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 nonsymmetric 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◦ (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,⟨·, ·⟩V, such that
(iii) ⟨x ◦ y, z⟩V =⟨y, x ◦ z⟩V for all x, y, z∈ V.
Given a Euclidean Jordan algebraA = (V, ◦, ⟨·, ·⟩V), we denote the set of squares as K :={
x2| 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 transformationT : 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 Rn with the (usual) inner product and Jordan product defined respectively as
⟨x, y⟩ =
∑n i=1
xiyi and x◦ y = x ∗ y ∀x, y ∈ Rn
where xidenotes the ith component of x, etc., and x∗y denotes the componentwise product of vectors x and y. Then,Rn is a Euclidean Jordan algebra with the nonnegative orthant Rn+as its cone of squares.
Example 1.2. LetSnbe the space of all n×n real symmetric matrices with the trace inner product and Jordan product, respectively, defined by
⟨X, Y ⟩T:= Tr(XY ) and X◦ Y := 1
2(XY + Y X) ∀X, Y ∈ Sn.
Then, (Sn,◦, ⟨·, ·⟩T) is a Euclidean Jordan algebra, and we write it as Sn. The cone of squaresSn+ inSn is the set of all positive semidefinite matrices.
Example 1.3. The Jordan spin algebra Ln. Consider Rn (n > 1) with the inner product
⟨·, ·⟩ and Jordan product
x◦ y :=
[ ⟨x, y⟩
x0y + y¯ 0x¯ ]
for any x = (x0, ¯x), y = (y0, ¯y) ∈ R × Rn−1. We denote the Euclidean Jordan algebra (Rn,◦, ⟨·, ·⟩) by Ln. The cone of squares, called the Lorentz cone (or second-order cone), is given by
L+n :={
(x0; ¯x)∈ R × Rn−1| x0≥ ∥¯x∥} .
For any given x∈ A, let ζ(x) be the degree of the minimal polynomial of x, i.e., ζ(x) := min{
k :{e, x, x2,· · · , xk} are linearly dependent} .
Then, the rank ofA 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 ci and 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 ̸= i for all j, i = 1, 2, · · · , k, and
∑k j=1
cj = 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, . . . , 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 eigenvalues and tr(x) =∑r
j=1λj(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 ⟨x, y⟩ := tr(x ◦ y) for any x, y ∈ V. The inner product⟨·, ·⟩ is associative by [19, Prop. II. 4.3], i.e., ⟨x, y ◦ z⟩ = ⟨y, x ◦ z⟩ 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 algebraLn.
(ii) The algebraSn of n× n real symmetric matrices.
(iii) The algebraHn of all n× n complex Hermitian matrices.
(iv) The algebraQn of all n× n quaternion Hermitian matrices.
(v) The algebraO3 of all 3× 3 octonion Hermitian matrices.
Given an n-dimensional Euclidean Jordan algebra A = (V, ⟨·, ·⟩, ◦) with K being its corresponding symmetric cone inV. For any scalar function f : R → R, we define a vector- valued function fsc(x) (called L¨owner function) onV as
fsc(x) = f (λ1(x))c1+ f (λ2(x))c2+· · · + f(λr(x))cr (1.1) where x∈ V has the spectral decomposition
x = λ1(x)c1+ λ2(x)c2+· · · + λr(x)cr.
When V is the space Sn which means n× n real symmetric matrices. The spectral decomposition reduces to the following: for any X∈ Sn,
X = P
λ1
. .. λn
PT,
where λ1, λ2,· · · , λn are eigenvalues of X and P is orthogonal (i.e., PT = P−1). Under this setting, for any function f :R → R, we define a corresponding matrix valued function associated with the Euclidean Jordan algebraSn:= Sym(n,R), denoted by fmat :Sn→ Sn, as
fmat(X) = P
f (λ1)
. ..
f (λn)
PT.
For this case, Chen, Qi and Tseng in [12] show that the function fmat inherits from f the properties of continuity, Lipschitz continuity, directional differentiability, Fr´echet differ- entiability, continuous differentiability, as well as semismoothness. We state them as below.
Theorem 1.3. (a) fmat is continuous if and only if f is continuous.
