DOI 10.1007/s10898-008-9373-z

**Some characterizations for SOC-monotone** **and SOC-convex functions**

**Jein-Shan Chen** **· Xin Chen · Shaohua Pan ·**
**Jiawei Zhang**

Received: 29 June 2007 / Accepted: 21 October 2008 / Published online: 7 November 2008

© Springer Science+Business Media, LLC. 2008

**Abstract** We provide some characterizations for SOC-monotone and SOC-convex
functions by using differential analysis. From these characterizations, we particularly obtain
that a continuously differentiable function defined in an open interval is SOC-monotone
*(SOC-convex) of order n* ≥ 3 if and only if it is 2-matrix monotone (matrix convex), and
*furthermore, such a function is also SOC-monotone (SOC-convex) of order n* ≤ 2 if it is
2-matrix monotone (matrix convex). In addition, we also prove that Conjecture 4.2 proposed
in Chen (Optimization 55:363–385, 2006) does not hold in general. Some examples are
included to illustrate that these characterizations open convenient ways to verify the SOC-
monotonicity and the SOC-convexity of a continuously differentiable function defined on an
open interval, which are often involved in the solution methods of the convex second-order
cone optimization.

**Keywords** Second-order cone· SOC-monotone function · SOC-convex function
**Mathematics Subject Classification (2000)** 26A48· 26A51 · 26B05 · 90C25

J.-S. Chen (

### B

^{)}

Department of Mathematics, National Taiwan Normal University, Taipei 11677, Taiwan e-mail: jschen@math.ntnu.edu.tw

X. Chen

Department of Industrial and Enterprise System Engineering, University of Illinois at Urbana–Champaign, Urbana 61801, IL, USA

e-mail: xinchen@uiuc.edu S. Pan

School of Mathematical Sciences, South China University of Technology, Guangzhou 510640, China e-mail: shhpan@scut.edu.cn

J. Zhang

Department of Information, Operations and Management Sciences, New York University, New York 10012-1126, NY, USA

e-mail: jzhang@stern.nyu.edu

**1 Introduction**

The second-order cone (SOC) in IR* ^{n}*, also called the Lorentz cone, is a set defined by

*K*

*:=*

^{n}*(x*1*, x*2*) ∈ IR × IR*^{n−1}*| x*2* ≤ x*1

*,* (1)

where · denotes the Euclidean norm, and*K*^{1}denotes the set of nonnegative reals IR_{+}. It
is known that*K** ^{n}*is a closed convex self-dual cone with nonempty interior int(

*K*

^{n}*). For any*

*x, y ∈ IR*

^{n}*, we write x*

_{Kn}*y if x− y ∈K*

^{n}*; and write x*

_{Kn}*y if x− y ∈ int(K*

^{n}*). In other*

*words, we have x*

_{Kn}*0 if and only if x*∈

*K*

^{n}*and x*

_{Kn}*0 if and only if x∈ int(K*

^{n}*). The*relation

*is a partial ordering, but not a linear ordering in*

_{Kn}*K*

^{n}*, i.e., there exist x, y ∈K*

^{n}*such that neither x*

_{Kn}*y nor y*

_{Kn}*x. To see this, let x*

*= (1, 1), y = (1, 0), and then we*

*have x− y = (0, 1) /∈K*

^{2}

*, y − x = (0, −1) /∈K*

^{2}.

*For any x= (x*1*, x*2*), y = (y*1*, y*2*) ∈ IR × IR*^{n−1}*, we define their Jordan product as*
*x◦ y = (x, y , y*1*x*_{2}*+ x*1*y*_{2}*).* (2)
*we write x*^{2}*to mean x◦x and write x+y to mean the usual componentwise addition of vectors.*

Then*◦, +, and e = (1, 0, . . . , 0)** ^{T}* ∈ IR

*have the following basic properties (see [7,8]): (1)*

^{n}*e◦ x = x for all x ∈ IR*

^{n}*. (2) x◦ y = y ◦ x for all x, y ∈ IR*

^{n}*. (3) x◦ (x*

^{2}

*◦ y) = x*

^{2}

*◦ (x ◦ y)*

*for all x, y ∈ IR*

*. (4)*

^{n}*(x + y) ◦ z = x ◦ z + y ◦ z for all x, y, z ∈ IR*

*. Note that the Jordan product is not associative. Besides,*

^{n}*K*

*is not closed under Jordan product.*

^{n}We recall from [7,8] that each x*= (x*1*, x*2*) ∈ IR × IR*^{n}^{−1}admits a spectral factorization,
associated with*K** ^{n}*, of the form

*x= λ*1*(x) · u*^{(1)}*x* *+ λ*2*(x) · u*^{(2)}*x* *,* (3)
where*λ*1*(x), λ*2*(x) and u*^{(1)}*x* *, u*^{(2)}*x* are the spectral values and the associated spectral vectors
*of x given by*

*λ**i**(x) = x*1*+ (−1)*^{i}*x*2*, u*^{(i)}* _{x}* = 1
2

1, (−1)^{i}*¯x*2

*for i= 1, 2,* (4)
with *¯x*2= _{x}^{x}^{2}_{2}_{}*if x*_{2}*= 0 and otherwise ¯x*2being any vector in IR* ^{n−1}*such that

*¯x*2 = 1. If

*x*2

*= 0, the factorization is unique. By the spectral factorization, for any f : IR → IR, we*can define a vector-valued function associated with

*K*

^{n}*(n*≥ 1) by

*f*^{soc}*(x) = f (λ*1*(x))u*^{(1)}_{x}*+ f (λ*2*(x))u*^{(2)}_{x}*, ∀x = (x*1*, x*2*) ∈ IR × IR*^{n−1}*,* (5)
*and call it the SOC-function induced by f . If f is defined only on a subset of IR, then f*^{soc}is
defined on the corresponding subset of IR^{n}*. The definition is unambiguous whether x*2= 0
*or x*2 *= 0. The cases of f*^{soc}*(x) = x*^{1/2}*, x*^{2}*, exp(x) were discussed in [7]. In fact, the above*
definition (5) is analogous to one associated with the semidefinite cone; see [19,20].

