https://doi.org/10.1007/s10898-019-00845-3

**The decompositions with respect to two core non-symmetric** **cones**

**Yue Lu**^{1}**· Ching-Yu Yang**^{2}**· Jein-Shan Chen**^{2}**· Hou-Duo Qi**^{3}

Received: 12 November 2018 / Accepted: 11 October 2019

© Springer Science+Business Media, LLC, part of Springer Nature 2019

**Abstract**

It is known that the analysis to tackle with non-symmetric cone optimization is quite different from the way to deal with symmetric cone optimization due to the discrepancy between these types of cones. However, there are still common concepts for both optimization problems, for example, the decomposition with respect to the given cone, smooth and nonsmooth analysis for the associated conic function, conic-convexity, conic-monotonicity and etc. In this paper, motivated by Chares’s thesis (Cones and interior-point algorithms for structured convex optimization involving powers and exponentials, 2009), we consider the decomposition issue of two core non-symmetric cones, in which two types of decomposition formulae will be proposed, one is adapted from the well-known Moreau decomposition theorem and the other follows from geometry properties of the given cones. As a byproduct, we also establish the conic functions of these cones and generalize the power cone case to its high-dimensional counterpart.

**Keywords Moreau decomposition theorem**· Power cone · Exponential cone ·
Non-symmetric cones

**Mathematics Subject Classification 49M27**· 90C25

### B

Jein-Shan Chen jschen@math.ntnu.edu.tw Yue Lujinjin403@sina.com Ching-Yu Yang

yangcy@math.ntnu.edu.tw Hou-Duo Qi

hdqi@soton.ac.uk

1 School of Mathematical Sciences, Tianjin Normal University, Tianjin 300387, China 2 Department of Mathematics, National Taiwan Normal University, Taipei 11677, Taiwan 3 School of Mathematics, University of Southampton, Southampton SO17 1BJ, UK

**1 Introduction**

Consider the following two core non-symmetric cones
*K**α* :=

*(x*1*, ¯x) ∈ R × R*^{2}

*|x*^{1}*| ≤ ¯x*_{1}^{α}^{1}*¯x*_{2}^{α}^{2}*, ¯x*1*≥ 0, ¯x*2≥ 0

*,* (1)

*K*exp := cl

*(x*1*, ¯x) ∈ R × R*^{2}

* x*^{1}*≥ ¯x*2· exp

*¯x*1

*¯x*2

*, ¯x*2*> 0, x*1≥ 0

*,* (2)

where*¯x := ( ¯x*1*, ¯x*2*)** ^{T}* ∈ R

^{2},

*α := (α*1

*, α*2

*)*

*∈ R*

^{T}^{2},

*α*1

*, α*2

*∈ (0, 1), α*1

*+α*2

*= 1 and cl(Ω) is*the closure of

*Ω. We callK*

*α*the power cone and

*K*expthe exponential cone

^{1}, whose graphs are depicted in Fig.1.

**1.1 Motivations and literatures**

Why do we pay attention to these two core non-symmetric cones? There are two main reasons.

In theory, Chares [5] proposes two important concepts (i.e.,*α-representable and extended α-*
representable, see “Appendix 6.1”) involving powers and exponentials and plenty of famous
cones can be generated from these two cones such as second-order cones [1,8,10,15,23,24],
*p-order cones [2,27,44], geometric cones [4,16,17,26], L**p* cones [18] and etc., one can
refer to [5, chapter 4] for more examples. In applications, many practical problems can
be cast into optimization models involving the power cone constraints and the exponential
cone constraints, such as location problems [5,21] and geometric programming problems
[4,31,34]. Therefore, it becomes quite obvious that there is a great demand for providing
systematic studies for these cones.

Location problem [5,21]: The generalized location problem is to find a point x∈ R* ^{n}*whose

*sum of weight distances from a given set of locations L*1

*, . . . , L*

*m*is minimized, which has the following form

*(P) min**x∈R*^{n}_{m}

*i*=1*w**i**x − L**i**p**i*

where · *p*_{i}*(p**i* *≥ 1) denotes the p**i*-norm defined onR^{n}*. If p**i*is equal to 2, then the above
*problem reduces to the classical Weber-Point problem. Denote by x:= (x*1*, . . . , x**n**)** ^{T}* ∈ R

^{n}*and a:= (a*1

*, . . . , a*

*n*

*)*

*∈ R*

^{T}*, Problem*

^{n}*(P) can be rewritten as*

min_{x,a,y}_{i}_{m}

*i=1**w**i**a**i*

*s.t.* *(y**i**, j**, a**i**, x**j**− L**i**, j**) ∈K*^{1}

*pi**, i = 1, . . . , m, j = 1, . . . , n,*

_{n}

*j=1**y*_{i, j}*= a**i**, i = 1, . . . , m,*

*where L*_{i, j}*and y*_{i, j}*stand for the j -th component of L** _{i}*∈ R

^{n}*and y*

*∈ R*

_{i}*, respectively.*

^{n}Geometric programming [4,31,34]: Let x *:= (x*1*, . . . , x**n**)** ^{T}* ∈ R

*be a vector with real*

^{n}*positive components x*

*i*

*. A real valued function m, of the form m(x) := c*

_{n}*i=1**x*_{i}^{α}* ^{i}*, is called

*a monomial function, where c> 0 and α*

*i*are its coefficient and exponents, respectively. A

*sum of one or more monomials, i.e., a function that looks like f(x) :=*

_{K}*k*=1*m**k**(x), is called*

1The definition of*K*expused in (2) comes from [5, Section 4.1], which has a slight difference from another
form in [34, Definition 2.1.2] as

*K*exp:= cl

*(x*1*, ¯x) ∈ R × R*^{2}

* x*^{1}*≥ ¯x*2· exp

*¯x*1

*¯x*2

*, ¯x*2*> 0*

*.*
However, one can observe that these two definitions coincide with each other.

