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◦ (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,⟨·, ·⟩*_{V}, such that

(iii) *⟨x ◦ y, z⟩*V =*⟨y, x ◦ z⟩*V *for all x, y, z∈ V.*

Given a Euclidean Jordan algebra*A = (V, ◦, ⟨·, ·⟩*_{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

*⟨x, y⟩ =*

∑*n*
*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

*+as its cone of squares.*

^{n}**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

*⟨X, Y ⟩*T*:= Tr(XY ) and X◦ Y :=* 1

2*(XY + Y X)* *∀X, Y ∈ S*^{n}*.*

Then, (S^{n}*,◦, ⟨·, ·⟩*T) is a Euclidean Jordan algebra, and we write it as S*n*. The cone of
squaresS* ^{n}*+ inS

*n*is the set of all positive semidefinite matrices.

**Example 1.3. The Jordan spin algebra** L*n*. Consider R^{n}*(n > 1) with the inner product*

*⟨·, ·⟩ and Jordan product*

*x◦ y :=*

[ *⟨x, y⟩*

*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}*,◦, ⟨·, ·⟩) by L*

*n*. 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*≥ ∥¯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, x*^{2}*,· · · , x*^{k}*} 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 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* *̸= i for all j, i = 1, 2, · · · , k, and*

∑*k*
*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 {c*1*, . . . , c**r**} and real numbers*
*λ*1*(x), . . . , λ**r**(x), arranged in the decreasing order λ*1*(x)≥ λ*2*(x)≥ · · · ≥ λ**r**(x), such that*

*x = λ*_{1}*(x)c*_{1}*+ λ*_{2}*(x)c*_{2}+*· · · + λ**r**(x)c*_{r}*.*

*The numbers λ*_{j}*(x) (counting multiplicities), which are uniquely determined by x, are called*
*the eigenvalues and tr(x) =*∑*r*

*j=1**λ*_{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 ⟨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 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, ⟨·, ·⟩, ◦) 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) onV as

*f*^{sc}*(x) = f (λ*1*(x))c*1*+ f (λ*2*(x))c*2+*· · · + f(λ**r**(x))c**r* (1.1)
*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*

*, as*

^{n}*f*^{mat}*(X) = P*

*f (λ*1)

. ..

*f (λ**n*)

* 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 diﬀer-
entiability, 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 Lipschitz*
*continuous with constant κ.*

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

WhenV 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}*| ∥¯x∥ ≤ x*1

*},*

*the function f*^{sc} *reduces to so-called SOC-function f*^{soc} studied in [4, 6, 7, 8]. More specifi-
*cally, 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}*,* (1.2)
*where λ*_{1}*(x), λ*_{2}*(x), u*^{(1)}*x* *and u*^{(2)}*x* with respect to*K** ^{n}* are given by

*λ**i**(x)* = *x*1+ (*−1)*^{i}*∥¯x∥,*

*u*^{(i)}* _{x}* =

1 2

(

*1, (−1)** ^{i}* ¯

*x*

*∥¯x∥*

)

if ¯*x̸= 0,*

1 2

(

*1, (−1)*^{i}*w*
)

if ¯*x = 0,*

*for i = 1, 2, with w being any vector in*R^{n}* ^{−1}*satisfying

*∥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 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}*.* (1.3)
The picture of second-order cone*K** ^{n}* in R

^{3}is depicted in Figure 1.