Recently, the concepts of SOC-monotone and SOC-convex functions are introduced in
[5]. Especially, a function f*: J → IR with J ⊆ IR is said to be SOC-monotone of order n if*

*x* _{Kn}*y* *⇒ f*^{soc}*(x) *_{Kn}*f*^{soc}*(y)* (6)

*for any x, y ∈ dom f*^{soc} ⊆ IR^{n}*, where dom f*^{soc}*denotes the domain of the function f*^{soc}; and
*f is said to be SOC-convex of order n if, for any x, y ∈ dom f*^{soc},

*f*^{soc}*(λx + (1 − λ)y) *_{Kn}*λf*^{soc}*(x) + (1 − λ) f*^{soc}*(y) λ ∈ [0, 1].* (7)

*The function f is said to be SOC-monotone (respectively, SOC-convex) if it is SOC-*
*monotone of all order n (respectively, SOC-convex of all order n), and f is SOC-convex on*
*J if and only if− f is SOC-concave on J. The concepts of SOC-monotone and SOC-convex*
functions are analogous to matrix monotone and matrix convex functions [2,10,11,14], and
are special cases of operator monotone and operator convex functions [1,3,12]. For example,
*the function f is said to be n-matrix convex on J if*

*f(λA + (1 − λ)B) λf (A) + (1 − λ) f (B) λ ∈ [0, 1]*

*for arbitrary Hermitian n× n matrices A and B with spectra in J. It is clear that the set*
of SOC-monotone functions and the set of SOC-convex functions are closed under positive
linear combinations and pointwise limits.

There has been systematic study on matrix monotone and matrix convex functions, and moreover, characterizations for such functions have been explored; see [4,10,11,13,14] and the references therein. To the contrast, the study on SOC-monotone and SOC-convex func- tions just starts its first step. One reason is that they were viewed as special cases of operator monotone and operator convex functions. However, we recently observed that SOC-monotone and SOC-convex functions play an important role in the design of solutions methods for con- vex second-order cone programs (SOCPs); for example, the proximal-like methods in [15]

and the augmented Lagrangian method introduced in Sect.5. On the other hand, we all know that the developments of matrix-valued functions have major contributions in the solution of optimization problems. Thus, we hope similar systematic study on SOC-functions can be exploited so that it can be readily adopted to optimization field. This is the main motivation of the paper.

Although some work was done in [5] for SOC-monotone and SOC-convex functions,
the focus there is to provide some specific examples by the definition and it seems difficult
to exploit the characterizations there to verify whether a given function is SOC-convex or
not. In this paper, we employ differential analysis to establish some useful characterizations
which will open convenient ways to verify the SOC-monotonicity and the SOC-convexity of a
function defined on an open interval. Particularly, from these characterizations, we obtain that
a continuously differentiable function defined on an open interval is SOC-monotone (SOC-
*convex) of order n* ≥ 3 if and only if it is 2-matrix monotone (matrix convex), and such a
*function is also SOC-monotone (SOC-convex) of order n* ≤ 2 if it is 2-matrix monotone
(matrix convex). Thus, if such functions are 2-matrix monotone (matrix convex), then it must
be SOC-monotone (SOC-convex). It should be pointed out that the analysis of this paper can
not be obtained from those for matrix-valued functions. One of the reasons is that the matrix
multiplication is associative whereas the Jordan product is not.

Throughout the paper,*·, · denotes the Euclidean inner product, IR** ^{n}* denotes the space

*of n-dimensional real column vectors, and IR*

^{n}^{1}× · · · × IR

^{n}*is identified with IR*

^{m}

^{n}^{1}

^{+···+n}*. Thus,*

^{m}*(x*1

*, . . . , x*

*m*

*) ∈ IR*

^{n}^{1}× · · · × IR

^{n}*is viewed as a column vector in IR*

^{m}

^{n}^{1}

^{+···+n}*. Also,*

^{m}*matrix or vector of suitable dimension. The notation*

**I represents an identity matrix of suitable dimension; J is a subset of IR; and 0 is a zero**

^{T}*means transpose and C*

^{(i)}*(J) denotes*

*the family of functions which are defined on J*

*⊆ IR to IR and have the i-th continuous*

*derivative. For a function f*

*: IR → IR, f*

^{(i)}*(x) represents the i-th order derivative of f at*

*x*

*∈ IR, and the first-order and the second-order derivative of f are also written as f*

^{}

*and f*

^{},

*respectively. For any f*: IR

^{n}*→ IR, ∇ f (x) denotes the gradient of f at x ∈ IR*

^{n}*and dom f*

*denotes the domain of f . For any differentiable mapping F= (F*1

*, . . . , F*

*m*

*)*

*: IR*

^{T}*→ IR*

^{n}*,*

^{m}*∇ F(x) = [∇ F*1*(x) · · · ∇ F**m**(x)] is an n × m matrix which denotes the transposed Jacobian*
*of F at x. For any symmetric matrices A, B ∈ IR*^{n×n}*, we write A B (respectively, A B)*
*to mean A− B is positive semidefinite (respectively, positive definite).*

**2 Preliminaries**

In this section, we develop the second-order Taylor’s expansion for the vector-valued SOC-
*function f*^{soc} defined as in (5) which is crucial in our subsequent analysis. To the end, we
*assume that f* *∈ C*^{(2)}*(J) with J being an open interval in IR and dom f*^{soc} ⊆ IR* ^{n}*.