**Fig. 1 The power cone***K**α*(left) and the exponential cone*K*exp(right)

*a posynomial function, where m*_{k}*(x) := c**k*

_{n}

*i=1**x*_{i}^{α}* ^{i,k}*. A geometric program is composed
of a posynomial objective with posynomial inequality constraints and monomial equality
constraints, which can be described as

*(G P)*

min_{x}*f*_{0}*(x)*

*s.t. f**s**(x) ≤ 1, s = 1, . . . , p,*
*g**t**(x) = 1, t = 1, . . . , q,*
*where f**s* := _{K}

*k*=1*c**k**,s*_{n}

*i*=1*x*_{i}^{α}^{i,k,s}*, s ∈ {0, 1, . . . , p} and g**t**(x) := c**t*_{n}

*i*=1*x*_{i}^{α}^{i,t}*, t ∈*
*{1, . . . , q}. Using the following change of variables as x**i* *:= exp(u**i**), c**k,i* *:= exp(d**k,i**), c**t* :=

exp*(d**t**) and adding some additional variables, Problem (GP) can be rewritten as*
min*u*_{i}*,w,ξ**k,0**,η**k,s* *w*

*s.t.* *(d**k**,0*+_{n}

*i*=1*u**i**· α**i**,k,0**, ξ**k**,0**, 1) ∈K*exp*,* _{K}

*k*=1*ξ**k**,0**= w,*
*(d**k,s*+_{n}

*i=1**u**i**· α**i,k,s**, η**k,s**, 1) ∈K*exp*,*_{K}

*k=1**η**k,s**= 1, s = 1, . . . , p,*
*d**t*+_{n}

*i*=1*u**i**· α**i**,t* *= 0, t = 1, . . . , q.*

In the past three decades, a great deal of mathematical effort in conic programming has been devoted to the study of symmetric cones and it has been made extensive progress [9,14,29,30,33,38], particularly for the second-order cone (SOC) [1,8,10,15,23,24] and the positive semidefinite cone [35,37,39–41]. For example, consider the second-order cone

L^{n}*:= {(x*1*, ¯x) ∈ R × R*^{n−1}*| x*1*≥ ¯x}.*

*For any given z= (z*1*, ¯z) ∈ R × R** ^{n−1}*, its decomposition with respect toL

*has the form*

^{n}*z= λ*1

*(z) · u*

^{(1)}

_{z}*+ λ*2

*(z) · u*

^{(2)}

_{z}*,*(3) where

*λ*

*i*

*(z) := z*1

*+ (−1)*

^{i}*¯z and u*

^{(i)}*z*is equal to

^{1}

_{2}

1*, (−1)*^{i}_{¯z}^{¯z}

, if*¯z = 0;*^{1}_{2}

1*, (−1)*^{i}*w*
,
*otherwise, which is applicable for i* *= 1, 2 with w ∈ R** ^{n−1}*being any unit vector. For any

*scalar function f*

*: R → R, the associated conic function f*

^{soc}*(z) (called the SOC function)*is given by

*f*^{soc}*(z) = f (λ*1*(z)) · u*^{(1)}*z* *+ f (λ*2*(z)) · u*^{(2)}*z* *.* (4)
In light of the decomposition formula and its conic function, one can further establish their
analytic properties (i.e., projection mapping, cone-convexity, conic-monotonicity) and design
numerical algorithms (i.e., proximal-like algorithms and interior-point algorithms), see Fig.2
for their relations and refer to the monograph [11] for more details. Similar results have

**Fig. 2 The relations between the decomposition with respect to SOC and other topics**

also been established for the positive semidefinite cone [40,43] and symmetric cones [14, 38]. Therefore, the past experience [11,14,43] indicate that how to derive the associated decomposition expression with respect to a given cone as the form (3) at a low cost becomes an important issue in the whole picture of researches.

As a fundamental tool in optimization, Moreau decomposition theorem [25] characterizes
the key relationship between the decomposition with respect to a closed convex cone and its
*projection mappings. More concretely, for any given z*∈ R* ^{n}*, it can be uniquely decomposed
into

*z= Π*_{K}*(z) + Π*_{K}^{◦}*(z) = Π*_{K}*(z) − Π*_{K}^{∗}*(−z),* (5)
where*Π*_{K}*(z) is the projection mapping of z ∈ R** ^{n}*onto

*K*and

*K*

^{◦}is the polar cone of

*K*, i.e.,

*K*^{◦}*:= {y ∈ R*^{n}*| x*^{T}*y≤ 0, ∀x ∈K}.*

In addition,*K*^{∗}is the dual cone of*K*and satisfies the relation*K*^{∗}= −*K*^{◦}. It follows from
(5) that if these projection mappings have closed-form expressions, the decomposition issue
can be simply solved by this classical theorem. However, for most non-symmetric cones
(except for the circular cone [7,45], see “Appendix 6.2”), their projection mappings are
usually not explicit, such as the power cone*K** _{α}*[21, section 2] and the exponential cone

*K*exp

[26, section 6]. Thus, one cannot employ the Moreau decomposition theorem directly and continue subsequent studies on optimization problems involving with these non-symmetric cones. This is a big hurdle for non-symmetric cone optimization problems.

In reality, there are plenty of non-symmetric cones in the literatures, such as homoge-
neous cones [6,20,42], matrix norm cones [12], p-order cones [2,18,27,44], hyperbolicity
cones [3,19,32], circular cones [7,45] and copositive cones [13], etc. Unlike the symmetric
cone optimization, there seems no systematic study due to the various features and very few
algorithms are proposed to solve optimization problems with these non-symmetric cones
constraints, except for some interior-point type methods [6,22,28,36,44]. For example, Xue
and Ye [44] study an optimization problem of minimizing a sum of p-norms, in which two
*new barrier functions are introduced for p-order cones and a primal-dual potential reduction*
algorithm is presented. Chua [6] combines the T-algebra with the primal-dual interior-point
algorithm to solve the homogeneous conic programming problems. Based on the concept of
self-concordant barriers and the efficient computational experience of the long path-following
steps, Nesterov [28] proposes a new predictor-corrector path-following method with an addi-
tional primal-dual lifting process (called Phase I). Skajaa and Ye [36] present a homogeneous
interior-point algorithm for non-symmetric convex conic optimization, in which no Phase
I method is needed. Recently, Karimi and Tuncel [22] present a primal-dual interior-point

methods for convex optimization problems, in which a new concept called Domain-Driven Setup plays a crucial role in their theoretical analysis.