(b) fmat is directionally differentiable if and only if f is directionally differentiable.
(c) fmat is Fr´echet-differentiable if and only if f is Fr´echet-differentiable.
(d) fmat is continuously differentiable if and only if f is continuously differentiable.
(e) fmat is locally Lipschitz continuous if and only if f is locally Lipschitz continuous.
(f) fmat is globally Lipschitz continuous with constant κ if and only if f is globally Lipschitz continuous with constant κ.
(g) fmat is semismooth if and only if f is semismooth.
WhenV is the Jordan spin algebra Ln in whichK corresponds to the second-order cone (SOC), which is defined as
Kn:={(x1, ¯x)∈ R × Rn−1| ∥¯x∥ ≤ x1},
the function fsc reduces to so-called SOC-function fsoc studied in [4, 6, 7, 8]. More specifi- cally, under such case, the spectral decomposition for any x = (x1, ¯x)∈ R × Rn−1 becomes
x = λ1(x)u(1)x + λ2(x)u(2)x , (1.2) where λ1(x), λ2(x), u(1)x and u(2)x with respect toKn are given by
λi(x) = x1+ (−1)i∥¯x∥,
u(i)x =
1 2
(
1, (−1)i ¯x
∥¯x∥
)
if ¯x̸= 0,
1 2
(
1, (−1)iw )
if ¯x = 0,
for i = 1, 2, with w being any vector inRn−1satisfying∥w∥ = 1. If ¯x ̸= 0, the decomposition (1.2) is unique. With this spectral decomposition, for any function f :R → R, the L¨owner function fsc associated withKn reduces to fsoc as below:
fsoc(x) = f (λ1(x))u(1)x + f (λ2(x))u(2)x ∀x = (x1, ¯x)∈ R × Rn−1. (1.3) The picture of second-order coneKn in R3is depicted in Figure 1.
Figure 1: The second-order cone inR3
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 fsoc 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) fsoc is continuous if and only if f is continuous.
(b) fsoc is directionally differentiable if and only if f is directionally differentiable.
(c) fsoc is Fr´echet-differentiable if and only if f is Fr´echet-differentiable.
(d) fsoc is continuously differentiable if and only if f is continuously differentiable.
(e) fsoc is locally Lipschitz continuous if and only if f is locally Lipschitz continuous.
(f) fsoc is globally Lipschitz continuous with constant κ if and only if f is globally Lipschitz continuous with constant κ.
g) fsoc is semismooth if and only if f is semismooth.
As for general symmetric cone case, Sun and Sun [38] uses ϕV to denote fsc defined as in (1.1). More specifically, for any function ϕ :R → R, they define a corresponding function associated with the Euclidean Jordan algebraV by
ϕV(x) = ϕ(λ1(x))c1+ ϕ(λ2(x))c2+· · · + ϕ(λr(x))cr,
where λ1(x), λ2(x),· · · , λr(x) and c1, c2,· · · , cr are the spectral values and spectral vectors of x, respectively. In addition, Sun and Sun [38] extend some of the aforementioned results to more general symmetric cone case regarding fsc (i.e., ϕV).
Theorem 1.5. Assume that the symmetric cone is simple in the Euclidean Jordan algebra V.
(a) ϕV is continuous if and only if ϕ is continuous.
(b) ϕV is directionally differentiable if and only if ϕ is directionally differentiable.
(c) ϕV is Fr´echet-differentiable if and only if ϕ is Fr´echet-differentiable.
(d) ϕV is continuously differentiable if and only if ϕ is continuously differentiable.
(e) ϕV 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 applications re- lated 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 continuity, 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 ⊆ Rn, we also recall its dual cone, polar cone, and the projection onto itself. For any a given closed convex coneK ⊆ Rn, its dual cone is defined by
K∗:={y ∈ Rn| ⟨y, x⟩ ≥ 0, ∀x ∈ K},
and its polar cone isK◦:=−K∗. Let ΠK(z) denote the Euclidean projection of z∈ Rnonto the closed convex coneK. Then, it follows that z = ΠK(z)− ΠK∗(−z) and
ΠK(z) = argminx∈K1
2∥x − z∥2.