*Given any x* *∈ dom f*^{soc} *and h* *= (h*1*, h*2*) ∈ IR × IR*^{n}^{−1}*, we have x+ th ∈ dom f*^{soc}
*for any sufficiently small t> 0. We wish to calculate the Taylor’s expansion of the function*
*f*^{soc}*(x + th) at x for any sufficiently small t > 0. In particular, we are interested in finding*
matrices*∇ f*^{soc}*(x) and A**i**(x) for i = 1, 2, . . . , n such that*

*f*^{soc}*(x + th) = f*^{soc}*(x) + t∇ f*^{soc}*(x)h +*1
2*t*^{2}

⎡

⎢⎢

⎢⎣

*h*^{T}*A*_{1}*(x)h*
*h*^{T}*A*2*(x)h*

*...*

*h*^{T}*A**n**(x)h*

⎤

⎥⎥

⎥⎦*+ o(t*^{2}*).* (8)

*For convenience, we omit the variable notion x inλ**i**(x) for i = 1, 2 in the discussions below.*

*It is known that f*^{soc}*is differentiable (respectively, smooth) if and only if f is differentiable*
(respectively, smooth); see [6,8]. Moreover, there holds that

*∇ f*^{soc}*(x) =*

⎡

⎢⎢

⎣

*b*^{(1)}*c*^{(1)}*x*_{2}^{T}

*x*2
*c*^{(1)}*x*2

*x*2 *a*^{(0)}*I+ (b*^{(1)}*− a*^{(0)}*)x*_{2}*x*_{2}^{T}

*x*2^{2}

⎤

⎥⎥

⎦ (9)

*if x*_{2}= 0, and otherwise

*∇ f*^{soc}*(x) = f*^{}*(x*1*)I* (10)

where

*a** ^{(0)}*=

*f(λ*2

*) − f (λ*1

*)*

*λ*2*− λ*1 *, b** ^{(1)}*=

*f*

^{}

*(λ*2

*) + f*

^{}

*(λ*1

*)*

2 *, c** ^{(1)}* =

*f*

^{}

*(λ*2

*) − f*

^{}

*(λ*1

*)*

2 *.* (11)

*Therefore, we only need to derive the formula of A*_{i}*(x) for i = 1, 2, . . . , n in (*8).

*We first consider the case where x*_{2} *= 0 and x*2*+ th*2= 0. By the definition (5),

*f*^{soc}*(x + th) =* 1

2*f(x*1*+ th*1*− x*2*+ th*2*)*

⎡

⎣ 1

− *x*_{2}*+ th*2

*x*2*+ th*2

⎤

⎦

+1

2*f(x*1*+ th*1*+ x*2*+ th*2*)*

⎡

⎣ 1

*x*_{2}*+ th*2

*x*2*+ th*2

⎤

⎦

=

⎡

⎢⎣

*f(x*1*+ th*1*− x*2*+ th*2*) + f (x*1*+ th*1*+ x*2*+ th*2*)*
*f(x*1*+ th*1*+ x*2*+ th*2*) − f (x*21*+ th*1*− x*2*+ th*2*)*

2

*x*_{2}*+ th*2

*x*2*+ th*2

⎤

⎥⎦

:=

1

2

*.* (12)

*To derive the Taylor’s expansion of f*^{soc}*(x + th) at x with x*2 = 0, we first write out and
expand*x*2*+ th*2. Notice that

*x*2*+ th*2 =

*x*2^{2}*+ 2tx*_{2}^{T}*h*_{2}*+ t*^{2}*h*2^{2}*= x*2

1*+ 2tx*_{2}^{T}*h*2

*x*2^{2} *+ t*^{2}*h*2^{2}

*x*2^{2}*.*
Therefore, using the fact that

√1*+ = 1 +*1
2* −* 1

8^{2}*+ o(*^{2}*),*
we may obtain

*x*2*+ th*2* = x*2

1*+ t* *α*

*x*2+1
2*t*^{2} *β*

*x*2^{2}

*+ o(t*^{2}*),* (13)

where

*α =* *x*_{2}^{T}*h*_{2}

*x*2*, β = h*2^{2}−*(x*_{2}^{T}*h*_{2}*)*^{2}

*x*2^{2} *= h*2^{2}*− α*^{2}*= h*^{T}_{2}*M*_{x}_{2}*h*_{2}*,*
with

*M*_{x}_{2} *= I −* *x*2*x*_{2}^{T}

*x*2^{2}*.*

Furthermore, from (13) and the fact that*(1 + )*^{−1}*= 1 − + *^{2}*+ o(*^{2}*), it follows that*

*x*2*+ th*2^{−1}*= x*2^{−1}

1*− t* *α*

*x*2+1
2*t*^{2}

2 *α*^{2}

*x*2^{2} − *β*

*x*2^{2}

*+ o(t*^{2}*)*

*.* (14)
Combining Eqs. (13) and (14) then yields that

*x*2*+ th*2

*x*2*+ th*2 = *x*2

*x*2*+ t*

*h*2

*x*2− *α*

*x*2
*x*2

*x*2

+1
2*t*^{2}

2 *α*^{2}

*x*2^{2} − *β*

*x*2^{2}

*x*_{2}

*x*2− 2 *h*_{2}

*x*2
*α*

*x*2

*+ o(t*^{2}*)*

= *x*_{2}

*x*2*+ t M**x*2

*h*_{2}

*x*2
+1

2*t*^{2}

3*h*^{T}_{2}*x*_{2}*x*_{2}^{T}*h*_{2}

*x*2^{4}
*x*2

*x*2−*h*2^{2}

*x*2^{2}
*x*2

*x*2− 2*h*_{2}*h*_{2}^{T}

*x*2^{2}
*x*2

*x*2

*+ o(t*^{2}*). (15)*

In addition, from (13), we have the following equalities
*f(x*1*+ th*1*− x*2*+ th*2*)*