In contrast to these interior-point type methods, we pay more attention to the decomposi-
tion issue of the given cones. It is worth noting that the decompositions with respect to the
second-order coneL* ^{n}*and the circular cone

*L*

*[see Eqs. (3) and (51)] show that any given*

_{θ}*point can be divided into two parts, one lies in the boundary of the given cone (i.e., u*

^{(1)}

_{z}*∈ ∂L*

*,*

^{n}*˜u*^{(1)}*z* *∈ ∂L** _{θ}*, where

*∂Ω is the boundary of Ω) and the other comes from the boundary of the*

*given cone (i.e., u*

^{(2)}

_{z}*∈ ∂L*

*) or its polar (i.e.,*

^{n}*˜u*

^{(2)}*z*

*∈ ∂L*

^{◦}

*). One can easily verify these results by the Moreau decomposition theorem in some cases (for example, the given point lies out the union of the given cone and its polar), but it is amazing that these decompositions are satisfied in all cases! These observations motivate us to study the boundary structures of the given cones more carefully.*

_{θ}**1.2 Contributions and contents**

In this paper, we successfully explore two new types of decompositions with respect to the
power cone_{K}* _{α}*and the exponential cone

_{K}_{exp}, one is adapted from the well-known Moreau decomposition theorem, which looks like

*z= s**x**· x + s**y**· y, x ∈ ∂K, y ∈ ∂K*^{◦}*, (s**x**, s**y**) = (0, 0)* (6)
and the other follows from geometric structures of the given cone, i.e.,

*z= s**x**· x + s**y**· y, x ∈ ∂K, y ∈ ∂K, (s**x**, s**y**) = (0, 0),* (7)
*where z*∈ R^{n}*, s*_{x}*, s**y**∈ R, x, y ∈ R** ^{n}*,

*has two choices, namely*

_{K}

_{K}*or*

_{α}

_{K}_{exp}, as defined in (1) and (2). In the sequel, we call (6) the Type I decomposition and (7) the Type II decomposition, respectively. To our best knowledge, no results about the decompositions with respect to these two non-symmetric cones have been reported. Hence, the purpose of this paper aims to fill this gap and the contributions of our research can be summarized as follows.

(a) We propose a more compact description of the boundary for these two cones.

(b) Two types of decompositions with respect to*K*_{α}*,K*expare presented, which are do-able
and computable.

As a byproduct, the decomposition expressions with respect to the high-dimensional power cone are also derived.

(c) We establish the conic functions of the power cone*K**α* and the exponential cone*K*exp

based on their decomposition formulae.

The remainder of this paper is organized as follows. In Sects.2and3, we present the
decomposition formulae with respect to the power cone*K** _{α}*and the exponential cone

*K*exp, respectively. In Sect.4, we discuss some applications of these decompositions. Finally, we draw some concluding remarks in Sect.5.

**2 The decompositions with respect to the power cone****K**_{˛}

In this section, we present two types of decompositions with respect to the power cone*K** _{α}*.
Before that, we present some analytic properties of

*K*

*in the following lemmas.*

_{α}**Lemma 1** *K*_{α}*is a closed convex cone.*

**Proof It can be easily verified by definition, see “Appendix 6.3” for more details.**

**Lemma 2 The dual cone**_{K}^{∗}_{α}*can be described as*
*K*^{∗}* _{α}*=

*(x*1*, ¯x) ∈ R × R*^{2}

*|x*^{1}| ≤

*¯x*1

*α*1

_{α}_{1}

*¯x*2

*α*2

_{α}_{2}

*, ¯x*1 *≥ 0, ¯x*2≥ 0

*,*
*where* *¯x := ( ¯x*1*, ¯x*2*)** ^{T}* ∈ R

^{2}

*,α := (α*1

*, α*2

*)*

*∈ R*

^{T}^{2}

*,α*1

*, α*2

*∈ (0, 1), α*1

*+ α*2

*= 1.*

**Proof We refer the readers to [5, Theorem 4.3.1] for its verification.**

From the relation*K*^{◦}* _{α}*= −

*K*

^{∗}

*and Lemma2, the polar cone*

_{α}*K*

^{◦}

*has the following closed- form expression.*

_{α}**Corollary 1 The polar cone**K^{◦}_{α}*is given by*
*K*^{◦}* _{α}* =

*(x*1*, ¯x) ∈ R × R*^{2}

*|x*^{1}| ≤

*− ¯x*1

*α*1

_{α}_{1}

*− ¯x*2

*α*2

_{α}_{2}

*, ¯x*1*≤ 0, ¯x*2≤ 0

*.*
We now proceed to identify the structures of the power cone*K** _{α}*, its dual

*K*

_{α}^{∗}and its polar

*K*

^{◦}

*more clearly, particularly for their interiors and boundaries.*

_{α}**Lemma 3 The interior of the power cone**K_{α}*, its dualK*^{∗}_{α}*and its polarK*^{◦}_{α}*, denoted by*
*intK**α**, intK*^{∗}_{α}*and intK*^{◦}_{α}*, are respectively given by*

*intK** _{α}* =

*(x*1*, ¯x) ∈ R × R*^{2}

*|x*^{1}*| < σ*_{α}*( ¯x), ¯x*1*> 0, ¯x*2*> 0*

*,* (8)