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

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.*

*(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 Lipschitz*
*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 (1.1). More specifically, for any function ϕ :R → R, they define a corresponding function*
associated with the Euclidean Jordan algebraV by

*ϕ*_{V}*(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 vectors
*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 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 ⊆ R** ^{n}*, we also recall its dual cone,
polar cone, and the projection onto itself. For any a given closed convex cone

*K ⊆ R*

*, its dual cone is defined by*

^{n}*K** ^{∗}*:=

*{y ∈ R*

^{n}*| ⟨y, x⟩ ≥ 0, ∀x ∈ K},*

and its polar cone is*K** ^{◦}*:=

*−K*

*. Let Π*

^{∗}

_{K}*(z) denote the Euclidean projection of z∈ R*

*onto the closed convex cone*

^{n}*K. Then, it follows that z = Π*

_{K}*(z)− Π*

_{K}*(*

^{∗}*−z) and*

Π_{K}*(z) = argmin*_{x}* _{∈K}*1

2*∥x − z∥*^{2}*.*

**2 Circular Cone**

The definition of the circular cone*L**θ* is defined as [42]:

*L**θ* := {

*x = (x*_{1}*, ¯x)∈ R × R*^{n}^{−1}*| ∥x∥ cos θ ≤ x*1

}

= {

*x = (x*1*, ¯x)∈ R × R*^{n}^{−1}*| ∥¯x∥ ≤ 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*

*̸= ∅),*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*

*. Moreover, its dual cone is given by*

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

= *{y = (y*1*, ¯y)∈ R × R*^{n}^{−1}*| ∥¯y∥ ≤ y*1*cot θ}*

= *L*^{π}_{2}_{−θ}*.*

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

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

In view of the expression of the dual cone *L*^{∗}* _{θ}*, we see that the dual cone

*L*

^{∗}*is also a solid, pointed, closed convex cone. By the reference [42], the explicit formula of projection onto the circular cone*

_{θ}*L*

*θ*can be expressed by in the following theorem.

* Theorem 2.1. ([42]) Let x = (x*1

*, ¯x)∈ R × R*

^{n}

^{−1}*and x*+

*denote the projection of x onto*

*the circular coneL*

*θ*

*. Then x*+

*is given below:*

*x*_{+}=

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

*u =*

*x*_{1}+*∥¯x∥ tan θ*
1 + tan^{2}*θ*
(*x*1+*∥¯x∥ tan θ*

1 + tan^{2}*θ* *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 = (x**_{1}*, ¯x)∈ R × R*^{n}^{−1}*, one has*
*x = λ*_{1}*(x)u*^{(1)}_{x}*+ λ*_{2}*(x)u*^{(2)}_{x}*,*

*where*

*λ*1*(x)* = *x*1*− ∥¯x∥ cot θ*
*λ*2*(x)* = *x*1+*∥¯x∥ 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 =* _{∥¯x∥}^{x}^{¯} *if ¯x̸= 0, and any vector in R*^{n}^{−1}*satisfying* *∥w∥ = 1 if ¯x = 0.*

**Theorem 2.3. ([42, Theorem 3.2]) For any x = (x**_{1}*, ¯x)∈ R*^{n}*× R, we have*
*x*_{+}*= (λ*_{1}*(x))*_{+}*u*^{(1)}_{x}*+ (λ*_{2}*(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}*.* (2.1)
*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 (2.1). 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 λ*_{1}*(x), λ*_{2}*(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 λ*1*(x), λ*2*(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 λ*_{1}*(x), λ*_{2}*(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 λ*1*(x), λ*2*(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. How-
ever, 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}*= AL**θ* 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**

**3 The p-Order Cone**

*The p-order cone in*R* ^{n}*, which is a generalization of the second-order cone

*K*

*[14], is defined as*

^{n}*K**p*:=

*x∈ R*^{n}*x*1*≥*

( _{n}

∑

*i=2*

*|x**i**|** ^{p}*
)

^{1}

**

_{p}

*(p≥ 1).* (3.1)

*In fact, the p-order coneK**p* can be equivalently expressed as
*K**p*={

*x = (x*1*, ¯x)∈ R × R*^{n}^{−1}*| x*1*≥ ∥¯x∥**p*

}*, (p≥ 1),*

where ¯*x := (x*_{2}*,· · · , x**n*)^{T}*∈ R*^{n}^{−1}*. From (3.1), 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}

∑

*i=2*

*|y**i**|** ^{q}*
)