*= f*

*x*1*+ th*1−

*x*2

1*+ t* *α*

*x*2+1
2*t*^{2} *β*

*x*2^{2}

*+ o(t*^{2}*)*

*= f*

*λ*1*+ t(h*1*− α) −*1
2*t*^{2} *β*

*x*2*+ o(t*^{2}*)*

*= f (λ*1*) + t f*^{}*(λ*1*)(h*1*− α) +* 1
2*t*^{2}

*− f*^{}*(λ*1*)* *β*

*x*2*+ f*^{}*(λ*1*)(h*1*− α)*^{2}

*+ o(t*^{2}*) (16)*

and

*f(x*1*+ th*1*+ x*2*+ th*2*)*

*= f*

*λ*2*+ t(h*1*+ α) +* 1
2*t*^{2} *β*

*x*2*+ o(t*^{2}*)*

*= f (λ*2*) + t f*^{}*(λ*2*)(h*1*+ α) +*1
2*t*^{2}

*f*^{}*(λ*2*)* *β*

*x*2*+ f*^{}*(λ*2*)(h*1*+ α)*^{2}

*+ o(t*^{2}*). (17)*
*For i= 0, 1, 2, we define*

*a** ^{(i)}*=

*f*

^{(i)}*(λ*2

*) − f*

^{(i)}*(λ*1

*)*

*λ*2*− λ*1 *, b** ^{(i)}*=

*f*

^{(i)}*(λ*2

*) + f*

^{(i)}*(λ*1

*)*

2 *, c** ^{(i)}*=

*f*

^{(i)}*(λ*2

*) − f*

^{(i)}*(λ*1

*)*

2 *,*

(18)
*where f*^{(i)}*means the i -th derivative of f and f*^{(0)}*is the same as the original f . Then, by*
the Eqs. (16)–(18), it can be verified that

1= 1 2

*f(x*1*+ th*1*+ x*2*+ th*2*) + f (x*1*+ th*1*− x*2*+ th*2*)*

*= b*^{(0)}*+ t*

*b*^{(1)}*h*_{1}*+ c*^{(1)}*α*
+1

2*t*^{2}

*a*^{(1)}*β + b*^{(2)}*(h*^{2}1*+ α*^{2}*) + 2c*^{(2)}*h*_{1}*α*
*+ o(t*^{2}*)*

*= b*^{(0)}*+ t*

*b*^{(1)}*h*1*+ c*^{(1)}*h*^{T}_{2} *x*2

*x*2

+1

2*t*^{2}*h*^{T}*A*1*(x)h + o(t*^{2}*),*
where

*A*1*(x) =*

⎡

⎢⎢

⎣

*b*^{(2)}*c*^{(2)}*x*_{2}^{T}

*x*2
*c*^{(2)}*x*_{2}

*x*2 *a*^{(1)}*I*+

*b*^{(2)}*− a*^{(1)}* x*2*x*_{2}^{T}

*x*2^{2}

⎤

⎥⎥

*⎦ .* (19)

Note that in the above expression for1*, b*^{(0)}*is exactly the first component of f*^{soc}*(x) and*

*b*^{(1)}*h*1*+ c*^{(1)}*h*^{T}_{2} _{x}^{x}^{2}

2

is the first component of*∇ f*^{soc}*(x)h. Using the same techniques again,*
1

2

*f(x*1*+ th*1*+ x*2*+ th*2*) − f (x*1*+ th*1*− x*2*+ th*2*)*

*= c*^{(0)}*+ t*

*c*^{(1)}*h*1*+ b*^{(1)}*α*
+1

2*t*^{2}

*b*^{(1)}*β*

*x*2*+ c*^{(2)}*(h*^{2}_{1}*+ α*^{2}*) + 2b*^{(2)}*h*1*α*

*+ o(t*^{2}*)*

*= c*^{(0)}*+ t*

*c*^{(1)}*h*1*+ b*^{(1)}*α*
+1

2*t*^{2}*h*^{T}*B(x)h + o(t*^{2}*),* (20)

where

*B(x) =*

⎡

⎢⎢

⎢⎣

*c*^{(2)}*b*^{(2)}*x*_{2}^{T}

*x*2
*b*^{(2)}*x*2

*x*2 *c*^{(2)}*I*+

*b*^{(1)}

*x*2*− c*^{(2)}

*M**x*2

⎤

⎥⎥

⎥⎦*.* (21)

Using Eqs. (15) and (20), we obtain that

2 = 1 2

*f(x*1*+ th*1*+ x*2*+ th*2*) − f (x*1*+ th*1*− x*2*+ th*2*)* *x*2*+ th*2

*x*2*+ th*2

*= c*^{(0)}*x*_{2}

*x*2*+ t*

*x*_{2}

*x*2*(c*^{(1)}*h*_{1}*+ b*^{(1)}*α) + c*^{(0)}*M*_{x}_{2} *h*_{2}

*x*2

+1

2*t*^{2}*W+ o(t*^{2}*),*

where

*W* = *x*_{2}

*x*2*h*^{T}*B(x)h + 2M**x*_{2}

*h*_{2}

*x*2

*c*^{(1)}*h*1*+ b*^{(1)}*α*
*+c*^{(0)}

3*h*_{2}^{T}*x*_{2}*x*_{2}^{T}*h*_{2}

*x*2^{4}
*x*2

*x*2−*h*2^{2}

*x*2^{2}
*x*2

*x*2− 2*h*_{2}*h*^{T}_{2}

*x*2^{2}
*x*2

*x*2

*.*

Now we denote

*d* := *b*^{(1)}*− a*^{(0)}

*x*2 = 2(b^{(1)}*− a*^{(0)}*)*
*λ*2*− λ*1

*, U := h*^{T}*C(x)h*

*V* := 2*c*^{(1)}*h*1*+ b*^{(1)}*α*

*x*2 *− c** ^{(0)}*2

*x*

_{2}

^{T}*h*

_{2}

*x*2^{3} *= 2a*^{(1)}*h*1*+ 2dx*_{2}^{T}*h*_{2}

*x*2*,*
where

*C(x) :=*

⎡

⎢⎢

⎣

*c*^{(2)}*(b*^{(2)}*− a*^{(1)}*)* *x*_{2}^{T}

*x*2
*(b*^{(2)}*− a*^{(1)}*)* *x*2

*x*2 *dI*+

*c*^{(2)}*− 3d x*2*x*_{2}^{T}

*x*2^{2}

⎤

⎥⎥

*⎦ .* (22)

