The decompositions with respect to two core non-symmetric cones

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

The decompositions with respect to two core non-symmetric cones

Yue Lu¹· Ching-Yu Yang²· Jein-Shan Chen² · Hou-Duo Qi³

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

jinjin403@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 Kα :=



(x1, ¯x) ∈ R × R²

|x¹| ≤ ¯x₁^α1¯x₂^α2, ¯x1≥ 0, ¯x2≥ 0



, (1)

Kexp := cl



(x1, ¯x) ∈ R × R²

 x¹≥ ¯x2· exp

¯x1

¯x2



, ¯x2> 0, x1≥ 0



, (2)

where¯x := ( ¯x1, ¯x2)^T ∈ R²,α := (α1, α2)^T ∈ R²,α1, α2∈ (0, 1), α1+α2= 1 and cl(Ω) is the closure ofΩ. We callKαthe power cone andKexpthe exponential cone¹, 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], Lp 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∈ Rⁿwhose sum of weight distances from a given set of locations L1, . . . , Lmis minimized, which has the following form

(P) minx∈Rⁿ _m

i=1wix − Lipi

where · p_i(pi ≥ 1) denotes the pi-norm defined onRⁿ. If piis equal to 2, then the above problem reduces to the classical Weber-Point problem. Denote by x:= (x1, . . . , xn)^T ∈ Rⁿ and a:= (a1, . . . , an)^T ∈ Rⁿ, Problem(P) can be rewritten as

min_x,a,yi _m

i=1wiai

s.t. (yi, j, ai, xj− Li, j) ∈K¹

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

_n

j=1y_{i, j} = ai, i = 1, . . . , m,

where L_{i, j}and y_{i, j}stand for the j -th component of L_i∈ Rⁿ and y_i∈ Rⁿ, respectively.

Geometric programming [4,31,34]: Let x := (x1, . . . , xn)^T ∈ Rⁿ be a vector with real positive components xi. A real valued function m, of the form m(x) := c_n

i=1x_i^αi, is called a monomial function, where c> 0 and αi are its coefficient and exponents, respectively. A sum of one or more monomials, i.e., a function that looks like f(x) :=_K

k=1mk(x), is called

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

Kexp:= cl



(x1, ¯x) ∈ R × R²

 x¹≥ ¯x2· exp

¯x1

¯x2

 , ¯x2> 0

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

Fig. 1 The power coneKα(left) and the exponential coneKexp(right)

a posynomial function, where m_k(x) := ck

_n

i=1x_i^αi,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 f₀(x)

s.t. fs(x) ≤ 1, s = 1, . . . , p, gt(x) = 1, t = 1, . . . , q, where fs := _K

k=1ck,s_n

i=1x_i^αi,k,s, s ∈ {0, 1, . . . , p} and gt(x) := ct_n

i=1x_i^αi,t, t ∈ {1, . . . , q}. Using the following change of variables as xi := exp(ui), ck,i := exp(dk,i), ct :=

exp(dt) and adding some additional variables, Problem (GP) can be rewritten as minu_i,w,ξk,0,ηk,s w

s.t. (dk,0+_n

i=1ui· αi,k,0, ξk,0, 1) ∈Kexp, _K

k=1ξk,0= w, (dk,s+_n

i=1ui· αi,k,s, ηk,s, 1) ∈Kexp,_K

k=1ηk,s= 1, s = 1, . . . , p, dt+_n

i=1ui· αi,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

Lⁿ := {(x1, ¯x) ∈ R × Rⁿ⁻¹| x1≥  ¯x}.

For any given z= (z1, ¯z) ∈ R × Rⁿ⁻¹, its decomposition with respect toLⁿhas the form z= λ1(z) · u⁽¹⁾_z + λ2(z) · u⁽²⁾_z , (3) whereλi(z) := z1+ (−1)ⁱ¯z and u⁽ⁱ⁾z is equal to¹₂

1, (−1)ⁱ_¯z^¯z

, if¯z = 0;¹₂

1, (−1)ⁱw , otherwise, which is applicable for i = 1, 2 with w ∈ Rⁿ⁻¹being 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 (λ1(z)) · u⁽¹⁾z + f (λ2(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∈ Rⁿ, it can be uniquely decomposed into

z= Π_K(z) + Π_K^◦(z) = Π_K(z) − Π_K^∗(−z), (5) whereΠ_K(z) is the projection mapping of z ∈ RⁿontoKandK^◦is the polar cone ofK, i.e.,

K^◦:= {y ∈ Rⁿ| x^Ty≤ 0, ∀x ∈K}.