^{1}

**

_{q}

or equivalently

*K*^{∗}*p*={

*y = (y*1*, ¯y)∈ R × R*^{n}^{−1}*| y*1*≥ ∥¯y∥**q*

}=*K**q*

with ¯*y := (y*_{2}*,· · · , y**n*)^{T}*∈ R*^{n}^{−1}*, where q≥ 1 and satisfies* ^{1}* _{p}*+

^{1}

*= 1. From the expression of the dual cone*

_{q}*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. 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 cones

*K*

*p*in R

^{3}are 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 in*R^{3}

* Theorem 3.1. ([33, Theorem 2.1]) For any z = (z*1

*, ¯z)∈ R × R*

^{n}

^{−1}*, then the projection of*

*z ontoK*

*p*

*is given by*

Π_{K}_{p}*(z) =*

*z,* *z∈ K**p*

*0,* *z∈ −K*^{∗}*p*=*−K**q*

*u,* *otherwise (i.e.,−∥¯z∥**q* *< z*1*<∥¯z∥**p*)
*where u = (u*1*, ¯u) with ¯u = (u*2*, u*3*,· · · , u**n*)^{T}*∈ R*^{n}^{−1}*satisfying*

*u*1=*∥¯u∥**p*= (*|u*2*|** ^{p}*+

*|u*3

*|*

*+*

^{p}*· · · + |u*

*n*

*|*

*)*

^{p}

^{p}^{1}

*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 decomposed*

*as*

*z = α*1*(z)· v*^{(1)}*(z) + α*2*(z)· v*^{(2)}*(z),*

*where*

*α*1*(z)* = *z*1+*∥¯z∥**p*

2
*α*2*(z)* = *z*_{1}*− ∥¯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]:

*G** ^{n}*:=

{

*(x, θ)∈ R** ^{n}*+

*× R*+

∑^{n}

*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*

*, we know that*

^{n}*G*

*is solid (i.e., int*

^{n}*G*

^{n}*̸= ∅), pointed (i.e.,*

*G*

^{n}*∩ −G*

*= 0), closed convex cone, and its dual cone is given by*

^{n}(*G** ^{n}*)

*= {*

^{∗}*(y, µ)∈ R** ^{n}*+

*× R*+

*µ≥*∑

*y*_{i}*>0*

*y** _{i}*ln

*y*

_{i}∑*n*
*i=1**y**i*

}

*where µ* *∈ R*+ *and y = (y*_{1}*,· · · , y**n*)^{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 coneG*

^{1}is just nonnegative octant coneR

^{2}+. In addition, by the expression of the geometric cone

*G*

*and its dual cone (*

^{n}*G*

*)*

^{n}*, it is not hard to verify that the boundary of the geometric cone*

^{∗}*G*

*and its dual cone (*

^{n}*G*

*)*

^{n}*can be respectively expressed as follows:*

^{∗}bd*G** ^{n}* =
{

*(x, θ)∈ R** ^{n}*+

*× R*+∑

^{n}*i=1*

*e*^{−}^{xi}* ^{θ}* = 1
}

and

bd (*G** ^{n}*)

*= {*

^{∗}*(y, µ)∈ R** ^{n}*+

*× R*+

*µ =*∑

*y*_{i}*>0*

*y**i*ln *y*_{i}

∑*n*
*i=1**y**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*

*and its dual cone (*

^{n}*G*

*)*

^{n}*inR*

^{∗}^{3}are depicted in Figure 4.