2 Circular Cone
The definition of the circular coneLθ is defined as [42]:
Lθ := {
x = (x1, ¯x)∈ R × Rn−1| ∥x∥ cos θ ≤ x1
}
= {
x = (x1, ¯x)∈ R × Rn−1| ∥¯x∥ ≤ x1tan θ} .
From the concept of the circular cone Lθ, we know that when θ = π4, the circular cone is exactly the second-order coneKn. In addition, we also see thatLθis solid (i.e., intLθ̸= ∅), pointed (i.e.,Lθ∩ −Lθ= 0), closed convex cone, and has a revolution axis which is the ray generated by the canonical vector e1:= (1, 0,· · · , 0)T ∈ Rn. Moreover, its dual cone is given by
L∗θ := {y = (y1, ¯y)∈ R × Rn−1| ∥y∥ sin θ ≤ y1}
= {y = (y1, ¯y)∈ R × Rn−1| ∥¯y∥ ≤ y1cot θ}
= Lπ2−θ.
The pictures of circular coneLθin R3are depicted in Figure 2.
Figure 2: Three different circular cones inR3.
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 coneLθcan be expressed by in the following theorem.
Theorem 2.1. ([42]) Let x = (x1, ¯x)∈ R × Rn−1 and x+ denote the projection of x onto the circular coneLθ. Then x+ is given below:
x+=
x if x∈ Lθ, 0 if x∈ −L∗θ, u otherwise, where
u =
x1+∥¯x∥ tan θ 1 + tan2θ (x1+∥¯x∥ tan θ
1 + tan2θ tan θ ) x¯
∥¯x∥
.
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 = (x1, ¯x)∈ R × Rn−1, one has x = λ1(x)u(1)x + λ2(x)u(2)x ,
where
λ1(x) = x1− ∥¯x∥ cot θ λ2(x) = x1+∥¯x∥ tan θ and
u(1)x = 1 1 + cot2θ
[ 1 0
0 cot θ
] ( 1
−w )
u(2)x = 1 1 + tan2θ
[ 1 0
0 tan θ ] ( 1
w )
with w = ∥¯x∥x¯ if ¯x̸= 0, and any vector in Rn−1 satisfying ∥w∥ = 1 if ¯x = 0.
Theorem 2.3. ([42, Theorem 3.2]) For any x = (x1, ¯x)∈ Rn× R, we have x+= (λ1(x))+u(1)x + (λ2(x))+u(2)x ,
where (a)+ := max{0, a}, λi(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 fcircassociated withLθis defined as below:
fcirc(x) = f (λ1(x))u(1)x + f (λ2(x))u(2)x ∀x = (x1, ¯x)∈ R × Rn−1. (2.1) In [15], Chang, Yang and Chen have obtained that many properties of the function fcircare 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 fcirc is defined by (2.1). Then, the following results hold.
(a) fcirc is continuous at x ∈ Rn with spectral values λ1(x), λ2(x) if and only if f is continuous at λ1(x), λ2(x).
(b) fcirc is directionally differentiable at x∈ Rn with spectral values λ1(x), λ2(x) if and only if f is directionally differentiable at λ1(x), λ2(x).
(c) fcirc is differentiable at x ∈ Rn with spectral values λ1(x), λ2(x) if and only if f is differentiable at λ1(x), λ2(x).
(d) fcirc is strictly continuous at x∈ Rn with spectral values λ1(x), λ2(x) if and only if f is strictly continuous at λ1(x), λ2(x).
(e) fcirc is semismooth at x ∈ Rn with spectral values λ1(x), λ2(x) if and only if f is semismooth at λ1(x), λ2(x).
(f) fcirc is continuously differentiable at x∈ Rn with spectral values λ1(x), λ2(x) if and only if f is continuously differentiable at λ1(x), λ2(x).
We point out that there is a close relation betweenLθ andKn (see [34, 42]) as below Kn= ALθ 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. How- ever, it is not possible to make bothLθ andKn are self-dual under a certain inner product.