In addition,K^∗is the dual cone ofKand satisfies the relationK^∗= −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 coneK_α[21, section 2] and the exponential coneKexp

[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 coneLⁿand the circular coneL_θ[see Eqs. (3) and (51)] show that any given point can be divided into two parts, one lies in the boundary of the given cone (i.e., u⁽¹⁾_z ∈ ∂Lⁿ,

˜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 ∈ ∂Lⁿ) or its polar (i.e.,˜u⁽²⁾z ∈ ∂L^◦_θ). 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.

1.2 Contributions and contents

In this paper, we successfully explore two new types of decompositions with respect to the power cone_Kαand the exponential cone_Kexp, one is adapted from the well-known Moreau decomposition theorem, which looks like

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

z= sx· x + sy· y, x ∈ ∂K, y ∈ ∂K, (sx, sy) = (0, 0), (7) where z∈ Rⁿ, s_x, sy∈ R, x, y ∈ Rⁿ,_Khas two choices, namely_Kαor_Kexp, 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 toK_α,Kexpare 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 coneKα and the exponential coneKexp

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 coneK_αand the exponential coneKexp, 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 coneK_˛

In this section, we present two types of decompositions with respect to the power coneK_α. Before that, we present some analytic properties ofK_αin the following lemmas.

Lemma 1 K_αis 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^∗_αcan be described as K^∗_α=



(x1, ¯x) ∈ R × R²

|x¹| ≤

¯x1

α1

_α1

¯x2

α2

_α2

, ¯x1 ≥ 0, ¯x2≥ 0

 , where ¯x := ( ¯x1, ¯x2)^T ∈ R²,α := (α1, α2)^T ∈ R²,α1, α2∈ (0, 1), α1+ α2 = 1.

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

From the relationK^◦_α= −K^∗_αand Lemma2, the polar coneK^◦_αhas the following closed- form expression.

Corollary 1 The polar coneK^◦_αis given by K^◦_α =



(x1, ¯x) ∈ R × R²

|x¹| ≤

− ¯x1

α1

_α1

− ¯x2

α2

_α2

, ¯x1≤ 0, ¯x2≤ 0

 . We now proceed to identify the structures of the power coneK_α, its dualK_α^∗and its polar K^◦_αmore clearly, particularly for their interiors and boundaries.

Lemma 3 The interior of the power coneK_α, its dualK^∗_α and its polarK^◦_α , denoted by intKα, intK^∗_αand intK^◦_α, are respectively given by

intK_α =



(x1, ¯x) ∈ R × R²

|x¹| < σ_α( ¯x), ¯x1> 0, ¯x2> 0



, (8)

intK^∗_α =



(x1, ¯x) ∈ R × R²

|x¹| < η_α( ¯x), ¯x1> 0, ¯x2> 0



, (9)

intK^◦_α =



(x1, ¯x) ∈ R × R²

|x¹| < ηα(− ¯x), ¯x1< 0, ¯x2< 0



, (10)

where

σ_α( ¯x) := ¯x^α₁¹¯x₂^α2, η_α( ¯x) :=

¯x1

α1

_α1

¯x2

α2

_α2

. (11)

Proof By definition, (x1, ¯x) is an element of intK_αif and only if there exists an open neigh- borhood of(x1, ¯x) ∈ R × R²entirely included inK_α. Let us take(x1, ¯x) ∈K_α. For any given strict positive scalars¯x1, ¯x2∈ R, it is easy to see that (0, 0, 0), (0, ¯x1, 0) and (0, 0, ¯x2) are all outside of intKα, due to the observation that every open neighborhood with respect to each of these points contains a point with the negative ¯x1or ¯x2 component. For a point (x1, ¯x1, ¯x2) ∈ R × R²such thatσ_α( ¯x) = |x1| with ¯x1, ¯x2 > 0, where σ_α( ¯x) is defined as in (11). In this case, we can take a point(x₁ , ¯x₁ , ¯x₂ ) with 0 < ¯x₁ < ¯x1, 0 < ¯x ₂ < ¯x2,

|x₁ | > |x1| in every open neighborhood of (x1, ¯x1, ¯x2) ∈ R × R², which implies that

|x₁ | > |x1| = σα( ¯x) > σα( ¯x ), i.e., the point (x₁ , ¯x₁ , ¯x₂ ) can not belong toKα and hence (x1, ¯x1, ¯x2) /∈ 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 ofK_α. For sufficiently small scalar ∈ (0, min{ ¯x1, ¯x2}), letB_(x

1, ¯x)be a neighborhood of(x1, ¯x) with the form B_{(x1, ¯x)}:=



(x₁ , ¯x ) ∈ R × R²

0 ≤ |x¹| −  ≤ |x₁ | ≤ |x1| + , 0< ¯xi−  ≤ ¯x_i ≤ ¯xi+ , i = 1, 2

.