*intK*^{∗}* _{α}* =

*(x*1*, ¯x) ∈ R × R*^{2}

*|x*^{1}*| < η*_{α}*( ¯x), ¯x*1*> 0, ¯x*2*> 0*

*,* (9)

*intK*^{◦}* _{α}* =

*(x*1*, ¯x) ∈ R × R*^{2}

*|x*^{1}*| < η**α**(− ¯x), ¯x*1*< 0, ¯x*2*< 0*

*,* (10)

*where*

*σ*_{α}*( ¯x) := ¯x*^{α}_{1}^{1}*¯x*_{2}^{α}^{2}*, η*_{α}*( ¯x) :=*

*¯x*1

*α*1

_{α}_{1}

*¯x*2

*α*2

_{α}_{2}

*.* (11)

* Proof By definition, (x*1

*, ¯x) is an element of intK*

*if and only if there exists an open neigh- borhood of*

_{α}*(x*1

*, ¯x) ∈ R × R*

^{2}entirely included in

*K*

*. Let us take*

_{α}*(x*1

*, ¯x) ∈K*

*. For any given strict positive scalars*

_{α}*¯x*1

*, ¯x*2

*∈ R, it is easy to see that (0, 0, 0), (0, ¯x*1

*, 0) and (0, 0, ¯x*2

*)*are all outside of int

*K*

*α*, due to the observation that every open neighborhood with respect to each of these points contains a point with the negative

*¯x*1or

*¯x*2 component. For a point

*(x*1

*, ¯x*1

*, ¯x*2

*) ∈ R × R*

^{2}such that

*σ*

_{α}*( ¯x) = |x*1

*| with ¯x*1

*, ¯x*2

*> 0, where σ*

_{α}*( ¯x) is defined as*in (11). In this case, we can take a point

*(x*

_{1}

^{ }

*, ¯x*

_{1}

^{ }

*, ¯x*

_{2}

^{ }

*) with 0 < ¯x*

_{1}

^{ }

*< ¯x*1, 0

*< ¯x*

^{ }

_{2}

*< ¯x*2,

*|x*_{1}^{
}*| > |x*1*| in every open neighborhood of (x*1*, ¯x*1*, ¯x*2*) ∈ R × R*^{2}, which implies that

*|x*_{1}^{
}*| > |x*1*| = σ**α**( ¯x) > σ**α**( ¯x*^{
}*), i.e., the point (x*_{1}^{
}*, ¯x*_{1}^{
}*, ¯x*_{2}^{
}*) can not belong toK**α* and hence
*(x*1*, ¯x*1*, ¯x*2*) /∈ intK** _{α}*.

Next, we turn to show that all the remaining points that do not satisfy the above two cases,
i.e., the points in the right-hand side of (8), belong to the interior of*K** _{α}*. For sufficiently small
scalar

*∈ (0, min{ ¯x*1

*, ¯x*2

*}), letB*

_{(x}

^{}1*, ¯x)*be a neighborhood of*(x*1*, ¯x) with the form*
*B*_{(x}^{}_{1}* _{, ¯x)}*:=

*(x*_{1}^{
}*, ¯x*^{
}*) ∈ R × R*^{2}

*0 ≤ |x*^{1}*| − ≤ |x*_{1}^{
}*| ≤ |x*1*| + ,*
0*< ¯x**i**− ≤ ¯x*_{i}^{
}*≤ ¯x**i**+ , i = 1, 2*

*.*

Taking*(x*1*, ¯x) ∈ R × R*^{2}from the right-hand side of (8), i.e.,*σ**α**( ¯x) > |x*1*|, ¯x**i* *> 0, i = 1, 2.*

For all elements*(x*_{1}^{
}*, ¯x*^{
}*) ∈B*^{}_{(x}

1*, ¯x)*, we have

*|x*_{1}^{
}*| − σ*_{α}*( ¯x*^{
}*) ≤ | ¯x*1*| + − ( ¯x*_{1}^{
}*)*^{α}^{1}*( ¯x*_{2}^{
}*)*^{α}^{2} *≤ | ¯x*1*| + − ( ¯x*1*− )*^{α}^{1}*( ¯x*2*− )*^{α}^{2}*.* (12)
In addition, letting* → 0, we obtain*

*→0*lim

*| ¯x*1*| + − ( ¯x*1*− )*^{α}^{1}*( ¯x*2*− )*^{α}^{2}

*= | ¯x*1*| − σ*_{α}*( ¯x) < 0.*

Therefore, there exists a scalar^{∗}such that*| ¯x*1*| + *^{∗}*− ( ¯x*1*− *^{∗}*)*^{α}^{1}*( ¯x*2*− *^{∗}*)*^{α}^{2} *< 0. This*
together with (12) imply that

*|x*^{
}_{1}*| − σ*_{α}*( ¯x*^{
}*) < 0, ∀(x*_{1}^{
}*, ¯x*^{
}*) ∈B*_{(x}^{}_{1}_{, ¯x)}*,*

which is sufficient to show that*B*_{(x}^{}_{1}* _{, ¯x)}*is entirely included in

*K*

*and hence*

_{α}*(x*1

*, ¯x) ∈ intK*

*. Applying a similar way to*

_{α}*K*

^{∗}

*and*

_{α}*K*

^{◦}

*, their interiors can also be verified as the right-hand*

_{α}side of (9) and (10).