Secondly, as shown in [43], the relation Kn = ALθ does not guarantee that there exists a similar close relation between fcircand fsoc. 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 inRn, which is a generalization of the second-order coneKn[14], is defined as
Kp:=
x∈ Rn x1≥
( n
∑
i=2
|xi|p )1p
(p≥ 1). (3.1)
In fact, the p-order coneKp can be equivalently expressed as Kp={
x = (x1, ¯x)∈ R × Rn−1| x1≥ ∥¯x∥p
}, (p≥ 1),
where ¯x := (x2,· · · , xn)T ∈ Rn−1. From (3.1), it is clear to see that when p = 2, K2 is exactly the second-order coneKn. That means that the second-order cone is a special case of p-order cone. Moreover, it is known that Kp is a convex cone and its dual cone is given by
K∗p=
y∈ Rn y1≥
( n
∑
i=2
|yi|q )1q
or equivalently
K∗p={
y = (y1, ¯y)∈ R × Rn−1| y1≥ ∥¯y∥q
}=Kq
with ¯y := (y2,· · · , yn)T ∈ Rn−1, where q≥ 1 and satisfies 1p+1q = 1. From the expression of the dual coneK∗p, we see that the coneKp∗ 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. Besides, 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 conesKp in R3are depicted in Figure 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.
Figure 3: Three different p-order cones inR3
Theorem 3.1. ([33, Theorem 2.1]) For any z = (z1, ¯z)∈ R × Rn−1, then the projection of z ontoKp is given by
ΠKp(z) =
z, z∈ Kp
0, z∈ −K∗p=−Kq
u, otherwise (i.e.,−∥¯z∥q < z1<∥¯z∥p) where u = (u1, ¯u) with ¯u = (u2, u3,· · · , un)T ∈ Rn−1 satisfying
u1=∥¯u∥p= (|u2|p+|u3|p+· · · + |un|p)p1 and
ui− zi+u1− z1
up1−1 |ui|p−2ui= 0, ∀i = 2, · · · , n.
Theorem 3.2. ([33, Theorem 2.2]) Let z = (z1, ¯z)∈ R × Rn−1. Then, z can be decomposed as
z = α1(z)· v(1)(z) + α2(z)· v(2)(z),
where
α1(z) = z1+∥¯z∥p
2 α2(z) = z1− ∥¯z∥p
2
and
v(1)(z) = ( 1
¯ w
)
v(2)(z) = ( 1
− ¯w )
with ¯w = ∥¯z∥z¯
p if ¯z̸= 0; while ¯w being an arbitrary element satisfying ∥ ¯w∥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]:
Gn:=
{
(x, θ)∈ Rn+× R+
∑n
i=1
e−xiθ ≤ 1 }
where x = (x1,· · · , xn)T ∈ Rn+ and we also use the convention e−xi0 = 0. From the definition of the geometric coneGn, we know thatGn is solid (i.e., intGn̸= ∅), pointed (i.e., Gn∩ −Gn= 0), closed convex cone, and its dual cone is given by
(Gn)∗= {
(y, µ)∈ Rn+× R+ µ≥ ∑
yi>0
yiln yi
∑n i=1yi
}
where µ ∈ R+ and y = (y1,· · · , yn)T ∈ Rn+. In view of the expression of the dual cone (Gn)∗, we see that the dual cone (Gn)∗ is also a solid, pointed, closed convex cone, and ((Gn)∗)∗=Gn. When n = 1, we note that the geometric coneG1is just nonnegative octant coneR2+. In addition, by the expression of the geometric cone Gn and its dual cone (Gn)∗, it is not hard to verify that the boundary of the geometric coneGn and its dual cone (Gn)∗ can be respectively expressed as follows:
bdGn = {
(x, θ)∈ Rn+× R+ ∑n
i=1
e−xiθ = 1 }
and
bd (Gn)∗= {
(y, µ)∈ Rn+× R+ µ = ∑
yi>0
yiln yi
∑n i=1yi
} .
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 Gn and (Gn)∗. The pictures of Gn and its dual cone (Gn)∗ inR3are depicted in Figure 4.
Figure 4: The geometric cone (left) and its dual cone (right) inR3 Next, we present the projection of (x, θ)∈ Rn× R onto the geometric cone Gn.
