{e} in R^{3}
are depicted in Figure 5.

For the geometric cone G^{n} and the exponential cone K_{e}, there exists the relationship
between these two types of cones, which is described in the following proposition.

Proposition 5.1. Under the suitable conditions, there is a corresponding relationship
between the geometric cone G^{n} and exponential cone K_{e}.

Figure 5: The exponential cone (left) and its dual cone (right) in R^{3}

Proof. For any (x, θ) ∈ G^{n} with x = (x_{1}, x_{2}, · · · , x_{n})^{T} ∈ R^{n}+, we have Pn

i=1e^{−}^{xi}^{θ} ≤ 1.

With this, it is equivalent to say

e^{−}^{xi}^{θ} ≤ z_{i}, and

n

X

i=1

z_{i} = 1.

Hence, we obtain that
(−x_{i}

θ, 1, z_{i})^{T} ∈ K_{e} (i = 1, 2, · · · , n) and

n

X

i=1

z_{i} = 1.

For the above analysis, it is clear to see that the proof is reversible.

Besides, we give another form of transformation for the exponential cone Ke. Indeed, for
any ˜x := (x_{1}, x_{2}, x_{3})^{T} := (ˆx^{T}, x_{3})^{T} ∈ K_{e} with ˆx := (x_{1}, x_{2})^{T}, we have two cases, i.e.,

(a) x_{2}e^{x1}^{x2} ≤ x_{3} and x_{2} > 0, or
(b) x1 ≤ 0, x2 = 0, x3 ≥ 0.

For the case (a), if x_{2} = x_{3} and x_{1} ≤ 0, it follows that e^{x1}^{x2} ≤ 1 and x_{2} > 0, which
yields (−x_{1}, x_{2})^{T} ∈ G^{1}. Under the condition x_{2} = x_{3}, if x_{1} > 0, we find that there is no
relationship between K_{e} and G^{1}. For the case (b), if x_{2} = x_{3}, then, we have x_{1} ≤ 0 and
x_{2} = x_{3} = 0. this implies that e^{x1}^{0} = 0. By this, we have ˆx = (−x_{1}, 0)^{T} ∈ G^{1}. 2

We also present the projection of x ∈ R^{3} onto the exponential cone K_{e}.

Theorem 5.1. Let x = (x_{1}, x_{2}, x_{3})^{T} ∈ R^{3}. Then the projection of x onto the exponential
cone Ke is given by

ΠK_{e}(x) =

x, if x ∈ K_{e},

0, if x ∈ (K_{e})^{◦} = −K^{∗}_{e},
v, otherwise,

(9)

where v = (v_{1}, v_{2}, v_{3})^{T} ∈ R^{3} has the following form:

(a) if x1 ≤ 0 and x2 ≤ 0, then v = (x1, 0,^{x}^{3}^{+|x}_{2} ^{3}^{|})^{T}.

(b) otherwise, the projection ΠKe(x) = v satisfies the equations:

v_{1} − x_{1} + e^{v1}^{v2}

v_{2}e^{v1}^{v2} − x_{3}

= 0,
v_{2}(v_{2}− x_{2}) − (v_{1}− x_{1})(v_{2}− v_{1}) = 0,
v_{2}e^{v1}^{v2} = v_{3}.

Proof. As the argument of Theorem 4.1, the first two cases of (9) are obvious. Hence,
we only need to consider the third case, i.e., x /∈ Ke∪ (Ke)^{◦}. For convenience, we denote

A :=n

(x_{1}, x_{2}, x_{3})^{T}

x_{2}e^{x1}^{x2} ≤ x_{3}, x_{2} > 0o

and B :=(x_{1}, 0, x_{3})^{T}

x_{1} ≤ 0, x_{3} ≥ 0 .
(a) If x1 ≤ 0 and x2 ≤ 0, since the exponential cone Ke is closed and convex, by
Proposition 2.2.1 in [3], we get that v is the projection of x onto K_{e} if and only if

hx − v, y − vi ≤ 0, ∀y ∈ K_{e}. (10)