*Then U can be further recast as*

*U* *= h*^{T}*B(x)h + c** ^{(0)}*3

*h*

_{2}

^{T}*x*2

*x*

_{2}

^{T}*h*2

*x*2^{4} *− c*^{(0)}*h*2^{2}

*x*2^{2} − 2*x*_{2}^{T}*h*2

*x*2^{2}*(c*^{(1)}*h*1*+ b*^{(1)}*α).*

Consequently,

*W* = *x*_{2}

*x*2*U+ h*2*V.*

*We next consider the case where x*_{2}*= 0 and x*2*+ th*2= 0. By definition (5),

*f*^{soc}*(x + th) =* *f(x*1*+ t(h*1*− h*2*))*
2

⎡

⎣ 1

− *h*_{2}

*h*2

⎤

⎦ + *f(x*1*+ t(h*1*+ h*2*))*
2

⎡

⎣ 1
*h*_{2}

*h*2

⎤

⎦

=

⎡

⎢⎣

*f(x*1*+ t(h*1*− h*2*)) + f (x*1*+ t(h*1*+ h*2*))*
*f(x*1*+ t(h*1*+ h*2*)) − f (x*2 1*+ t(h*1*− h*2*))*

2

*h*_{2}

*h*2

⎤

⎥*⎦ .* (23)

*Using the Taylor expansion of f at x*1, we can obtain that
1

2

*f(x*1*+ t(h*1*− h*2*)) + f (x*1*+ t(h*1*+ h*2*))*

*= f (x*1*) + t f*^{(1)}*(x*1*)h*1+1

2*t*^{2}*f*^{(2)}*(x*1*)h*^{T}*h+ o(t*^{2}*),*
1

2

*f(x*1*+ t(h*1*− h*2*)) − f (x*1*+ t(h*1*+ h*2*))*

*= t f*^{(1)}*(x*1*)h*2+1

2*t*^{2}*f*^{(2)}*(x*1*)2h*1*h*2*+ o(t*^{2}*).*

Therefore,

*f*^{soc}*(x + th) = f*^{soc}*(x) + t f*^{(1)}*(x*1*)h +* 1

2*t*^{2}*f*^{(2)}*(x*1*)*
*h*^{T}*h*

*2h*_{1}*h*_{2}

*.* (24)

Thus, under this case, we have that

*A*_{1}*(x) = f*^{(2)}*(x*1*)I, A**i**(x) = f*^{(2)}*(x*1*)*

0 *¯e*_{i−1}^{T}

*¯e**i−1* **0**

*i= 2, . . . , n,* (25)
where *¯e**j* ∈ IR^{n−1}*is the vector whose j th component is 1 and the others are 0.*

Summing up the above discussions, we may obtain the following conclusion.

**Proposition 2.1 Let f***∈ C*^{(2)}*(J) with J being an open interval in IR and dom f*^{soc} ⊆ IR^{n}*.*
*Then, for given x∈ dom f*^{soc}*, h*∈ IR^{n}*and any sufficiently small t> 0,*

*f*^{soc}*(x + th) = f*^{soc}*(x) + t∇ f*^{soc}*(x)h +*1
2*t*^{2}

⎡

⎢⎢

⎢⎣

*h*^{T}*A*1*(x)h*
*h*^{T}*A*_{2}*(x)h*

*...*

*h*^{T}*A*_{n}*(x)h*

⎤

⎥⎥

⎥⎦*+ o(t*^{2}*),*

*where∇ f*^{soc}*(x) and A**i**(x), i = 1, 2, . . . , n are given by (10) and (25) if x*_{2} *= 0; and*
*otherwise∇ f*^{soc}*(x) and A*1*(x) are given by (9) and (19), respectively, and for i* *≥ 2,*

*A**i**(x) = C(x)* *x**2i*

*x*2*+ B**i**(x)*
*where*

*B*_{i}*(x) = ve*^{T}_{i}*+ e**i**v*^{T}*, v =*

*a*^{(1)}*d* *x*_{2}^{T}

*x*2

_{T}*.*

From Proposition 4.3 of [5] and Proposition2.1, we readily have the following result.

**Proposition 2.2 Let f***∈ C*^{(2)}*(J) with J being an open interval in IR and dom f*^{soc} ⊆ IR^{n}*.*
*Then, f is SOC-convex if and only if for any x∈ dom f*^{soc}*and h*∈ IR^{n}*, the vector*

⎡

⎢⎢

⎢⎣

*h*^{T}*A*1*(x)h*
*h*^{T}*A*_{2}*(x)h*

*...*

*h*^{T}*A*_{n}*(x)h*

⎤

⎥⎥

⎥⎦∈*K*^{n}*.*

**3 Characterizations of SOC-monotone functions**

Now we are ready to show our main result concerning the characterization of SOC-monotone functions. We need the following technical lemmas for the proof. The first one is so-called S-Lemma whose proof can be found in [16,18].

