SOC-monotone and SOC-convex functions vs.

Contents lists available atSciVerse ScienceDirect

Linear Algebra and its Applications

journal homepage:w w w . e l s e v i e r . c o m / l o c a t e / l a a

SOC-monotone and SOC-convex functions vs.

matrix-monotone and matrix-convex functions<

Shaohua Pan

^a

, Yungyen Chiang

^b

, Jein-Shan Chen

^c,∗,1,2

aDepartment of Mathematics, South China University of Technology, Wushan Road 381, Tianhe District, Guangzhou 510641, China bDepartment of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 80424, Taiwan

cDepartment of Mathematics, National Taiwan Normal University, Taipei 11677, Taiwan

A R T I C L E I N F O A B S T R A C T

Article history:

Received 30 September 2010 Accepted 9 April 2012 Available online 12 May 2012 Submitted by R.A. Brualdi AMS classification:

26B05 26B35 90C33 65K05

Keywords:

Hilbert space Second-order cone SOC-monotonicity SOC-convexity

The SOC-monotone function (respectively, SOC-convex function) is a scalar valued function that induces a map to preserve the monotone order (respectively, the convex order), when imposed on the spectral factorization of vectors associated with second-order cones (SOCs) in general Hilbert spaces. In this paper, we provide the sufficient and necessary characterizations for the two classes of functions, and particularly establish that the set of continuous SOC-monotone (re- spectively, SOC-convex) functions coincides with that of continuous matrix monotone (respectively, matrix convex) functions of order 2.

© 2012 Elsevier Inc. All rights reserved.

1. Introduction

Let

H

be a real Hilbert space of dimension dim

(H) ≥

3 endowed with an inner product

·, ·

and its induced norm

 · 

. Fix a unit vector e

∈ H

and denote by



^e



^⊥the orthogonal complementary

< This work was supported by National Young Natural Science Foundation (No. 10901058) and the Fundamental Research Funds for the Central Universities.

∗ Corresponding author.

E-mail addresses:shhpan@scut.edu.cn(S. Pan),chiangyy@math.nsysu.edu.tw(Y. Chiang),jschen@math.ntnu.edu.tw(J.-S. Chen).

1 The author’s work is supported by National Science Council of Taiwan, Department of Mathematics, National Taiwan Normal University, Taipei 11677, Taiwan.

2 Also a Member of Mathematics Division, National Center for Theoretical Sciences, Taipei Office.

0024-3795/$ - see front matter © 2012 Elsevier Inc. All rights reserved.

http://dx.doi.org/10.1016/j.laa.2012.04.030

space of e, i.e.,



^e



^⊥

= {

^x

∈ H | 

^x

,

^e

 =

⁰

} .

Then each x can be written as x

=

xe

+

x0e for some xe

∈ 

e



^⊥and x0

∈ R.

The second-order cone (SOC) in

H

, also called the Lorentz cone, is a set defined by K

:=



x

∈ H | 

^x

,

^e

 ≥ √

¹ 2



^x





=

^xe

+

^x0e

∈ H |

^x0

≥ 

^xe



^

.

From [7, Section 2], we know that K is a pointed closed convex self-dual cone. Hence,

H

^{becomes a} partially ordered space via the relation

K. In the sequel, for any x

,

y

∈ H

, we always write x

K y (respectively, x

K y) when x

−

^y

∈

K (respectively, x

−

^y

∈

intK); and denote x_eby the vector_^x_x^e

e if x_e

=

0, and otherwise by any unit vector from



^e



^⊥.

Associated with the second-order cone K, each x

=

^xe

+

^x0e

∈H

can be decomposed as

x

= λ

1

(

^x

)

^u1

(

^x

) + λ

2

(

^x

)

^u2

(

^x

),

⁽¹⁾

where

λ

i

(

x

) ∈ R

and ui

(

x

) ∈ H

for i

=

1

,

2 are the spectral values and the associated spectral vectors of x, defined by

λ

i

(

^x

) =

^x0

+ (−

¹

)

ⁱ



^xe

,

^ui

(

^x

) =

¹ 2

e

+ (−

¹

)

^ixe



.