Taking(x1, ¯x) ∈ R × R²from the right-hand side of (8), i.e.,σα( ¯x) > |x1|, ¯xi > 0, i = 1, 2.

For all elements(x₁ , ¯x ) ∈B_(x

1, ¯x), we have

|x₁ | − σ_α( ¯x ) ≤ | ¯x1| +  − ( ¯x₁ )^α1( ¯x₂ )^α2 ≤ | ¯x1| +  − ( ¯x1− )^α1( ¯x2− )^α2. (12) In addition, letting → 0, we obtain

→0lim

| ¯x1| +  − ( ¯x1− )^α1( ¯x2− )^α2

= | ¯x1| − σ_α( ¯x) < 0.

Therefore, there exists a scalar^∗such that| ¯x1| + ^∗− ( ¯x1− ^∗)^α1( ¯x2− ^∗)^α2 < 0. This together with (12) imply that

|x ₁| − σ_α( ¯x ) < 0, ∀(x₁ , ¯x ) ∈B_{(x1, ¯x)},

which is sufficient to show thatB_{(x1, ¯x)}is entirely included inK_αand hence(x1, ¯x) ∈ intK_α. Applying a similar way toK^∗_αandK^◦_α, their interiors can also be verified as the right-hand

side of (9) and (10). 

From the proof of Lemma 3, we further define the following sets S₁ :=

(x1, ¯x) ∈ R × R²| x1= 0, ¯x1> 0, ¯x2 = 0 , S2 :=

(x1, ¯x) ∈ R × R²| x1 = 0, ¯x1 = 0, ¯x2 > 0 , S₃ :=

(x1, ¯x) ∈ R × R²| |x1| = σα( ¯x), ¯x1> 0, ¯x2> 0 , S4 :=

(x1, ¯x) ∈ R × R²| |x1| = η_α( ¯x), ¯x1> 0, ¯x2> 0 , T₁ :=

(x1, ¯x) ∈ R × R²| x1 = 0, ¯x1 < 0, ¯x2 = 0

= −S1, T2 :=

(x1, ¯x) ∈ R × R²| x1= 0, ¯x1= 0, ¯x2< 0

= −S2, T₃ :=

(x1, ¯x) ∈ R × R²| |x1| = ηα(− ¯x), ¯x1< 0, ¯x2< 0

= −S4.

(13)

Then, the boundary ofK_α,K^∗_αandK_α^◦can be stated in a more compact form.

Lemma 4 The boundary of_Kαand_K^∗_α, denoted by∂Kαand∂K^∗_α, are respectively given by

∂Kα := S1∪ S2∪ S3∪ {0}, ∂K^∗_α:= S1∪ S2∪ S4∪ {0}.

Similarly, the boundary ofK^◦_α, denoted by∂K^◦_α, can be formulated as

∂K_α^◦ := T1∪ T2∪ T3∪ {0}.

Remark 1 It follows that the union setK_α∪K_α^◦can be divided into seven parts K_α∪K^◦_α = S1∪ S2∪ T1∪ T2∪ P1∪ P2∪ {0},

where

P₁:=

(x1, ¯x) ∈ R × R²| |x1| ≤ σα( ¯x), ¯x1> 0, ¯x2> 0 , P2:=

(x1, ¯x) ∈ R × R²| |x1| ≤ η_α(− ¯x), ¯x1 < 0, ¯x2 < 0 . In addition, the boundary ofK_αand its polarK^◦_αare depicted in Fig.3.

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

z:= (z1, ¯z) ∈ R × R², ¯z := (¯z1, ¯z2)^T ∈ R², ¯zmin:= min{¯z1, ¯z2}, ¯zmax:= max{¯z1, ¯z2}.