From the proof of Lemma 3, we further define the following sets
*S*_{1} :=

*(x*1*, ¯x) ∈ R × R*^{2}*| x*1*= 0, ¯x*1*> 0, ¯x*2 = 0
*,*
*S*2 :=

*(x*1*, ¯x) ∈ R × R*^{2}*| x*1 *= 0, ¯x*1 *= 0, ¯x*2 *> 0*
*,*
*S*_{3} :=

*(x*1*, ¯x) ∈ R × R*^{2}*| |x*1*| = σ**α**( ¯x), ¯x*1*> 0, ¯x*2*> 0*
*,*
*S*4 :=

*(x*1*, ¯x) ∈ R × R*^{2}*| |x*1*| = η*_{α}*( ¯x), ¯x*1*> 0, ¯x*2*> 0*
*,*
*T*_{1} :=

*(x*1*, ¯x) ∈ R × R*^{2}*| x*1 *= 0, ¯x*1 *< 0, ¯x*2 = 0

*= −S*1*,*
*T*2 :=

*(x*1*, ¯x) ∈ R × R*^{2}*| x*1*= 0, ¯x*1*= 0, ¯x*2*< 0*

*= −S*2*,*
*T*_{3} :=

*(x*1*, ¯x) ∈ R × R*^{2}*| |x*1*| = η**α**(− ¯x), ¯x*1*< 0, ¯x*2*< 0*

*= −S*4*.*

(13)

Then, the boundary of*K*_{α}*,K*^{∗}* _{α}*and

*K*

_{α}^{◦}can be stated in a more compact form.

**Lemma 4 The boundary of**_{K}_{α}*and _{K}*

^{∗}

_{α}*, denoted by∂K*

*α*

*and∂K*

^{∗}

_{α}*, are respectively given by*

*∂K**α* *:= S*1*∪ S*2*∪ S*3*∪ {0}, ∂K*^{∗}_{α}*:= S*1*∪ S*2*∪ S*4*∪ {0}.*

*Similarly, the boundary ofK*^{◦}_{α}*, denoted by∂K*^{◦}_{α}*, can be formulated as*

*∂K*_{α}^{◦} *:= T*1*∪ T*2*∪ T*3*∪ {0}.*

**Remark 1 It follows that the union set**K* _{α}*∪

*K*

_{α}^{◦}can be divided into seven parts

*K*

*∪*

_{α}*K*

^{◦}

_{α}*= S*1

*∪ S*2

*∪ T*1

*∪ T*2

*∪ P*1

*∪ P*2

*∪ {0},*

where

*P*_{1}:=

*(x*1*, ¯x) ∈ R × R*^{2}*| |x*1*| ≤ σ**α**( ¯x), ¯x*1*> 0, ¯x*2*> 0*
*,*
*P*2:=

*(x*1*, ¯x) ∈ R × R*^{2}*| |x*1*| ≤ η*_{α}*(− ¯x), ¯x*1 *< 0, ¯x*2 *< 0*
*.*
In addition, the boundary of*K** _{α}*and its polar

*K*

^{◦}

*are depicted in Fig.3.*

_{α}In order to make the classifications clear and neat, we adapt some notations as follows:

*z:= (z*1*, ¯z) ∈ R × R*^{2}*, ¯z := (¯z*1*, ¯z*2*)** ^{T}* ∈ R

^{2}

*, ¯z*min

*:= min{¯z*1

*, ¯z*2

*}, ¯z*max

*:= max{¯z*1

*, ¯z*2

*}.*

(14)

**Fig. 3 The different parts of***∂K**α*(left) and*∂K*^{◦}* _{α}*(right)

Consequently, we divide the spaceR × R^{2}into the following four blocks
Block I: *B*1:=

*(z*1*, ¯z) ∈ R × R*^{2}*| ¯z*min*· ¯z*max*> 0 or (z*1*= 0 and ¯z*min*= ¯z*max*= 0)*
*.*
Block II*: B*2:=

*(z*1*, ¯z) ∈ R × R*^{2}*| ¯z*min*· ¯z*max*= 0 and ¯z*min*+ ¯z*max= 0
*.*
Block III*: B*3:=

*(z*1*, ¯z) ∈ R × R*^{2}*| ¯z*min*· ¯z*max*< 0*
*.*
Block IV*: B*4:=

*(z*1*, ¯z) ∈ R × R*^{2}*| z*1*= 0 and ¯z*min*= ¯z*max= 0
*.*

(15)
The subcases of these blocks with respect to*K** _{α}*can be found in Table1.

**2.1 The Type I decomposition with respect to the power cone****K**_{˛}

In this subsection, we present the Type I decomposition with respect to the power cone
*K*_{α}*. To proceed, we discuss four cases, in which the sets S** _{i}* ⊂

*K*

*(i = 1, 2, 3, 4) and*

*T*

_{j}*⊂ ∂K*

^{◦}

*( j = 1, 2, 3) are defined as in (*13).

**Case 1:***(z*1*, ¯z) ∈ B*1.

(a) *¯z*min*> 0. In this subcase, (z*1*, ¯z) ∈ B*11, i.e.,*¯z*1 *> 0, ¯z*2 *> 0, which implies σ*_{α}*(¯z) > 0*
and*η*_{α}*(¯z) > 0. Then, we take x = ˙x*^{(B}^{1}^{,a)}*, y= ˙y*^{(B}^{1}^{,a)}*and s**x* *= ˙s*^{(B}*x* ^{1}^{,a)}*, s**y* *= ˙s*^{(B}*y* ^{1}* ^{,a)}*,
where

*˙x*^{(B}^{1}* ^{,a)}* :=

1

*σ**α**¯z**(¯z)*

*∈ S*3*, ˙y*^{(B}^{1}* ^{,a)}*:=

1

−_{η}_{α}^{¯z}_{(¯z)}

*∈ T*3*.* (16)

*˙s*^{(B}_{x}^{1}* ^{,a)}* :=

*z*1

*+ η*

*α*

*(¯z)*

*σ**α**(¯z) + η**α**(¯z)· σ**α**(¯z), ˙s*_{y}^{(B}^{1}* ^{,a)}*:=

*z*1

*− σ*

*α*

*(¯z)*

*σ**α**(¯z) + η**α**(¯z)· η**α**(¯z).* (17)
It is easy to show that the above setting satisfies the decomposition formula (6).