**Lemma 3.1 Let A, B be symmetric matrices and y**^{T}*Ay> 0 for some y. Then, the implica-*
*tion*

*z*^{T}*Az≥ 0 ⇒ z*^{T}*Bz*≥ 0

*is valid if and only if B λA for some λ ≥ 0.*

**Lemma 3.2 Given**θ ∈ IR, a ∈ IR^{n}^{−1}*, and a symmetric matrix A* ∈ IR^{n}^{×n}*. LetB*^{n}^{−1} :=

*{z ∈ IR*^{n−1}*| z ≤ 1}. Then, the following results hold:*

*(a) For any h*∈*K*^{n}*, Ah*∈*K*^{n}*is equivalent to A*
1

*z*

∈*K*^{n}*for any z*∈*B*^{n−1}*.*
*(b) For any z*∈*B*^{n}^{−1}*,θ + a*^{T}*z≥ 0 is equivalent to θ ≥ a.*

*(c) If A* =
*θ a*^{T}

*a H*

*with H being an(n − 1) × (n − 1) symmetric matrix, then for any*
*h*∈*K*^{n}*, Ah*∈*K*^{n}*is equivalent toθ ≥ a and there exists λ ≥ 0 such that the matrix*

*θ*^{2}*− a*^{2}*− λ* *θa*^{T}*− a*^{T}*H*
*θa − H*^{T}*a* *aa*^{T}*− H*^{T}*H+ λI*

* O.*

*Proof (a)* *For any h*∈*K*^{n}*, suppose that Ah*∈*K*^{n}*. Let h*=
1

*z*

*where z*∈*B** ^{n−1}*. Then

*h*∈

*K*

^{n}*and the desired result follows. For the other direction, if h*= 0, the conclusion

*is obvious. Now let h*

*:= (h*1

*, h*2

*) be any nonzero vector inK*

^{n}*. Then, h*1

*> 0 and*

*h*2* ≤ h*1. Consequently, *h*2

*h*1 ∈*B*^{n−1}*and A*

⎡

⎣1
*h*_{2}
*h*_{1}

⎤

⎦ ∈*K** ^{n}*. Since

*K*

*is a cone, we have*

^{n}*h*1*A*

⎡

⎣1
*h*2

*h*1

⎤

*⎦ = Ah ∈K*^{n}*.*

(b) *For z*∈*B** ^{n−1}*, suppose

*θ + a*

^{T}*z≥ 0. If a = 0, then the result is clear since θ ≥ 0. If*

*a*

*= 0, let z := −a/a. Clearly, z ∈B*

*and hence*

^{n−1}*θ +*

*−a*

^{T}*a*

*a* ≥ 0 which gives
*θ − a ≥ 0. For the other direction, the result follows from the Cauchy Schwarz*
inequality:

*θ + a*^{T}*z≥ θ − a · z ≥ θ − a ≥ 0.*

(c) *From part (a), Ah*∈*K*^{n}*for any h*∈*K*^{n}*is equivalent to A*
1

*z*

∈*K*^{n}*for any z*∈*B** ^{n−1}*.
Notice that

*A*
1

*z*

=
*θ a*^{T}

*a H*
1

*z*

=

*θ + a*^{T}*z*
*a+ Hz*

*.*
*Then, Ah*∈*K*^{n}*for any h*∈*K** ^{n}* is equivalent to the following two things:

*θ + a*^{T}*z≥ 0, for any z ∈B** ^{n−1}* (26)

and

*(a + Hz)*^{T}*(a + Hz) ≤ (θ + a*^{T}*z)*^{2}*, for any z ∈B*^{n−1}*.* (27)
By part (b), (26) is equivalent to*θ ≥ a. Now, we write the expression of (*27) as below:

*z*^{T}*(aa*^{T}*− H*^{T}*H)z + 2(θa*^{T}*− a*^{T}*H)z + θ*^{2}*− a*^{T}*a≥ 0, for any z ∈B*^{n−1}*,*
which can be further simplified as

*1 z*^{T}* θ*^{2}*− a*^{2} *θa*^{T}*− a*^{T}*H*
*θa − H*^{T}*a aa*^{T}*− H*^{T}*H*

1
*z*

*≥ 0, for any z ∈B*^{n−1}*.*

*Observe that z*∈*B** ^{n−1}*is the same as

*1 z** ^{T}* 1 0
0

*−I*

1
*z*

*≥ 0.*

Thus, by applying the S-Lemma (Lemma3.1), there exists*λ ≥ 0 such that*
*θ*^{2}*− a*^{2} *θa*^{T}*− a*^{T}*H*

*θa − H*^{T}*a aa*^{T}*− H*^{T}*H*

*− λ*
1 0

0*−I*

* O*

This completes the proof of part (c).

**Theorem 3.1 Let f***∈ C*^{(1)}*(J) with J being an open interval and dom f*^{soc}⊆ IR^{n}*. Then,*
*(i) when n= 2, f is SOC-monotone if and only if f*^{}*(τ) ≥ 0 for any τ ∈ J;*

*(ii) when n≥ 3, f is SOC-monotone if and only if the 2 × 2 matrix*

⎡

⎢⎣

*f*^{(1)}*(t*1*)* *f(t*2*) − f (t*1*)*
*t*_{2}*− t*1

*f(t*2*) − f (t*1*)*
*t*_{2}*− t*1

*f*^{(1)}*(t*2*)*

⎤

⎥*⎦ O for all t*^{1}*, t*2*∈ J.*

*Proof By the definition of SOC-monotonicity, f is SOC-monotone if and only if*

*f*^{soc}*(x + h) − f*^{soc}*(x) ∈K** ^{n}* (28)

*for any x* *∈ dom f*^{soc} *and h* ∈ *K*^{n}*such that x+ h ∈ dom f*^{soc}. By the first-order Taylor
*expansion of f*^{soc}, i.e.,

*f*^{soc}*(x + h) = f*^{soc}*(x) + ∇ f*^{soc}*(x + th)h* *for some t* *∈ (0, 1),*

it is clear that (28) is equivalent to*∇ f*^{soc}*(x + th)h ∈K*^{n}*for any x* *∈ dom f*^{soc}*and h* ∈*K*^{n}*such that x+ h ∈ dom f*^{soc}*, and some t* *∈ (0, 1). Let y := x + th = µ*1*v*^{(1)}*+ µ*2*v** ^{(2)}*for

*such x, h and t. We next proceed the arguments by the two cases of y*2

*= 0 and y*2= 0.