Figure 4: The geometric cone (left) and its dual cone (right) inR^{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*

Π_{G}^{n}**(x) =**

**x,****if x***∈ G*^{n}*,*
*0,* **if x***∈ (G** ^{n}*)

^{◦}*,*

**u,***otherwise,*

(4.1)

**where u = (u, λ)**∈ R* ^{n}*+

*× R*+

*with u = (u*

_{1}

*, u*

_{2}

*,· · · , u*

*n*)

^{T}*∈ R*

*+*

^{n}*satisfying*

*u**i**− x**i*+ *λ(λ− θ)*

∑*n*

*j=1**e*^{−}^{uj}^{λ}*u**j*

*e*^{−}^{ui}^{λ}*= 0,* *i = 1, 2,· · · , n* (4.2)

*and* ∑*n*

*i=1*

*e*^{−}^{ui}^{λ}*= 1.* (4.3)

**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 Π

_{G}

^{n}**(x) if and only if**

**u***∈ G*^{n}**, x****− u ∈ (G*** ^{n}*)

^{◦}*, 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 of*G*^{n}**, it is obvious that u***∈ G** ^{n}*. In addition, from
(4.2), we obtain that∑

*n*

*i=1**u*_{i}*(u*_{i}*− x**i**) + λ(λ − θ) = 0, which explains that ⟨x − u, u⟩ = 0.*

**Next, we argue that x****− u ∈ (G*** ^{n}*)

*. To see this, by (4.2) and (4.3), we have*

^{◦}∑*n*
*i=1*

*(u**i**− x**i*) =*−* *λ(λ− θ)*

∑*n*

*j=1**e*^{−}^{uj}^{λ}*u**j*

*.*

Together with (4.2) again, it follows that∑_{n}^{u}^{i}^{−x}^{i}

*j=1**(u**j**−x**j*) *= e*^{−}^{ui}* ^{λ}*, which leads to ln∑

_{n}

^{u}

^{i}

^{−x}

^{i}*j=1**(u**j**−x**j*) =

*−*^{u}_{λ}* ^{i}*. Hence, we have

∑

*u*_{i}*−x**i**>0*

*(u**i**− x**i*) ln∑*n**u**i**− x**i*
*j=1**(u*_{j}*− x**j*)

= *−* ∑

*u*_{i}*−x**i**>0*

*(u**i**− x**i*)*u**i*

*λ*

= *−*1
*λ*

∑

*u**i**−x**i**>0*

*(u**i**− x**i**)u**i*

*≤* 1

*λ· λ(λ − θ) = λ − θ,*
where the inequality holds since ∑*n*

*i=1**u**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 (4.2) and 4.3 cannot be easily solved. Moreover,*the decomposition associated with the geometric cone

*G*

*and the corresponding nonsmooth analysis for its cone-functions are not established.*

^{n}**5 The Exponential Cone**

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

*K**e*:= cl
{

*(x*1*, x*2*, x*3)^{T}*∈ R*^{3}*x*2*e*^{x1}^{x2}*≤ x*3*, x*2*> 0*
}

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

*K**e*:=

{

*(x*1*, x*2*, x*3)^{T}*∈ R*^{3}*x*2*e*^{x1}^{x2}*≤ x*3*, x*2*> 0*
}*∪*{

*(x*1*, 0, x*3)^{T}*x*1*≤ 0, x*3*≥ 0*}
*.*
As shown in [5], the dual cone*K*^{∗}*e* of the exponential cone*K**e* is given by

*K*^{∗}*e*= cl
{

*(y*1*, y*2*, y*3)^{T}*∈ R*^{3}* −y*1*e*^{y2}^{y1}*≤ ey*3*, y*1*< 0*
}

*.*

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

*K**e** ^{∗}*=
{

*(y*_{1}*, y*_{2}*, y*_{3})^{T}*∈ R*^{3}* −y*_{1}*e*^{y2}^{y1}*≤ ey*3*, y*_{1}*< 0*
}*∪*{

*(0, y*_{2}*, 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 *K**e* and its dual cone*K*^{∗}*e* are closed convex cone inR^{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*K**e*:=

{

*(x*1*, x*2*, x*3)^{T}*∈ R*^{3}*x*2*e*^{x1}^{x2}*= x*3*, x*2*> 0*
}*∪*{

*(x*1*, 0, x*3)^{T}*x*1*≤ 0, x*3*≥ 0*}
*.*
and

bd*K*^{∗}*e*:=

{

*(y*1*, y*2*, y*3)^{T}*∈ R*^{3}* −y*1*e*^{y2}^{y1}*= ey*3*, y*1*< 0*
}*∪*{

*(0, y*2*, y*3)^{T}*y*2*≥ 0, y*3*≥ 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*K**e*and its dual cone*K*^{∗}*e*inR^{3}are depicted
in Figure 5.