⁽²⁾

Clearly, when xe

=

0, the spectral factorization of x is unique by definition.

Let f

:

^J

⊆ R → R

be a scalar valued function, where J is an interval (finite or infinite, closed or open) in

R

. Let S be the set of all x

∈ H

whose spectral values

λ

1

(

^x

)

^and

λ

2

(

^x

)

belong to J. Unless otherwise stated, in this paper S is always taken in this way. By the spectral factorization of x in (1) and (2), it is natural to define f^soc

:

^S

⊆ H → H

^by

f^soc

(

^x

) :=

^f

(λ

1

(

^x

))

^u1

(

^x

) +

^f

(λ

2

(

^x

))

^u2

(

^x

), ∀

^x

∈

^S

.

⁽³⁾ It is easy to see that the function f^socis well defined whether x_e

=

0 or not. For example, by taking f

(

t

) =

t², we have that f^soc

(

x

) =

x²

=

x

◦

x, where “

◦

” means the Jordan product and the detailed definition is see in the next section. Note that

(λ

1

(

^x

) − λ

1

(

^y

))

²

+ (λ

2

(

^x

) − λ

2

(

^y

))

²

=

²

(

^x



²

+ 

^y



²

−

^2x0y0

−

²



^xe



^ye

)

≤

^2



^x



²

+ 

^y



²

−

²



^x

,

^y



=

²



^x

−

^y



²

.

We may verify that the domain S of f^socis open in

H

if and only if J is open in

R

. Also, S is always convex since, for any x

=

^xe

+

^x0e

,

^y

=

^ye

+

^y0e

∈

^{S and}

β ∈ [

⁰

,

¹

]

^,

λ

1[

β

^x

+ (

¹

− β)

^y]

=

^

β

^x0

+ (

¹

− β)

^y0



− β

^xe

+ (

¹

− β)

^ye

 ≥

^min

{λ

1

(

^x

), λ

1

(

^y

)}, λ

2[

β

^x

+ (

¹

− β)

^y]

=

^

β

^x0

+ (

¹

− β)

^y0



+ β

^xe

+ (

¹

− β)

^ye

 ≤

^max

{λ

2

(

^x

), λ

2

(

^y

)},

which implies that

β

^x

+ (

¹

− β)

^y

∈

^{S. Thus, fsoc}

(β

^x

+ (

¹

− β)

^y

)

is well defined.

In this paper we are interested in two classes of special scalar valued functions that induce the maps via (3) to preserve the monotone order and the convex order, respectively.

Definition 1.1. A function f

:

^J

→ R

is said to be SOC-monotone if for any x

,

^y

∈

^S,

x

K y

⇒

^fsoc

(

^x

)

Kf^soc

(

^y

);

⁽⁴⁾

and f is said to be SOC-convex if, for any x

,

^y

∈

^{S and any}

β ∈ [

⁰

,

¹

]

^,

f^soc

(β

^x

+ (

¹

− β)

^y

) 

K

β

^fsoc

(

^x

) + (

¹

− β)

^fsoc

(

^y

).

⁽⁵⁾ From Definition1.1and Eq. (3), it is easy to see that the set of SOC-monotone and SOC-convex functions are closed under positive linear combinations and pointwise limits.

The concept of SOC-monotone (respectively, SOC-convex) functions above is a direct extension of those given by [5,6] to general Hilbert spaces, and is analogous to that of matrix monotone (respec- tively, matrix convex) functions and more general operator monotone (respectively, operator convex) functions; see, e.g., [17,15,14,2,11,23]. Just as the importance of matrix monotone (respectively, matrix convex) functions to the solution of convex semidefinite programming [19,4], SOC-monotone (respec- tively, SOC-convex) functions also play a crucial role in the design and analysis of algorithms for convex second-order cone programming [3,22]. For matrix monotone and matrix convex functions, after the seminal work of Löwner [17] and Kraus [15], there have been systematic studies and perfect char- acterizations for them; see [8,16,4,13,12,21,20] and the references therein. However, the study on SOC-monotone and SOC-convex functions just begins with [5], and the characterizations for them are still imperfect. Particularly, it is not clear what is the relation between the SOC-monotone (respectively, SOC-convex) functions and the matrix monotone (respectively, matrix convex) functions.