Theorem 4.1. Let x = (x, θ)∈ Rn× R. Then the projection of x onto the geometric cone Gn is given by
ΠGn(x) =
x, if x∈ Gn, 0, if x∈ (Gn)◦, u, otherwise,
(4.1)
where u = (u, λ)∈ Rn+× R+ with u = (u1, u2,· · · , un)T ∈ Rn+ satisfying
ui− xi+ λ(λ− θ)
∑n
j=1e−ujλuj
e−uiλ = 0, i = 1, 2,· · · , n (4.2)
and ∑n
i=1
e−uiλ = 1. (4.3)
Proof. From Projection Theorem [3, Prop. 2.2.1], we know that, for every x = (x, θ) ∈ Rn× R, a vector u ∈ Gn is equal to the projection point ΠGn(x) if and only if
u∈ Gn, x− u ∈ (Gn)◦, and ⟨x − u, u⟩ = 0.
With this, the first two cases of (4.1) are obvious. Hence, we only need to consider the third case. Based on (4.3) and the definition ofGn, it is obvious that u∈ Gn. In addition, from (4.2), we obtain that∑n
i=1ui(ui− xi) + λ(λ− θ) = 0, which explains that ⟨x − u, u⟩ = 0.
Next, we argue that x− u ∈ (Gn)◦. To see this, by (4.2) and (4.3), we have
∑n i=1
(ui− xi) =− λ(λ− θ)
∑n
j=1e−ujλ uj
.
Together with (4.2) again, it follows that∑nui−xi
j=1(uj−xj) = e−uiλ, which leads to ln∑nui−xi
j=1(uj−xj) =
−uλi. Hence, we have
∑
ui−xi>0
(ui− xi) ln∑nui− xi j=1(uj− xj)
= − ∑
ui−xi>0
(ui− xi)ui
λ
= −1 λ
∑
ui−xi>0
(ui− xi)ui
≤ 1
λ· λ(λ − θ) = λ − θ, where the inequality holds since ∑n
i=1ui(ui − xi) + λ(λ− θ) = 0. This explains that u− x ∈ (Gn)∗, i.e, x− u ∈ (Gn)◦. Then, the proof is complete. 2
For the projection onto geometric cone Gn, we notice again that this projection is not an explicit formula since the equations (4.2) and 4.3 cannot be easily solved. Moreover, the decomposition associated with the geometric coneGnand 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 {
(x1, x2, x3)T ∈ R3 x2ex1x2 ≤ x3, x2> 0 }
. In fact, the exponential cone can be expressed as the union of two sets, i.e.,
Ke:=
{
(x1, x2, x3)T ∈ R3 x2ex1x2 ≤ x3, x2> 0 }∪{
(x1, 0, x3)T x1≤ 0, x3≥ 0} . As shown in [5], the dual coneK∗e of the exponential coneKe is given by
K∗e= cl {
(y1, y2, y3)T ∈ R3 −y1ey2y1 ≤ ey3, y1< 0 }
.
In addition, the dual coneK∗e is expressed as the union of the two following sets:
Ke∗= {
(y1, y2, y3)T ∈ R3 −y1ey2y1 ≤ ey3, y1< 0 }∪{
(0, y2, y3)T y2≥ 0, y3≥ 0} . From the expression of the exponential coneKeand its dual coneK∗e, it is known that the exponential cone Ke and its dual coneK∗e are closed convex cone inR3. Moreover, based on the expression ofKe andK∗e, it is easy to verify that their boundary can be respectively expressed as follows:
bdKe:=
{
(x1, x2, x3)T ∈ R3 x2ex1x2 = x3, x2> 0 }∪{
(x1, 0, x3)T x1≤ 0, x3≥ 0} . and
bdK∗e:=
{
(y1, y2, y3)T ∈ R3 −y1ey2y1 = ey3, y1< 0 }∪{
(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 coneKeand its dual coneK∗einR3are depicted in Figure 5.
Figure 5: The exponential cone (left) and its dual cone (right) inR3
For the geometric cone Gn and the exponential cone Ke, 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 be- tween the geometric coneGn and exponential coneKe.
Proof. For any (x, θ)∈ Gn with x = (x1, x2,· · · , xn)T ∈ Rn+, we have∑n
i=1e−xiθ ≤ 1. With this, it is equivalent to say
e−xiθ ≤ zi, and
∑n i=1
zi= 1.