(14)

Fig. 3 The different parts of∂Kα(left) and∂K^◦_α(right)

Consequently, we divide the spaceR × R²into the following four blocks Block I: B1:=

(z1, ¯z) ∈ R × R²| ¯zmin· ¯zmax> 0 or (z1= 0 and ¯zmin= ¯zmax= 0) . Block II: B2:=

(z1, ¯z) ∈ R × R²| ¯zmin· ¯zmax= 0 and ¯zmin+ ¯zmax= 0 . Block III: B3:=

(z1, ¯z) ∈ R × R²| ¯zmin· ¯zmax< 0 . Block IV: B4:=

(z1, ¯z) ∈ R × R²| z1= 0 and ¯zmin= ¯zmax= 0 .

(15) The subcases of these blocks with respect toK_αcan be found in Table1.

2.1 The Type I decomposition with respect to the power coneK_˛

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:(z1, ¯z) ∈ B1.

(a) ¯zmin> 0. In this subcase, (z1, ¯z) ∈ B11, i.e.,¯z1 > 0, ¯z2 > 0, which implies σ_α(¯z) > 0 andη_α(¯z) > 0. Then, we take x = ˙x^(B1,a), y= ˙y^(B1,a) and sx = ˙s^(Bx ^1,a), sy = ˙s^(By ^1,a), where

˙x^(B1,a) :=

 1

σα¯z(¯z)



∈ S3, ˙y^(B1,a):=

 1

−_ηα^¯z_(¯z)



∈ T3. (16)

˙s^(B_x ^1,a) := z1+ ηα(¯z)

σα(¯z) + ηα(¯z)· σα(¯z), ˙s_y^(B1,a):= z1− σα(¯z)

σα(¯z) + ηα(¯z)· ηα(¯z). (17) It is easy to show that the above setting satisfies the decomposition formula (6).

(b) ¯zmax< 0. Similar to the argument in Case 1 (a), (z1, ¯z) ∈ B12, i.e.,¯z1< 0, ¯z2< 0, which impliesσα(−¯z) > 0 and ηα(−¯z) > 0. In this subcase, we set x = ˙x^(B1,b), y= ˙y^(B1,b) and sx = ˙sx^(B1,b), sy= ˙s^(By ^1,b), where

˙x^(B1,b):=

 1

σα−¯z(−¯z)



∈ S3, ˙y^(B1,b):=

 1

ηα(−¯z)¯z



∈ T3. (18)

˙s^(B_x ^1,b):= z1− η_α(−¯z)

σ_α(−¯z) + η_α(−¯z) · σα(−¯z),

Table1Thesubcasesofeachblockin(15)withrespecttoKα B1B2B3B4 (B11)z1free,¯z1>0,¯z2>0(B21)z1free,¯z1=0,¯z2>0(B31)z1free,¯z1<0,¯z2>0(B4)z1=0,¯z1=0,¯z2=0 (B12)z1free,¯z1<0,¯z2<0(B22)z1free,¯z1>0,¯z2=0(B32)z1free,¯z1>0,¯z2<0 (B13)z1=0,¯z1=0,¯z2=0(B23)z1free,¯z1=0,¯z2<0 (B24)z1free,¯z1<0,¯z2=0

˙s^(B_y ^1,b):= z1+ σ_α(−¯z)

σ_α(−¯z) + η_α(−¯z) · η_α(−¯z). (19)

(c) z₁ = 0 and ¯zmin = ¯zmax= 0. In this subcase, (z1, ¯z) ∈ B13, which impliesσα(¯z) = 0 andηα(¯z) = 0. Therefore, we set x = ˙x^(B1,c), y= ˙y^(B1,c)and s_x = ˙s^(Bx ^1,c), s_y= ˙s^(By ^1,c), where

˙x^(B1,c):=

 1

σα1(1)



∈ S3, ˙y^(B1,c):=

 1

−_ηα¹₍₁₎



∈ T3, (20)

˙s_x^(B1,c):= z1

σ_α(1) + η_α(1)· σ_α(1), ˙s^(B_y ^1,c):= z1

σ_α(1) + η_α(1)· η_α(1) (21)

with 1:= (1, 1)^T ∈ R². Case 2:(z1, ¯z) ∈ B2.