In this work, we provide the sufficient and necessary characterizations for SOC-monotone and SOC- convex functions in the setting of Hilbert spaces, and show that the set of continuous SOC-monotone (SOC-convex) functions coincides with that of continuous matrix monotone (matrix convex) functions of order 2. Some of these results generalize those of [5,6] (see Propositions3.2and4.2), and some are new, which are difficult to achieve by using the techniques of [5,6] (see, for example, Proposition 4.4). In addition, we also discuss the relations between SOC-monotone functions and SOC-convex functions, verify Conjecture 4.2 in [5] under a little stronger condition (see Proposition6.2), and present a counterexample to show that Conjecture 4.1 in [5] generally does not hold. It is worthwhile to point out that the analysis in this paper depends only on the inner product of Hilbert spaces, whereas most of the results in [5,6] are obtained with the help of matrix operations.

Throughout this paper, all differentiability means Fréchet differentiability. If F

: H → H

^{is (twice)} differentiable at x

∈ H

, we denote by F^

(

^x

)

^(F

(

^x

)

) the first-order F-derivative (the second-order F-derivative) of F at x. In addition, we use Cⁿ

(

J

)

and C^∞

(

J

)

to denote the set of n times and infinite times continuously differentiable real functions on J, respectively. When f

∈

^C1

(

^J

)

, we denote by f^[1] the function on J

×

J defined by

f^[1]

(λ, μ) :=

⎧⎨

⎩

f(λ)−^f(μ)

λ−μ if

λ = μ,

f^

(λ)

^if

λ = μ;

and when f

∈

^C2

(

^J

)

, denote by f^[2]the function on J

×

^J

×

J defined by f^[2]

(τ

1

, τ

2

, τ

3

) :=

^f[1]

(τ

1

, τ

2

) −

^f[1]

(τ

1

, τ

3

)

τ

2

− τ

3

if

τ

1

, τ

2

, τ

3are distinct, and for other values of

τ

1

, τ

2

, τ

3, f^[2]is defined by continuity; e.g., f^[2]

(τ

1

, τ

1

, τ

3

) =

^f

(τ

3

) −

f

(τ

1

) −

f^

(τ

1

)(τ

3

− τ

1

)

(τ

3

− τ

1

)

²

,

^f[2]

(τ

1

, τ

1

, τ

1

) =

¹ 2f^

(τ

1

).

For a linear operatorL from

H

^into

H

^{, we write}L

≥

0 (respectively,L

>

0) to mean thatL is positive semidefinite (respectively, positive definite), i.e.,



h

,

Lh

 ≥

0 for any h

∈ H

(respectively,



h

,

Lh

 >

0 for any 0

=

^h

∈ H

^).

2. Preliminaries

This section recalls some background material and gives several lemmas that will be used in the subsequent sections. We start with the definition of Jordan product [9]. For any x

=

xe

+

x0e

,

y

=

y_e

+

^y0e

∈ H

, the Jordan product of x and y is defined as

x

◦

y

:= (

x0ye

+

y0xe

) + 

x

,

y



e

.

A simple computation can verify that for any x

,

^y

,

^z

∈ H

and the unit vector e, (i) e

◦

^e

=

^{e and e}

◦

^x

=

^x;

(ii) x

◦

y

=

y

◦

x; (iii) x

◦ (

x²

◦

y

) =

x²

◦ (

x

◦

y

)

, where x²

=

x

◦

x; (iv)

(

x

+

y

) ◦

z

=

x

◦

z

+

y

◦

z. For any x

∈ H

, define its determinant by

det

(

^x

) := λ

1

(

^x

)λ

2

(

^x

) =

^x₀²

− 

^xe



²

.