From this, we need to verify that v = (x_{1}, 0,^{x}^{3}^{+|x}_{2} ^{3}^{|})^{T} satisfies (10). For any y :=

(y_{1}, y_{2}, y_{3})^{T} ∈ K_{e}, it follows that

hx − v, y − vi = x_{2}y_{2}+x3− |x3|
2

y_{3}− x3+ |x3|
2

= x_{2}y_{2}+ y_{3}x_{3}− |x_{3}|

2 .

If y ∈ A, we have y2 > 0 and y3 ≥ y2e^{y1}^{y2} > 0, which leads to
hx − v, y − vi = x_{2}y_{2}+ y_{3}x_{3}− |x_{3}|

2 ≤ 0.

If y ∈ B, we have y_{2} = 0 and y_{3} ≥ 0, which implies that
hx − v, y − vi = y_{3}x_{3}− |x_{3}|

2 ≤ 0.

Hence, under the conditions of x_{1} ≤ 0 and x_{2} ≤ 0, we can obtain that ΠK_{e}(x) = v =
(x_{1}, 0,^{x}^{3}^{+|x}_{2} ^{3}^{|})^{T}.

(b) If x belongs to other cases, we assert that the projection ΠKe(x) of x onto K_{e} lies in
the set A. Suppose not, i.e., ΠKe(x) ∈ B. Then, for any x = (x1, x2, x3)^{T} ∈ R^{3}, it follows
that ΠK_{e}(x) = v = (min{x_{1}, 0}, 0,^{x}^{3}^{+|x}_{2} ^{3}^{|})^{T} ∈ B. By Projection Theorem [3, Prop. 2.2.1],
we know that the projection v should satisfy the condition

v ∈ K_{e}, x − v ∈ (K_{e})^{◦}, and hx − v, vi = 0.

However, we see that there exists x_{1} > 0 or x_{2} 6= 0 such that
v − x = (min{x_{1}, 0} − x_{1}, −x_{2},|x_{3}| − x_{3}

2 )^{T} ∈ K/ _{e}^{∗},

i.e., x − v /∈ (K_{e})^{◦}. For example, when x_{1} = 1, x_{2} = 0 and x_{3} = 1, we have v − x =
(−1, 0, 0)^{T} ∈ K/ ^{∗}_{e}. This contradicts with x − v ∈ (Ke)^{◦}. Hence, the projection ΠKe(x) ∈ A.

To obtain the expression of ΠK_{e}(x), we look into the following problem:

min f (x) = ^{1}_{2}kv − xk^{2}

s.t. v ∈ A. (11)

In light of the convexity of the function f and the set A, it is easy to verify that the problem (11) is a convex optimization problem. Moreover, it follows from v ∈ A that

v_{1}

v_{2} − ln v3+ ln v2 ≤ 0.

Thus, the KKT conditions of the problem (11) are recast as

v1− x1+_{v}^{µ}

2 = 0,
v_{2}− x_{2}+ µ(−^{v}_{v}^{1}2

2

+ _{v}^{1}

2) = 0,
v3− x3− _{v}^{µ}

3 = 0,
µ ≥ 0, ^{v}_{v}^{1}

2 − ln v_{3}+ ln v_{2} ≤ 0, µ(^{v}_{v}^{1}

2 − ln v_{3}+ ln v_{2}) = 0.

(12)

From (12), by the fact that the projection of x ∈ /∈ K_{e}∪ (K^{∗}_{e})^{◦} must be a point in the
boundary, it is not hard to see that ^{v}_{v}^{1}

2 − ln v_{3}+ ln v_{2} = 0 and µ > 0, i.e., v_{3} = v_{2}e^{v1}^{v2} and
µ > 0. In addition, by the first and third equations in (12), we have

v_{1}− x_{1}+v_{3}(v_{3}− x_{3})
v2

= 0.