(a) ¯zmin= 0, ¯zmax> 0. In this subcase, (z1, ¯z) ∈ B21or(z1, ¯z) ∈ B22. Therefore, we set x = ˙x^(B2,a), y = ˙y^(B2,a) and sx = 1, sy = 1, where ˙x^(B2,a) = ( ˙x₁^(B2,a), ˙¯x^(B2,a)) and

˙y^(B2,a)= ( ˙y₁^(B2,a), ˙¯y^(B2,a)) with

˙x₁^(B2,a) := z1, ˙¯x^(B2,a) :=

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎩

⎡

⎢⎣

|z1|

¯z^α2₂

¹

α1

¯z2

⎤

⎥⎦ if (z¹, ¯z) ∈ B21,

⎡

⎢⎣

¯z1

|z1|

¯z^α1₁

¹

α2

⎤

⎥⎦ if (z1, ¯z) ∈ B22,

(22)

˙y₁^(B2,a):= 0, ˙¯y^(B2,a) :=

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎩

⎡

⎢⎣ −

|z1|

¯z₂^α2

_α1¹

0

⎤

⎥⎦ if (z1, ¯z) ∈ B21,

⎡

⎢⎣ 0

−

|z1|

¯z^α1₁

¹

α2

⎤

⎥⎦ if (z¹, ¯z) ∈ B22.

(23)

It is easy to see that

(a)(z¹, ¯z) ∈ B²¹, z¹= 0 ⇒ x ∈ S², y = 0; (b) (z¹, ¯z) ∈ B²¹, z¹= 0 ⇒ x ∈ S³, y ∈ T¹; (c)(z1, ¯z) ∈ B22, z1= 0 ⇒ x ∈ S1, y = 0; (d) (z1, ¯z) ∈ B22, z1= 0 ⇒ x ∈ S3, y ∈ T2.

(b) ¯zmin< 0, ¯zmax= 0. In this subcase, (z1, ¯z) ∈ B23or(z1, ¯z) ∈ B24. We set x = ˙x^(B2,b), y = ˙y^(B2,b)and sx = −1, sy = −1, where ˙x^(B2,b) = ( ˙x₁^(B2,b), ˙¯x^(B2,b)) and ˙y^(B2,b) = ( ˙y^(B₁ ^2,b), ˙¯y^(B2,b)) with

˙x₁^(B2,b):= −z1, ˙¯x^(B2,b) :=

⎧⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎩

⎡

⎣ |z1|

(−¯z2)^α2

¹

α1

−¯z2

⎤

⎦ if (z1, ¯z) ∈ B23,

⎡

⎣ −¯z1

|z1| (−¯z1)^α1

¹

α2

⎤

⎦ if (z1, ¯z) ∈ B24,

(24)

˙y^(B₁ ^2,b):= 0, ˙¯y^(B2,b) :=

⎧⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎩

⎡

⎣ −

|z1| (−¯z2)^α2

¹

α1

0

⎤

⎦ if (z1, ¯z) ∈ B23,

⎡

⎣ 0

−

|z1| (−¯z1)^α1

¹

α2

⎤

⎦ if (z1, ¯z) ∈ B24.

(25)

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

(a)(z1, ¯z) ∈ B23, z1= 0 ⇒ x ∈ S2, y = 0; (b) (z1, ¯z) ∈ B23, z1= 0 ⇒ x ∈ S3, y ∈ T1; (c)(z1, ¯z) ∈ B24, z1= 0 ⇒ x ∈ S1, y = 0; (d) (z1, ¯z) ∈ B24, z1= 0 ⇒ x ∈ S3, y ∈ T2. Case 3:(z1, ¯z) ∈ B3. In this subcase,(z1, ¯z) ∈ B31or(z1, ¯z) ∈ B32. We set x = ˙x^(B3)∈ ∂K_α, y = ˙y^(B3) ∈ ∂K^◦_α and s_x = 1, sy = 1, where ˙x^(B3) = ( ˙x₁^(B3), ˙¯x^(B3)) and ˙y^(B3) = ( ˙y₁^(B3),

˙¯y^(B3)) with

˙x₁^(B3):= z1, ˙¯x^(B3):=

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎩

⎡

⎢⎣

|z1|

¯z^α2₂

_α1¹

¯z2

⎤

⎥⎦ if z ∈ B³¹,

⎡

⎢⎣

¯z1

|z1|

¯z^α1₁

¹

α2

⎤

⎥⎦ if z ∈ B³²,

(26)

˙y^(B₁ ³⁾:= 0, ˙¯y^(B3):=

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎩

⎡

⎢⎣ ¯z¹−

|z1|

¯z^α2₂

¹

α1

0

⎤

⎥⎦ if z ∈ B³¹,

⎡

⎢⎣ 0

¯z2−

|z1|

¯z^α1₁

¹

α2

⎤

⎥⎦ if z ∈ B32.