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

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 be-***tween the geometric coneG*^{n}*and exponential coneK**e**.*

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

*n*

*i=1**e*^{−}^{xi}^{θ}*≤ 1. With*
this, it is equivalent to say

*e*^{−}^{xi}^{θ}*≤ z**i**,* and

∑*n*
*i=1*

*z*_{i}*= 1.*

Hence, we obtain that
(*−x**i*

*θ, 1, z** _{i}*)

^{T}*∈ K*

*e*

*(i = 1, 2,· · · , n) and*

∑*n*
*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 *K**e*. Indeed, for
any ˜*x := (x*_{1}*, x*_{2}*, x*_{3})* ^{T}* := (ˆ

*x*

^{T}*, x*

_{3})

^{T}*∈ K*

*e*with ˆ

*x := (x*

_{1}

*, x*

_{2})

*, we have two cases, i.e.,*

^{T}*(a) x*2*e*^{x1}^{x2}*≤ x*3 *and x*2*> 0, or*
*(b) x*_{1}*≤ 0, x*2*= 0, x*_{3}*≥ 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 ˆ*x = (−x*1*, 0)*^{T}*∈ G*^{1}. *2*

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

**Theorem 5.2. Let x = (x**_{1}*, x*_{2}*, x*_{3})^{T}*∈ R*^{3}*. Then the projection of x onto the exponential*
*coneK**e**is given by*

Π_{K}_{e}*(x) =*

*x,* *if x∈ K**e**,*

*0,* *if x∈ (K**e*)* ^{◦}*=

*−K*

^{∗}*e*

*,*

*v,*

*otherwise,*

(5.1)

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

*(a) if x*1*≤ 0 and x*2*≤ 0, then v = (x*1*, 0,*^{x}^{3}^{+}_{2}^{|x}^{3}* ^{|}*)

^{T}*.*

*(b) otherwise, the projection Π*_{K}_{e}*(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 (5.1) are obvious. Hence, we*
*only need to consider the third case, i.e., x /∈ K**e**∪ (K**e*)* ^{◦}*. For convenience, we denote

*A :=*

{

*(x*_{1}*, x*_{2}*, x*_{3})^{T}*x*_{2}*e*^{x1}^{x2}*≤ x*3*, x*_{2}*> 0*
}

and *B :=*{

*(x*_{1}*, 0, x*_{3})^{T}*x*_{1}*≤ 0, x*3*≥ 0*}
*.*

*(a) If x*1*≤ 0 and x*2*≤ 0, since the exponential cone K**e*is closed and convex, by Proposition
*2.2.1 in [3], we get that v is the projection of x ontoK**e* if and only if

*⟨x − v, y − v⟩ ≤ 0,* *∀y ∈ K**e**.* (5.2)

*From this, we need to verify that v = (x*1*, 0,*^{x}^{3}^{+}_{2}^{|x}^{3}* ^{|}*)

*satisfies (5.2).*

^{T}*For any y :=*

*(y*1*, y*2*, y*3)^{T}*∈ K**e*, it follows that

*⟨x − v, y − v⟩ = x*2*y*2+*x*3*− |x*3*|*
2

(

*y*3*−x*3+*|x*3*|*
2

)