Combining with v3 = v2e^{v1}^{v2}, this implies that
v_{1} − x_{1} + e^{v1}^{v2}

v_{2}e^{v1}^{v2} − x_{3}

= 0.

On the other hand, by the first and second equations in (12), we have
v_{2}(v_{2}− x_{2}) = (v_{1}− x_{1})(v_{2}− v_{1}).

Therefore, we obtain that the projection ΠKe(x) = v satisfies the following equations:

v_{1}− x_{1}+ e^{v1}^{v2}

v_{2}e^{v1}^{v2} − x_{3}

= 0,
v_{2}(v_{2}− x_{2}) − (v_{1}− x_{1})(v_{2} − v_{1}) = 0,
v2e^{v1}^{v2} = v3.
Then, the proof is complete. 2

Here, we say a few words about Theorem 5.1. Unfortunately, unlike second-order
cone or circular cone cases, we do not obtain an explicit formula for the projection onto
the exponential cone, since there are nonlinear transcendental equations in Theorem
5.1. For example, when we examine the projection onto the exponential cone K_{e}. Let
x = (1, −2, 3). For the case in Theorem 5.1(b), using the second condition v_{2}(v_{2}− x_{2}) −
(v_{1}− x_{1})(v_{2}− v_{1}) = 0, we have

v_{2} = v_{1}− 3 +p−3v_{1}^{2}− 2v_{1}+ 9

2 .

Combining with the first condition v_{1} − x_{1} + e^{v1}^{v2}

v_{2}e^{v1}^{v2} − x_{3}

= 0 in the case (b), this yields a nonlinear transcendental equations as bellow:

v_{1}− 1 + e

2v1 v1−3+

√

−3v21−2v1+9 v_{1}− 3 +p−3v_{1}^{2}− 2v_{1}+ 9

2 e

2v1 v1−3+

√

−3v21−2v1+9 − 3

!

= 0.

From this equation, we do not have the specific expression of v_{1}. Hence, the explicit for-
mula for the projection onto exponential cone cannot be obtained. Moreover, analogous
to the geometric cone G^{n}, the decomposition for x associated with the exponential cone
K_{e} and the corresponding nonsmooth analysis for its cone-functions are not established.

### 6 The power cone

The high dimensional power cone is defined as bellow [25, 39]:

K^{α}_{m,n} :=

(

(x, z) ∈ R^{m}+ × R^{n}
kzk ≤

m

Y

i=1

x^{α}_{i}^{i}
)

,

where α_{i} > 0, Pm

i=1α_{i} = 1 and x = (x_{1}, · · · , x_{m})^{T}. For the power cone, when m =
2, n = 1, Truong and Tuncel [39] have discussed the homogeneity of the power cone.

However, Hien [25] states that the power cone is not homogeneous in general case, and
the power cone is self-dual cone. Moreover, when m = 2 and α_{1} = α_{2} = ^{1}_{2}, we see that
the power cone K_{m,n}^{α} is exactly the rotated second-order cone, which has a broad range

of applications. In [25], Hien provides the expression of the dual cone of the power cone
K^{α}_{m,n} as below:

(K^{α}_{m,n})^{∗} =
(

(s_{1}, · · · , s_{m}, ω_{1}, · · · , ω_{n}) ∈ R^{m}+ × R^{n}

m

Y

i=1

s_{i}
α_{i}

αi

≥ kωk )

,

where ω = (ω_{1}, · · · , ω_{n})^{T} ∈ R^{n}. For an application of power cone programming, we
refer the readers to [5], in which a lot of practical applications such as location problems
and geometric programming can be modelled using K^{α}_{m,n} and its limiting case K_{e}. The
pictures of the power cone K^{α}_{m,n} and its dual cone (K_{m,n}^{α} )^{∗} in R^{3} are depicted in Figure
6, where the parameters (m, n) = (2, 1) and (α_{1}, α_{2}) = (0.8, 0.2).