Hence, we obtain that (−xi
θ, 1, zi)T ∈ Ke (i = 1, 2,· · · , n) and
∑n i=1
zi= 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 := (x1, x2, x3)T := (ˆxT, x3)T ∈ Ke with ˆx := (x1, x2)T, we have two cases, i.e.,
(a) x2ex1x2 ≤ x3 and x2> 0, or (b) x1≤ 0, x2= 0, x3≥ 0.
For the case (a), if x2 = x3 and x1 ≤ 0, it follows that ex1x2 ≤ 1 and x2 > 0, which yields (−x1, x2)T ∈ G1. Under the condition x2= x3, if x1> 0, we find that there is no relationship betweenKe and G1. For the case (b), if x2 = x3, then, we have x1 ≤ 0 and x2= x3 = 0.
this implies that ex10 = 0. By this, we have ˆx = (−x1, 0)T ∈ G1. 2
We also present the projection of x∈ R3 onto the exponential coneKe.
Theorem 5.2. Let x = (x1, x2, x3)T ∈ R3. Then the projection of x onto the exponential coneKeis given by
ΠKe(x) =
x, if x∈ Ke,
0, if x∈ (Ke)◦=−K∗e, v, otherwise,
(5.1)
where v = (v1, v2, v3)T ∈ R3 has the following form:
(a) if x1≤ 0 and x2≤ 0, then v = (x1, 0,x3+2|x3|)T.
(b) otherwise, the projection ΠKe(x) = v satisfies the equations:
v1− x1+ ev1v2 (
v2ev1v2 − x3
)
= 0, v2(v2− x2)− (v1− x1)(v2− v1) = 0, v2ev1v2 = v3.
Proof. As the argument of Theorem 4.1, the first two cases of (5.1) are obvious. Hence, we only need to consider the third case, i.e., x /∈ Ke∪ (Ke)◦. For convenience, we denote
A :=
{
(x1, x2, x3)T x2ex1x2 ≤ x3, x2> 0 }
and B :={
(x1, 0, x3)T x1≤ 0, x3≥ 0} .
(a) If x1≤ 0 and x2≤ 0, since the exponential cone Keis closed and convex, by Proposition 2.2.1 in [3], we get that v is the projection of x ontoKe if and only if
⟨x − v, y − v⟩ ≤ 0, ∀y ∈ Ke. (5.2)
From this, we need to verify that v = (x1, 0,x3+2|x3|)T satisfies (5.2). For any y :=
(y1, y2, y3)T ∈ Ke, it follows that
⟨x − v, y − v⟩ = x2y2+x3− |x3| 2
(
y3−x3+|x3| 2
)
= x2y2+ y3x3− |x3|
2 .
If y∈ A, we have y2> 0 and y3≥ y2ey1y2 > 0, which leads to
⟨x − v, y − v⟩ = x2y2+ y3
x3− |x3| 2 ≤ 0.
If y∈ B, we have y2= 0 and y3≥ 0, which implies that
⟨x − v, y − v⟩ = y3
x3− |x3| 2 ≤ 0.
Hence, under the conditions of x1 ≤ 0 and x2 ≤ 0, we can obtain that ΠKe(x) = v = (x1, 0,x3+2|x3|)T.
(b) If x belongs to other cases, we assert that the projection ΠKe(x) of x onto Ke lies in the set A. Suppose not, i.e., ΠKe(x)∈ B. Then, for any x = (x1, x2, x3)T ∈ R3, it follows that ΠKe(x) = v = (min{x1, 0}, 0,x3+2|x3|)T ∈ B. By Projection Theorem [3, Prop. 2.2.1], we know that the projection v should satisfy the condition
v∈ Ke, x− v ∈ (Ke)◦, and ⟨x − v, v⟩ = 0.
However, we see that there exists x1> 0 or x2̸= 0 such that v− x = (min{x1, 0} − x1,−x2,|x3| − x3
2 )T ∈ K/ e∗,
i.e., x− v /∈ (Ke)◦. For example, when x1 = 1, x2 = 0 and x3 = 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 ΠKe(x), we look into the following problem:
min f (x) = 12∥v − x∥2
s.t. v∈ A. (5.3)
In light of the convexity of the function f and the set A, it is easy to verify that the problem (5.3) is a convex optimization problem. Moreover, it follows from v∈ A that
v1
v2 − ln v3+ ln v2≤ 0.