Then each x

=

xe

+

x0e with det

(

x

) =

0 is invertible with respect to the Jordan product, i.e., there is a unique x⁻¹

= (−

^xe

+

^x0e

)/

^det

(

^x

)

such that x

◦

^x−1

=

^e.

We next give several lemmas where Lemma 2.1 is used in Section 3 to characterize SOC- monotonicity, and Lemmas2.2and2.3are used in Section4to characterize SOC-convexity.

Lemma 2.1. Let

B := {

^z

∈ 

^e



^⊥

| 

^z

 ≤

¹

}

. Then, for any given u

∈ 

^e



^⊥with



^u

 =

^{1 and}

θ, λ ∈ R

^, the following results hold.

(a)

θ + λ

^u

,

^z

 ≥

0 for any z

∈ B

if and only if

θ ≥ |λ|

^.

(b)

θ − λ

^z



²

≥ (θ − λ

²

)

^u

,

^z



^{2for any z}

∈ B

if and only if

θ − λ

²

≥

^0.

Proof. (a) Suppose that

θ + λ

^u

,

^z

 ≥

0 for any z

∈ B

^{. If}

λ =

^{0, then}

θ ≥ |λ|

clearly holds. If

λ =

^0, take z

= −

^sign

(λ)

^{u. Since}



^u

 =

1, we have z

∈ B

, and consequently,

θ + λ

^u

,

^z

 ≥

0 reduces to

θ − |λ| ≥

0. Conversely, if

θ ≥ |λ|

, then using the Cauchy–Schwartz inequality yields

θ + λ

u

,

z

 ≥

0 for any z

∈ B

^.

(b) Suppose that

θ − λ

^z



²

≥ (θ − λ

²

)

^u

,

^z



^{2for any z}

∈ B

. Then we must have

θ − λ

²

≥

^{0. If not,} for those z

∈ B

with



z

 =

1 but



u

,

z

 = 

u



z



, it holds that

(θ − λ

²

)

^u

,

^z



²

> (θ − λ

²

)

^u



²



^z



²

= θ − λ

^z



²

,

which contradicts the given assumption. Conversely, if

θ − λ

²

≥

0, the Cauchy–Schwartz inequality implies that

(θ − λ

²

)

^u

,

^z



²

≤ θ − λ

^z



^{2for any z}

∈ B

^. 

Lemma 2.2. For any given a

,

^b

,

^c

∈ R

^{and x}

=

^xe

+

^x0e with x_e

=

0, the inequality a



^he



²

− 

^he

,

^xe



^2

+

^bh0

+ 

^xe

,

^he



^2

+

^ch0

− 

^xe

,

^he



^2

≥

^{0 (6)} holds for all h

=

he

+

h0e

∈ H

if and only if a

≥

0

,

b

≥

0 and c

≥

0.

Proof. Suppose that (6) holds for all h

=

^he

+

^h0e

∈ H

. By letting h_e

=

^xe

,

^h0

=

^{1 and h}e

=

−

xe

,

h0

=

1, respectively, we get b

≥

0 and c

≥

0 from (6). If a

≥

0 does not hold, then by taking h_e

=

^b+_|^c_a⁺_| ¹_^z_z^e_e ^with



^ze

,

^xe

 =

^{0 and h}0

=

1, (6) gives a contradiction

−

¹

≥

0. Conversely, if a

≥

⁰

,

^b

≥

^{0 and c}

≥

0, then (6) clearly holds for all h

∈ H

^. 

Lemma 2.3. Let f

∈

^C2

(

^J

)

^{and u}e

∈

^e



^⊥with



^ue

=

1. For any h

=

^he

+

^h0e

∈ H

^{, define}

μ

1

(

^h

) :=

^h0

− 

ue

,

he



√

2

, μ

2

(

^h

) :=

^h0

+ 

ue

,

he



√

2

, μ(

^h

) :=

^



^he



²

− 

^ue

,

^he



²

.