(27)

More concretely, we obtain

(a)(z1, ¯z) ∈ B31, z1= 0 ⇒ x ∈ S2, y ∈ T1; (b) (z1, ¯z) ∈ B31, z1= 0 ⇒ x ∈ S3, y ∈ T1; (c)(z1, ¯z) ∈ B32, z1 = 0 ⇒ x ∈ S1, y ∈ T2; (d) (z1, ¯z) ∈ B32, z1= 0 ⇒ x ∈ S3, y ∈ T2. Case 4:(z1, ¯z) ∈ B4. In this subcase, we set x= ˙x^(B4), y= ˙y^(B4)and s_x= 1, sy= 1, where

˙x^(B4):=

⎡

⎣0 1 0

⎤

⎦ ∈ S1, ˙y^(B4):=

⎡

⎣ 0

−1 0

⎤

⎦ ∈ T1, (28)

or

˙x^(B4):=

⎡

⎣0 0 1

⎤

⎦ ∈ S2, ˙y^(B4):=

⎡

⎣ 0 0

−1

⎤

⎦ ∈ T2. (29)

Table 2 The locations of the x-part and y-part in the Type I decomposition with respect toKα

¯B₁ ¯B₂ ¯B₃ ¯B₄

¯B₂₁ ¯B₂₂ ¯B₂₃ ¯B₂₄ ¯B₃₁ ¯B₃₂

x_loc S₃ S₂∪ S3 S₁∪ S3 S₂∪ S3 S₁∪ S3 S₂∪ S3 S₁∪ S3 S₁∪ S2 yloc T3 {0} ∪ T1 {0} ∪ T2 {0} ∪ T1 {0} ∪ T2 T1 T2 T1∪ T2

To sum up these discussions, we present the Type I decomposition with respect to the power coneK_αin the following theorem.

Theorem 1 For any given z= (z1, ¯z) ∈ R × R², its Type I decomposition with respect toK_α is given by

(a) If z∈ B1, then

z=

⎧⎪

⎨

⎪⎩

˙s^(Bx ^1,a)· ˙x^(B1,a)+ ˙s^(By ^1,a)· ˙y^(B1,a), if z ∈ B11,

˙s^(Bx ^1,b)· ˙x^(B1,b)+ ˙s^(By ^1,b)· ˙y^(B1,b), if z ∈ B12,

˙s^(Bx ^1,c)· ˙x^(B1,c)+ ˙s^(By ^1,c)· ˙y^(B1,c), if z ∈ B13,

where˙x^(B1,a),˙y^(B1,a),˙sx^(B1,a),˙s^(By ^1,a)are defined as in (16)–(17), ˙x^(B1,b),˙y^(B1,b),˙sx^(B1,b),

˙sy^(B1,b)are defined as in (18)–(19) and ˙x^(B1,c), ˙y^(B1,c),˙sx^(B1,c),˙s^(By ^1,c)are defined as in (20)–(21).

(b) If z∈ B2, then z=

˙x^(B2,a)+ ˙y^(B2,a), if z∈ B21or z∈ B22, (−1) · ˙x^(B2,b)+ (−1) · ˙y^(B2,b), if z ∈ B23or z∈ B24,

where ˙x^(B2,a),˙y^(B2,a)are defined as in (22)–(23), ˙x^(B2,b),˙y^(B2,b)are defined as in (24)–

(25).

(c) If z∈ B3, then z= ˙x^(B3)+ ˙y^(B3), where˙x^(B3), ˙y^(B3)are defined as in (26)–(27).

(d) If z∈ B4, then z= ˙x^(B4)+ ˙y^(B4), where˙x^(B4)and ˙y^(B4)are defined as in (28) or (29).

In addition, the locations of the x-part and y-part in each case are shown in Table2, where S_i, Ti (i = 1, 2, 3, 4) are defined as in (13) and x_loc, yloc denote the locations of x and y, respectively.

2.2 The Type II decomposition with respect to the power coneK_˛

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

Case 1:(z1, ¯z) ∈ B1.

(a) ¯zmin > 0. In this subcase, (z1, ¯z) ∈ B11 andσα(¯z) > 0. Then, we take x = ¨x^(B1,a), y= ¨y^(B1,a)and sx = ¨sx^(B1,a), sy= ¨s^(By ^1,a), where

¨x^(B1,a):=

 1

σα¯z(¯z)



∈ S3, ¨y^(B1,a):=

 −1

σα¯z(¯z)



∈ S3. (30)

¨s^(Bx ^1,a):= z₁+ σα(¯z)

2 , ¨s^(By ^1,a):= σα(¯z) − z1

2 . (31)

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