*Case (1): y*_{2}= 0. Under this case, we notice that

*∇ f*^{soc}*(y) =*
*θ a*^{T}

*a H*

*,*

where

*θ = ˜b*^{(1)}*, a = ˜c*^{(1)}*y*2

*y*2*, and H = ˜a*^{(0)}*I+ ( ˜b*^{(1)}*− ˜a*^{(0)}*)y*2*y*_{2}^{T}

*y*2^{2}*,*
with

*˜a** ^{(0)}*=

*f(µ*2

*) − f (µ*1

*)*

*µ*2*− µ*1 *, ˜b** ^{(1)}*=

*f*

^{}

*(µ*2

*) + f*

^{}

*(µ*1

*)*

2 *, ˜c** ^{(1)}* =

*f*

^{}

*(µ*2

*) − f*

^{}

*(µ*1

*)*

2 *. (29)*

In addition, we also observe that

*θ*^{2}*− a*^{2}*= ( ˜b*^{(1)}*)*^{2}*− (˜c*^{(1)}*)*^{2}*,* *θa*^{T}*− a*^{T}*H*= 0
and

*aa*^{T}*− H*^{T}*H= −(˜a*^{(0)}*)*^{2}*I*+

*(˜c*^{(1)}*)*^{2}*− ( ˜b*^{(1)}*)*^{2}*+ (˜a*^{(0)}*)*^{2}* y*2*y*_{2}^{T}

*y*2^{2}*.*
Thus, by Lemma3.2, f is SOC-monotone if and only if

(a) *˜b*^{(1)}*≥ |˜c** ^{(1)}*|;

(b) and there exists*λ ≥ 0 such that the matrix*

⎡

⎣*( ˜b*^{(1)}*)*^{2}*− (˜c*^{(1)}*)*^{2}*− λ* 0
0 *(λ − (˜a*^{(0)}*)*^{2}*)I +*

*(˜c*^{(1)}*)*^{2}*− ( ˜b*^{(1)}*)*^{2}*+ (˜a*^{(0)}*)*^{2}* y*_{2}*y*_{2}^{T}

*y*2^{2}

⎤

*⎦ O.*

*When n= 2, (a) together with (b) is equivalent to saying that f*^{}*(µ*1*) ≥ 0 and f*^{}*(µ*2*) ≥ 0.*

*Then we conclude that f is SOC-monotone if and only if f*^{}*(τ) ≥ 0 for any τ ∈ J.*

*When n≥ 3, (b) is equivalent to saying that ( ˜b*^{(1)}*)*^{2}*−(˜c*^{(1)}*)*^{2}*= λ ≥ 0 and λ−(˜a*^{(0)}*)*^{2}≥ 0,
i.e.,*( ˜b*^{(1)}*)*^{2}*− (˜c*^{(1)}*)*^{2}*≥ (˜a*^{(0)}*)*^{2}. Therefore, (a) together with (b) is equivalent to

⎡

⎢⎣

*f*^{(1)}*(µ*1*)* *f(µ*2*) − f (µ*1*)*
*µ*2*− µ*1

*f(µ*2*) − f (µ*1*)*
*µ*2*− µ*1

*f*^{(1)}*(µ*2*)*

⎤

⎥*⎦ O*

*for any x*∈ IR^{n}*, h ∈K*^{n}*such that x+ h ∈ dom f*^{soc}*, and some t∈ (0, 1). Thus, we conclude*
*that f is SOC-monotone if and only if*

⎡

⎢⎣

*f*^{(1)}*(t*1*)* *f(t*2*) − f (t*1*)*
*t*_{2}*− t*1

*f(t*2*) − f (t*1*)*
*t*_{2}*− t*1

*f*^{(1)}*(t*2*)*

⎤

⎥*⎦ O for all t*^{1}*, t*2*∈ J.*

*Case (2): y*2 *= 0. Now we have µ*1 *= µ*2 and*∇ f*^{soc}*(y) = f*^{(1)}*(µ*1*)I = f*^{(1)}*(µ*2*)I .*
*Hence, f is SOC-monotone is equivalent to f*^{(1)}*(µ*1*) ≥ 0, which is also equivalent to*

⎡

⎢⎣

*f*^{(1)}*(µ*1*)* *f(µ*2*) − f (µ*1*)*
*µ*2*− µ*1

*f(µ*2*) − f (µ*1*)*
*µ*2*− µ*1

*f*^{(1)}*(µ*2*)*

⎤

⎥*⎦ O*

*since f*^{(1)}*(µ*1*) = f*^{(1)}*(µ*2*) and* ^{f}^{(µ}_{µ}^{2}_{2}^{)− f (µ}_{−µ}_{1} ^{1}^{)}*= f*^{(1)}*(µ*1*) = f*^{(1)}*(µ*2*) by the Taylor formula*
and*µ*1*= µ*2. Thus, similar to Case (1), the conclusion also holds under this case.

From Theorem3.1and [11, Theorem 6.6.36], we immediately have the following results.

**Corollary 3.1 Let f***∈ C*^{(1)}*(J) with J being an open interval in IR. Then,*

*(a) f is SOC-monotone of order n* *≥ 3 if and only if it is 2-matrix monotone, and f is*
*SOC-monotone of order n≤ 2 if it is 2-matrix monotone.*

*(b) Suppose that n≥ 3 and f is SOC-monotone of order n. Then, f*^{}*(t*0*) = 0 for some t*0*∈ J*
*if and only if f(·) is a constant function on J.*

Note that the SOC-monotonicity of order 2 does not imply the 2-matrix monotonicity.