(b) *¯z*max*< 0. Similar to the argument in Case 1 (a), (z*1*, ¯z) ∈ B*12, i.e.,*¯z*1*< 0, ¯z*2*< 0, which*
implies*σ**α**(−¯z) > 0 and η**α**(−¯z) > 0. In this subcase, we set x = ˙x*^{(B}^{1}^{,b)}*, y= ˙y*^{(B}^{1}^{,b)}*and s**x* *= ˙s**x*^{(B}^{1}^{,b)}*, s**y**= ˙s*^{(B}*y* ^{1}* ^{,b)}*, where

*˙x*^{(B}^{1}* ^{,b)}*:=

1

*σ**α**−¯z**(−¯z)*

*∈ S*3*, ˙y*^{(B}^{1}* ^{,b)}*:=

1

*η**α**(−¯z)**¯z*

*∈ T*3*.* (18)

*˙s*^{(B}_{x}^{1}* ^{,b)}*:=

*z*1

*− η*

_{α}*(−¯z)*

*σ*_{α}*(−¯z) + η*_{α}*(−¯z)* *· σ**α**(−¯z),*

**Table****1**Thesubcasesofeachblockin(15)withrespectto*K**α* *B*1*B*2*B*3*B*4 *(B*11)*z*1free,*¯z*1*>*0,*¯z*2*>*0(*B*21)*z*1free,*¯z*1=0,*¯z*2*>*0(*B*31)*z*1free,*¯z*1*<*0,*¯z*2*>*0(*B*4)*z*1=0,*¯z*1=0,*¯z*2=0 *(B*12)*z*1free,*¯z*1*<*0,*¯z*2*<*0(*B*22)*z*1free,*¯z*1*>*0,*¯z*2=0(*B*32)*z*1free,*¯z*1*>*0,*¯z*2*<*0 *(B*13)*z*1=0,*¯z*1=0,*¯z*2=0(*B*23)*z*1free,*¯z*1=0,*¯z*2*<*0 *(B*24)*z*1free,*¯z*1*<*0,*¯z*2=0

*˙s*^{(B}_{y}^{1}* ^{,b)}*:=

*z*1

*+ σ*

_{α}*(−¯z)*

*σ*_{α}*(−¯z) + η*_{α}*(−¯z)* *· η*_{α}*(−¯z).* (19)

*(c) z*_{1} *= 0 and ¯z*min *= ¯z*max*= 0. In this subcase, (z*1*, ¯z) ∈ B*13, which implies*σ**α**(¯z) = 0*
and*η**α**(¯z) = 0. Therefore, we set x = ˙x*^{(B}^{1}^{,c)}*, y= ˙y*^{(B}^{1}^{,c)}*and s*_{x}*= ˙s*^{(B}*x* ^{1}^{,c)}*, s*_{y}*= ˙s*^{(B}*y* ^{1}* ^{,c)}*,
where

*˙x*^{(B}^{1}* ^{,c)}*:=

1

*σ**α***1****(1)**

*∈ S*3*, ˙y*^{(B}^{1}* ^{,c)}*:=

1

−_{η}_{α}^{1}_{(1)}

*∈ T*3*,* (20)

*˙s*_{x}^{(B}^{1}* ^{,c)}*:=

*z*1

*σ*_{α}**(1) + η**_{α}**(1)**· σ_{α}**(1), ˙s**^{(B}_{y}^{1}* ^{,c)}*:=

*z*1

*σ*_{α}**(1) + η**_{α}**(1)**· η_{α}* (1)* (21)

**with 1***:= (1, 1)** ^{T}* ∈ R

^{2}.

**Case 2:**

*(z*1

*, ¯z) ∈ B*2.

(a) *¯z*min*= 0, ¯z*max*> 0. In this subcase, (z*1*, ¯z) ∈ B*21or*(z*1*, ¯z) ∈ B*22. Therefore, we set
*x* *= ˙x*^{(B}^{2}^{,a)}*, y* *= ˙y*^{(B}^{2}^{,a)}*and s**x* *= 1, s**y* *= 1, where ˙x*^{(B}^{2}^{,a)}*= ( ˙x*_{1}^{(B}^{2}^{,a)}*, ˙¯x*^{(B}^{2}^{,a)}*) and*

*˙y*^{(B}^{2}^{,a)}*= ( ˙y*_{1}^{(B}^{2}^{,a)}*, ˙¯y*^{(B}^{2}^{,a)}*) with*

*˙x*_{1}^{(B}^{2}^{,a)}*:= z*1*, ˙¯x*^{(B}^{2}* ^{,a)}* :=

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎩

⎡

⎢⎣

*|z*1|

*¯z*^{α2}_{2}

^{1}

*α1*

*¯z*2

⎤

⎥*⎦ if (z*^{1}*, ¯z) ∈ B*21*,*

⎡

⎢⎣

*¯z*1

*|z*1|

*¯z*^{α1}_{1}

^{1}

*α2*

⎤

⎥*⎦ if (z*1*, ¯z) ∈ B*22*,*

(22)

*˙y*_{1}^{(B}^{2}^{,a)}*:= 0, ˙¯y*^{(B}^{2}* ^{,a)}* :=

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎩

⎡

⎢⎣ −

*|z*1|

*¯z*_{2}^{α2}

_{α1}^{1}

0

⎤

⎥*⎦ if (z*1*, ¯z) ∈ B*21*,*

⎡

⎢⎣ 0

−

*|z*1|

*¯z*^{α1}_{1}

^{1}

*α2*

⎤

⎥*⎦ if (z*^{1}*, ¯z) ∈ B*22*.*

(23)

It is easy to see that

(a)*(z*^{1}*, ¯z) ∈ B*^{21}*, z*^{1}*= 0 ⇒ x ∈ S*^{2}*, y = 0; (b) (z*^{1}*, ¯z) ∈ B*^{21}*, z*^{1}*= 0 ⇒ x ∈ S*^{3}*, y ∈ T*^{1};
(c)*(z*1*, ¯z) ∈ B*22*, z*1*= 0 ⇒ x ∈ S*1*, y = 0; (d) (z*1*, ¯z) ∈ B*22*, z*1*= 0 ⇒ x ∈ S*3*, y ∈ T*2*.*