Figure 6: The power cone (left) and its dual cone (right) in R^{3}.

The projection onto the power cone K_{m,n}^{α} is already figured out by Hien in [25], which
is presented in the following theorem.

Theorem 6.1. ([25, Proposition 2.2]) Let (x, z) ∈ R^{m}×R^{n} with x = (x_{1}, · · · , x_{m})^{T} ∈ R^{m}
and z = (z1, · · · , zn)^{T} ∈ R^{n}. Set (ˆx, ˆz) be the projection of (x, z) onto the power cone
K^{α}_{m,n}. Denote

Φ(x, z, r) = 1 2

m

Y

i=1

x_{i}+

q

x^{2}_{i} + 4α_{i}r(kzk − r)

αi

− r.

(a) If (x, z) /∈ K^{α}_{m,n}∪ −(K_{m,n}^{α} )^{∗} and z 6= 0, then its projection onto K_{m,n}^{α} is
( xˆ_{i} = ^{1}_{2}

x_{i}+px^{2}_{i} + 4α_{i}r(kzk − r)

, i = 1, · · · , m, ˆ

z_{l}= z_{l}_{kzk}^{r} , l = 1, · · · , n,

where r = r(x, z) is the unique solution of the following system:

E(x, z) : Φ(x, z, r) = 0, 0 < r < kzk.

(b) If (x, z) /∈ K^{α}_{m,n}∪ −(K^{α}_{m,n})^{∗} and z = 0, then its projection onto K^{α}_{m,n} is

xˆ_{i} = (x_{i})_{+}= max{0, x_{i}}, i = 1, · · · , m,
ˆ

z_{l} = 0, l = 1, · · · , n.

(c) If (x, z) ∈ K^{α}_{m,n}, then its projection onto K_{m,n}^{α} is itself, i.e., (ˆx, ˆz) = (x, z).

(d) If (x, z) ∈ −(K^{α}_{m,n})^{∗}, then its projection onto K_{m,n}^{α} is zero vector, i.e., (ˆx, ˆz) = 0.

Nonetheless, Hein does not obtain an explicit formula for the projection onto the
power cone K_{m,n}^{α} in [25]. Accordingly, analogous to the geometric cone G^{n} and the
exponential cone K_{e}, the decomposition for (x, z) associated with the power cone K^{α}_{m,n}
and the corresponding nonsmooth analysis for its cone-functions are not established yet.

### 7 Conclusion

According to the authors’ earlier experience on symmetric cone optimization, we believe that spectral decomposition associated with cones, nonsmooth analysis regarding cone- functions, projections onto cones, and cone-convexity are the bridges between symmetric cone programs and nonsymmetric cone programs. Therefore, in this paper, we survey some related results about circular cone, p-order cone, geometric cone, exponential cone, and the power cone. Although the results are not quite complete due to the difficulty of handling nonsymmetric cones, they are very crucial to subsequent study towards nonsym- metric cone optimization. Further investigations are definitely desirable. We summarize and list out some future topics as below.

1. Exploring more structures and properties for each non-symmetric cone. Also look- ing for more non-symmetric cones, e.g., EDM cone.

2. For geometric cone, exponential cone, and power cone, etc., figuring out their spec-
tral decompositions, projections, and doing nonsmooth analysis for their corre-
sponding cone-functions like f^{sc}, f^{mat} and f^{circ}. We point out that through appro-
priate transformations (for example, α-representation and extended α-representation
defined in [5]), the aforementioned geometric cone, exponential cone, and power
cone can be generated from the 3-dimensional power cone and the exponential
cone in Figure 5 and 6. More recently, Lu et al. [31] propose two types of decom-
position approaches for these cones. We believe their results yield a possibility to
construct the corresponding cone-functions.

3. Designing appropriate algorithms based on the structures of non-symmetric cones.

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