Thus, the KKT conditions of the problem (5.3) are recast as
v1− x1+vµ
2 = 0, v2− x2+ µ(−vv12
2
+v1
2) = 0, v3− x3−vµ3 = 0,
µ≥ 0, vv12 − ln v3+ ln v2≤ 0, µ(vv12 − ln v3+ ln v2) = 0.
(5.4)
From (5.4), by the fact that the projection of x ∈/∈ Ke∪ (K∗e)◦ must be a point in the boundary, it is not hard to see that vv1
2 − ln v3+ ln v2 = 0 and µ > 0, i.e., v3= v2ev1v2 and µ > 0. In addition, by the first and third equations in (5.4), we have
v1− x1+v3(v3− x3) v2
= 0.
Combining with v3= v2ev1v2, this implies that v1− x1+ ev1v2
(
v2ev1v2 − x3
)
= 0.
On the other hand, by the first and second equations in (5.4), we have v2(v2− x2) = (v1− x1)(v2− v1).
Therefore, we obtain that the projection ΠKe(x) = v satisfies the following equations:
v1− x1+ ev1v2 (
v2ev1v2 − x3
)
= 0, v2(v2− x2)− (v1− x1)(v2− v1) = 0, v2ev1v2 = v3. Then, the proof is complete. 2
Here, we say a few words about Theorem 5.2. 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 5.2. For example, when we examine the projection onto the exponential coneKe. Let x = (1,−2, 3).
For the case in Theorem 5.2(b), using the second condition v2(v2−x2)−(v1−x1)(v2−v1) = 0, we have
v2=v1− 3 +√
−3v21− 2v1+ 9
2 .
Combining with the first condition v1−x1+ ev1v2 (
v2ev1v2 − x3
)
= 0 in the case (b), this yields a nonlinear transcendental equations as bellow:
v1− 1 + e
2v1 v1−3+√
−3v21 −2v1+9
(
v1− 3 +√
−3v12− 2v1+ 9
2 e
2v1 v1−3+√
−3v21 −2v1+9 − 3 )
= 0.
From this equation, we do not have the specific expression of v1. Hence, the explicit formula for the projection onto exponential cone cannot be obtained. Moreover, analogous to the geometric cone Gn, the decomposition for x associated with the exponential cone Ke 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]:
Kαm,n:=
{
(x, z)∈ Rm+ × Rn ∥z∥ ≤
∏m i=1
xαii }
,
where αi> 0,∑m
i=1αi= 1 and x = (x1,· · · , xm)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 α1= α2= 12, we see that the power coneKαm,n
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 coneKαm,nas below:
(Kαm,n)∗= {
(s1,· · · , sm, ω1,· · · , ωn)∈ Rm+ × Rn ∏m
i=1
(si αi
)αi
≥ ∥ω∥
} ,
where ω = (ω1,· · · , ωn)T ∈ Rn. 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 usingKαm,nand its limiting caseKe. The pictures of the power coneKαm,nand its dual cone (Km,nα )∗ inR3are depicted in Figure 6, where the parameters (m, n) = (2, 1) and (α1, α2) = (0.8, 0.2).
Figure 6: The power cone (left) and its dual cone (right) inR3.
The projection onto the power cone Km,nα 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)∈ Rm× Rn with x = (x1,· · · , xm)T ∈ Rm and z = (z1,· · · , zn)T ∈ Rn. Set (ˆx, ˆz) be the projection of (x, z) onto the power coneKm,nα . Denote
Φ(x, z, r) = 1 2
∏m i=1
( xi+
√
x2i + 4αir(∥z∥ − r) )αi
− r.
a) If (x, z) /∈ Kαm,n∪ −(Kαm,n)∗ and z̸= 0, then its projection onto Kαm,n is {
ˆ xi= 12
( xi+√
x2i + 4αir(∥z∥ − r))
, i = 1,· · · , m, ˆ
zl= zl r
∥z∥, 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 <∥z∥.