(b) *¯z*min*< 0, ¯z*max*= 0. In this subcase, (z*1*, ¯z) ∈ B*23or*(z*1*, ¯z) ∈ B*24*. We set x* *= ˙x*^{(B}^{2}* ^{,b)}*,

*y*

*= ˙y*

^{(B}^{2}

^{,b)}*and s*

*x*

*= −1, s*

*y*

*= −1, where ˙x*

^{(B}^{2}

^{,b)}*= ( ˙x*

_{1}

^{(B}^{2}

^{,b)}*, ˙¯x*

^{(B}^{2}

^{,b)}*) and ˙y*

^{(B}^{2}

*=*

^{,b)}*( ˙y*

^{(B}_{1}

^{2}

^{,b)}*, ˙¯y*

^{(B}^{2}

^{,b)}*) with*

*˙x*_{1}^{(B}^{2}^{,b)}*:= −z*1*, ˙¯x*^{(B}^{2}* ^{,b)}* :=

⎧⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎩

⎡

⎣
*|z*1|

*(−¯z*2*)*^{α2}

^{1}

*α1*

*−¯z*2

⎤

*⎦ if (z*1*, ¯z) ∈ B*23*,*

⎡

⎣ *−¯z*1

*|z*1|
*(−¯z*1*)*^{α1}

^{1}

*α2*

⎤

*⎦ if (z*1*, ¯z) ∈ B*24*,*

(24)

*˙y*^{(B}_{1} ^{2}^{,b)}*:= 0, ˙¯y*^{(B}^{2}* ^{,b)}* :=

⎧⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎩

⎡

⎣ −

*|z*1|
*(−¯z*2*)*^{α2}

^{1}

*α1*

0

⎤

*⎦ if (z*1*, ¯z) ∈ B*23*,*

⎡

⎣ 0

−

*|z*1|
*(−¯z*1*)*^{α1}

^{1}

*α2*

⎤

*⎦ if (z*1*, ¯z) ∈ B*24*.*

(25)

Similar to the arguments in Case 2 (a), we obtain

(a)*(z*1*, ¯z) ∈ B*23*, z*1*= 0 ⇒ x ∈ S*2*, y = 0; (b) (z*1*, ¯z) ∈ B*23*, z*1*= 0 ⇒ x ∈ S*3*, y ∈ T*1;
(c)*(z*1*, ¯z) ∈ B*24*, z*1*= 0 ⇒ x ∈ S*1*, y = 0; (d) (z*1*, ¯z) ∈ B*24*, z*1*= 0 ⇒ x ∈ S*3*, y ∈ T*2*.*
**Case 3:***(z*1*, ¯z) ∈ B*3. In this subcase,*(z*1*, ¯z) ∈ B*31or*(z*1*, ¯z) ∈ B*32*. We set x* *= ˙x*^{(B}^{3}^{)}*∈ ∂K** _{α}*,

*y*

*= ˙y*

^{(B}^{3}

^{)}*∈ ∂K*

^{◦}

_{α}*and s*

_{x}*= 1, s*

*y*

*= 1, where ˙x*

^{(B}^{3}

^{)}*= ( ˙x*

_{1}

^{(B}^{3}

^{)}*, ˙¯x*

^{(B}^{3}

^{)}*) and ˙y*

^{(B}^{3}

^{)}*= ( ˙y*

_{1}

^{(B}^{3}

^{)}*,*

*˙¯y*^{(B}^{3}^{)}*) with*

*˙x*_{1}^{(B}^{3}^{)}*:= z*1*, ˙¯x*^{(B}^{3}* ^{)}*:=

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎩

⎡

⎢⎣

*|z*1|

*¯z*^{α2}_{2}

_{α1}^{1}

*¯z*2

⎤

⎥*⎦ if z ∈ B*^{31}*,*

⎡

⎢⎣

*¯z*1

*|z*1|

*¯z*^{α1}_{1}

^{1}

*α2*

⎤

⎥*⎦ if z ∈ B*^{32}*,*

(26)

*˙y*^{(B}_{1} ^{3}^{)}*:= 0, ˙¯y*^{(B}^{3}* ^{)}*:=

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎩

⎡

⎢*⎣ ¯z*^{1}−

*|z*1|

*¯z*^{α2}_{2}

^{1}

*α1*

0

⎤

⎥*⎦ if z ∈ B*^{31}*,*

⎡

⎢⎣ 0

*¯z*2−

*|z*1|

*¯z*^{α1}_{1}

^{1}

*α2*

⎤

⎥*⎦ if z ∈ B*32*.*

(27)

More concretely, we obtain

(a)*(z*1*, ¯z) ∈ B*31*, z*1*= 0 ⇒ x ∈ S*2*, y ∈ T*1*; (b) (z*1*, ¯z) ∈ B*31*, z*1*= 0 ⇒ x ∈ S*3*, y ∈ T*1;
(c)*(z*1*, ¯z) ∈ B*32*, z*1 *= 0 ⇒ x ∈ S*1*, y ∈ T*2*; (d) (z*1*, ¯z) ∈ B*32*, z*1*= 0 ⇒ x ∈ S*3*, y ∈ T*2*.*
**Case 4:***(z*1*, ¯z) ∈ B*4*. In this subcase, we set x= ˙x*^{(B}^{4}^{)}*, y= ˙y*^{(B}^{4}^{)}*and s*_{x}*= 1, s**y*= 1, where

*˙x*^{(B}^{4}* ^{)}*:=

⎡

⎣0 1 0

⎤

*⎦ ∈ S*1*, ˙y*^{(B}^{4}* ^{)}*:=

⎡

⎣ 0

−1 0

⎤

*⎦ ∈ T*1*,* (28)

or

*˙x*^{(B}^{4}* ^{)}*:=

⎡

⎣0 0 1

⎤

*⎦ ∈ S*2*, ˙y*^{(B}^{4}* ^{)}*:=

⎡

⎣ 0 0

−1

⎤

*⎦ ∈ T*2*.* (29)