= *x*_{2}*y*_{2}*+ y*_{3}*x*_{3}*− |x*3*|*

2 *.*

*If y∈ A, we have y*2*> 0 and y*_{3}*≥ y*2*e*^{y1}^{y2}*> 0, which leads to*

*⟨x − v, y − v⟩ = 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*

*⟨x − v, y − v⟩ = y*3

*x*_{3}*− |x*3*|*
2 *≤ 0.*

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

*.*

^{T}*(b) If x belongs to other cases, we assert that the projection Π*_{K}_{e}*(x) of x onto* *K**e* lies in
*the set A. Suppose not, i.e., Π*_{K}_{e}*(x)∈ B. Then, for any x = (x*1*, x*2*, x*3)^{T}*∈ R*^{3}, it follows
that Π_{K}_{e}*(x) = v = (min{x*1*, 0}, 0,*^{x}^{3}^{+}_{2}^{|x}^{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* *⟨x − v, v⟩ = 0.*

*However, we see that there exists x*1*> 0 or x*2*̸= 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 ∈ (K**e*)* ^{◦}*. Hence, the projection Π

_{K}

_{e}*(x)∈ A.*

To obtain the expression of Π_{K}_{e}*(x), we look into the following problem:*

min *f (x) =* ^{1}_{2}*∥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*

*v*1

*v*2 *− ln v*3*+ ln v*2*≤ 0.*

Thus, the KKT conditions of the problem (5.3) are recast as

*v*_{1}*− x*1+_{v}^{µ}

2 *= 0,*
*v*2*− x*2*+ µ(−*^{v}_{v}^{1}2

2

+_{v}^{1}

2*) = 0,*
*v*3*− x*3*−*_{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.*

(5.4)

*From (5.4), 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 (5.4), we have*

*v*1*− x*1+*v*_{3}*(v*_{3}*− x*3)
*v*2

*= 0.*

*Combining with v*_{3}*= v*_{2}*e*^{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 (5.4), we have
*v*2*(v*2*− x*2*) = (v*1*− x*1*)(v*2*− v*1*).*

Therefore, we obtain that the projection Π_{K}_{e}*(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,*
*v*2*e*^{v1}* ^{v2}* =

*v*3

*.*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 cone*K**e**. Let x = (1,−2, 3).*

*For the case in Theorem 5.2(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 +*√

*−3v*^{2}1*− 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+**√*

*−3v2**1 −2v1+9*

(

*v*_{1}*− 3 +*√

*−3v*1^{2}*− 2v*1+ 9

2 *e*

*2v1*
*v1−3+**√*

*−3v2**1 −2v1+9* *− 3*
)

*= 0.*

*From this equation, we do not have the specific expression of v*_{1}. Hence, the explicit formula
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}*∥z∥ ≤*

∏*m*
*i=1*

*x*^{α}_{i}* ^{i}*
}

*,*

*where α**i**> 0,*∑*m*

*i=1**α**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*^{α}*m,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 cone*K*^{α}*m,n*as below:

(*K*^{α}*m,n*)* ^{∗}*=
{

*(s*_{1}*,· · · , s**m**, ω*_{1}*,· · · , ω**n*)*∈ R** ^{m}*+

*× R*

*∏*

^{n}

^{m}*i=1*

(*s*_{i}*α**i*

)*α**i*

*≥ ∥ω∥*

}
*,*

*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}*inR*

^{∗}^{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) inR^{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 = (z*_{1}*,· · · , z**n*)^{T}*∈ R*^{n}*. Set (ˆx, ˆz) be the projection of (x, z) onto the power coneK**m,n*^{α}*.*
*Denote*

*Φ(x, z, r) =* 1
2

∏*m*
*i=1*

(
*x**i*+

√

*x*^{2}_{i}*+ 4α**i**r(∥z∥ − r)*
)*α*_{i}

*− r.*

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

ˆ
*x** _{i}*=

^{1}

_{2}

(
*x** _{i}*+√

*x*^{2}_{i}*+ 4α*_{i}*r(∥z∥ − r)*)

*,* *i = 1,· · · , m,*
ˆ

*z**l**= z**l* *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∥.*