*For example, f(t) = t*^{2}is SOC-monotone of order 2 on*(0, +∞) by Example 3.2 (a) in [5],*
but by [11, Theorem 6.6.36] we can verify that it is not 2-matrix monotone. Corollary3.1
(a) implies that a continuously differentiable function defined on an open interval must be
SOC-monotone if it is 2-matrix monotone. In addition, from the following proposition, we
also have that the compound of two simple SOC-monotone functions is SOC-monotone.

**Proposition 3.1 If f***: J*1*→ J and g : J → IR with J*1*, J ⊆ IR are SOC-monotone on J*1

*and J , respectively, then the function g◦ f : J*1*→ IR is SOC-monotone on J.*

*Proof It is easy to verify that for all x, y ∈ IR*^{n}*, x*_{K}^{n}*y if and only ifλ**i**(x) ≥ λ**i**(y) with*
*i= 1, 2. In addition, g is monotone on J since it is SOC-monotone. From the two facts, we*

immediately obtain the result.

**4 Characterizations of SOC-convex functions**

In this section, we exploit Peirce decomposition to derive some characterizations for SOC-
*convex functions. Let f* *∈ C*^{(2)}*(J) with J being an open interval in IR and dom f*^{soc}⊆ IR* ^{n}*.

*For any x∈ dom f*^{soc} *and h*∈ IR^{n}*, if x*2= 0, then from Proposition2.1we have that

⎡

⎢⎢

⎢⎣

*h*^{T}*A*1*(x)h*
*h*^{T}*A*_{2}*(x)h*

*...*

*h*^{T}*A*_{n}*(x)h*

⎤

⎥⎥

⎥⎦*= f*^{(2)}*(x*1*)*
*h*^{T}*h*

*2h*1*h*2

*.*

Since*(h*^{T}*h, 2h*1*h*_{2}*) ∈K** ^{n}*, from Proposition2.2

*it follows that f is SOC-convex if and only*

*if f*

^{(2)}*(x*1

*) ≥ 0. By the arbitrariness of x*1

*, f is SOC-convex if and only if f is convex on J .*

*In what follows, we assume that x*

_{2}

*= 0. Let x = λ*1

*(x)u*

^{(1)}*x*

*+ λ*2

*(x)u*

^{(2)}*x*

*, where u*

^{(1)}

_{x}*and u*

^{(2)}*are given by (4) with*

_{x}*¯x*2 =

_{x}

^{x}^{2}

_{2}

_{}

*. Let u*

^{(i)}

_{x}*= (0, υ*

_{2}

^{(i)}*) for i = 3, . . . , n, where*

*υ*

_{2}

^{(3)}*, . . . , υ*

_{2}

*is any orthonormal set of vectors that span the subspace of IR*

^{(n)}*orthogonal*

^{n−2}*to x*2

*. It is easy to verify that the vectors u*

^{(1)}

_{x}*, u*

^{(2)}*x*

*, u*

^{(3)}*x*

*, . . . , u*

^{(n)}*x*are linearly independent.

*Hence, for any given h= (h*1*, h*2*) ∈ IR × IR*^{n}^{−1}, there exists*µ**i**, i = 1, 2, . . . , n such that*
*h* *= µ*1

√*2u*^{(1)}_{x}*+ µ*2

√*2u*^{(2)}* _{x}* +

*n*
*i=3*

*µ**i**u*^{(i)}_{x}*.*

From (19), we can verify that b^{(2)}*+ c*^{(2)}*and b*^{(2)}*− c*^{(2)}*are the eigenvalues of A*_{1}*(x) with*
*u*^{(2)}_{x}*and u*^{(1)}_{x}*being the corresponding eigenvectors, and a** ^{(1)}*is the eigenvalue of multiplicity

*n−2 with u*

^{(i)}*x*

*= (0, υ*

_{2}

^{(i)}*) for i = 3, . . . , n being the corresponding eigenvectors. Therefore,*

*h*^{T}*A*_{1}*(x)h = µ*^{2}_{1}*(b*^{(2)}*− c*^{(2)}*) + µ*^{2}_{2}*(b*^{(2)}*+ c*^{(2)}*) + a*^{(1)}

*n*
*i*=3

*µ*^{2}_{i}

*= f*^{(2)}*(λ*1*)µ*^{2}_{1}*+ f*^{(2)}*(λ*2*)µ*^{2}_{2}*+ a*^{(1)}*µ*^{2}*,* (30)
where

*µ*^{2}=_{n}

*i*=3*µ*^{2}_{i}*.*

*Similarly, we can verify that c*^{(2)}*+ b*^{(2)}*− a*^{(1)}*and c*^{(2)}*− b*^{(2)}*+ a** ^{(1)}*are the eigenvalues of

⎡

⎢⎢

⎣

*c*^{(2)}*(b*^{(2)}*− a*^{(1)}*)* *x*_{2}^{T}

*x*2
*(b*^{(2)}*− a*^{(1)}*)* *x*2

*x*2 *dI*+

*c*^{(2)}*− d x*_{2}*x*_{2}^{T}

*x*2^{2}

⎤

⎥⎥

⎦

*with u*^{(2)}_{x}*and u*^{(1)}_{x}*being the corresponding eigenvectors, and d is the eigenvalue of multiplicity*
*n− 2 with u*^{(i)}*x* *= (0, υ*_{2}^{(i)}*) for i = 3, . . . , n being the corresponding eigenvectors. Notice*
*that C(x) in (*22) can be decomposed the sum of the above matrix and

⎡

⎣0 0

0*−2dx*_{2}*x*_{2}^{T}

*x*2^{2}

⎤

*⎦ .*