**Table 2 The locations of the x-part and y-part in the Type I decomposition with respect to***K**α*

*¯B*_{1} *¯B*_{2} *¯B*_{3} *¯B*_{4}

*¯B*_{21} *¯B*_{22} *¯B*_{23} *¯B*_{24} *¯B*_{31} *¯B*_{32}

*x*_{loc}*S*_{3} *S*_{2}*∪ S*3 *S*_{1}*∪ S*3 *S*_{2}*∪ S*3 *S*_{1}*∪ S*3 *S*_{2}*∪ S*3 *S*_{1}*∪ S*3 *S*_{1}*∪ S*2
*y**loc* *T*3 *{0} ∪ T*1 *{0} ∪ T*2 *{0} ∪ T*1 *{0} ∪ T*2 *T*1 *T*2 *T*1*∪ T*2

To sum up these discussions, we present the Type I decomposition with respect to the
power cone*K** _{α}*in the following theorem.

* Theorem 1 For any given z= (z*1

*, ¯z) ∈ R × R*

^{2}

*, its Type I decomposition with respect toK*

_{α}*is given by*

*(a) If z∈ B*1*, then*

*z*=

⎧⎪

⎨

⎪⎩

*˙s*^{(B}*x* ^{1}^{,a)}*· ˙x*^{(B}^{1}^{,a)}*+ ˙s*^{(B}*y* ^{1}^{,a)}*· ˙y*^{(B}^{1}^{,a)}*, if z ∈ B*11*,*

*˙s*^{(B}*x* ^{1}^{,b)}*· ˙x*^{(B}^{1}^{,b)}*+ ˙s*^{(B}*y* ^{1}^{,b)}*· ˙y*^{(B}^{1}^{,b)}*, if z ∈ B*12*,*

*˙s*^{(B}*x* ^{1}^{,c)}*· ˙x*^{(B}^{1}^{,c)}*+ ˙s*^{(B}*y* ^{1}^{,c)}*· ˙y*^{(B}^{1}^{,c)}*, if z ∈ B*13*,*

*where˙x*^{(B}^{1}^{,a)}*,˙y*^{(B}^{1}^{,a)}*,˙s**x*^{(B}^{1}^{,a)}*,˙s*^{(B}*y* ^{1}^{,a)}*are defined as in (16)–(17),* *˙x*^{(B}^{1}^{,b)}*,˙y*^{(B}^{1}^{,b)}*,˙s**x*^{(B}^{1}^{,b)}*,*

*˙s**y*^{(B}^{1}^{,b)}*are defined as in (18)–(19) and* *˙x*^{(B}^{1}^{,c)}*,* *˙y*^{(B}^{1}^{,c)}*,˙s**x*^{(B}^{1}^{,c)}*,˙s*^{(B}*y* ^{1}^{,c)}*are defined as in*
(20)–(21).

*(b) If z∈ B*2*, then*
*z*=

*˙x*^{(B}^{2}^{,a)}*+ ˙y*^{(B}^{2}^{,a)}*,* *if z∈ B*21*or z∈ B*22*,*
*(−1) · ˙x*^{(B}^{2}^{,b)}*+ (−1) · ˙y*^{(B}^{2}^{,b)}*, if z ∈ B*23*or z∈ B*24*,*

*where* *˙x*^{(B}^{2}^{,a)}*,˙y*^{(B}^{2}^{,a)}*are defined as in (22)–(23),* *˙x*^{(B}^{2}^{,b)}*,˙y*^{(B}^{2}^{,b)}*are defined as in (24)–*

(25).

*(c) If z∈ B*3*, then z= ˙x*^{(B}^{3}^{)}*+ ˙y*^{(B}^{3}^{)}*, where˙x*^{(B}^{3}^{)}*,* *˙y*^{(B}^{3}^{)}*are defined as in (26)–(27).*

*(d) If z∈ B*4*, then z= ˙x*^{(B}^{4}^{)}*+ ˙y*^{(B}^{4}^{)}*, where˙x*^{(B}^{4}^{)}*and* *˙y*^{(B}^{4}^{)}*are defined as in (28) or (29).*

*In addition, the locations of the x-part and y-part in each case are shown in Table*2, where
*S*_{i}*, T**i* *(i = 1, 2, 3, 4) are defined as in (*13) and x_{loc}*, y**loc* *denote the locations of x and y,*
*respectively.*

**2.2 The Type II decomposition with respect to the power cone****K**_{˛}

In this subsection, we present the Type II decomposition with respect to the power cone*K**α*.
Similarly, we consider the following four cases.

**Case 1:***(z*1*, ¯z) ∈ B*1.

(a) *¯z*min *> 0. In this subcase, (z*1*, ¯z) ∈ B*11 and*σ**α**(¯z) > 0. Then, we take x = ¨x*^{(B}^{1}* ^{,a)}*,

*y= ¨y*

^{(B}^{1}

^{,a)}*and s*

*x*

*= ¨s*

*x*

^{(B}^{1}

^{,a)}*, s*

*y*

*= ¨s*

^{(B}*y*

^{1}

*, where*

^{,a)}*¨x*^{(B}^{1}* ^{,a)}*:=

1

*σ**α**¯z**(¯z)*

*∈ S*3*, ¨y*^{(B}^{1}* ^{,a)}*:=

−1

*σ**α**¯z**(¯z)*

*∈ S*3*.* (30)

*¨s*^{(B}*x* ^{1}* ^{,a)}*:=

*z*

_{1}

*+ σ*

*α*

*(¯z)*

2 *, ¨s*^{(B}*y* ^{1}* ^{,a)}*:=

*σ*

*α*

*(¯z) − z*1

2 *.* (31)

Similarly, we can show that the above setting satisfies the decomposition formula (7).