Then, for any given a

,

^d

∈ R

^and

λ

1

, λ

2

∈

J, the following inequality 4f^

(λ

1

)

^f

(λ

2

)μ

1

(

^h

)

²

μ

2

(

^h

)

²

+

²

(

^a

−

^d

)

^f

(λ

2

)μ

2

(

^h

)

²

μ(

^h

)

²

+

²

(

^a

+

^d

)

^f

(λ

1

)μ

1

(

^h

)

²

μ(

^h

)

²

+

^a2

−

^d2

μ(

^h

)

⁴

−

^{2 [}

(

^a

−

^d

) μ

1

(

^h

) + (

^a

+

^d

) μ

2

(

^h

)

^]2

μ(

^h

)

²

≥

^{0 (7)} holds for all h

=

he

+

h0e

∈ H

if and only if

a²

−

^d2

≥

⁰

,

^f

(λ

2

)(

^a

−

^d

) ≥ (

^a

+

^d

)

^{2and f}

(λ

1

)(

^a

+

^d

) ≥ (

^a

−

^d

)

²

.

⁽⁸⁾

Proof. Suppose that (7) holds for all h

=

^he

+

^h0e

∈ H

^{. Taking h}0

=

^{0 and h}e

=

^{0 with}



^he

,

^ue

 =

^0, we have

μ

1

(

h

) =

0

, μ

2

(

h

) =

0 and

μ(

h

) = 

he

 >

0, and then (7) gives a²

−

d²

≥

0. Taking h_e

=

0 such that

|

^ue

,

^he

| < 

^he



^{and h}0

= 

^ue

,

^he

 =

^{0, we have}

μ

1

(

^h

) =

⁰

, μ

2

(

^h

) = √

2h₀and

μ(

^h

) >

0, and then (7) reduces to the following inequality

4

(

^a

−

^d

)

^f

(λ

2

) − (

^a

+

^d

)

^2h20

+ (

^a2

−

^d2

)(

^he



²

−

^h20

) ≥

⁰

.

This implies that

(

^a

−

^d

)

^f

(λ

2

) − (

^a

+

^d

)

²

≥

0. If not, by letting h₀be sufficiently close to



^he



^{, the} last inequality yields a contradiction. Similarly, taking h with he

=

0 satisfying

|

^ue

,

^he

| < 

^he



^and h₀

= −

^ue

,

^he



^{, we get f}

(λ

1

)(

^a

+

^d

) ≥ (

^a

−

^d

)

^{2from (7).}

Next, suppose that (8) holds. Then, the inequalities f^

(λ

2

)(

^a

−

^d

) ≥ (

^a

+

^d

)

^{2and f}

(λ

1

)(

^a

+

^d

) ≥ (

^a

−

^d

)

²imply that the left-hand side of (7) is greater than

4f^

(λ

1

)

^f

(λ

2

)μ

1

(

^h

)

²

μ

2

(

^h

)

²

−

⁴

(

^a2

−

^d2

)μ

1

(

^h

)μ

2

(

^h

)μ(

^h

)

²

+

^a2

−

^d2

μ(

^h

)

⁴

,

which is obviously nonnegative if

μ

1

(

^h

)μ

2

(

^h

) ≤

0. Now assume that

μ

1

(

^h

)μ

2

(

^h

) >

^{0. If a2}

−

^d2

=

^0, then the last expression is clearly nonnegative, and if a²

−

^d2

>

0, then the last two inequalities in (8) imply that f^

(λ

1

)

f^

(λ

2

) ≥ (

a²

−

d²

) >

0

,

and therefore,

4f^

(λ

1

)

^f

(λ

2

)μ

1

(

^h

)

²

μ

2

(

^h

)

²

−

⁴

(

^a2

−

^d2

)μ

1

(

^h

)μ

2

(

^h

)μ(

^h

)

²

+

^a2

−

^d2

μ(

^h

)

⁴

≥

⁴

(

^a2

−

^d2

)μ

1

(

^h

)

²

μ

2

(

^h

)

²

−

⁴

(

^a2

−

^d2

)μ

1

(

^h

)μ

2

(

^h

)μ(

^h

)

²

+

^a2

−

^d2

μ(

^h

)

⁴

= (

^a2

−

^d2

)

²

μ

1

(

^h

)μ

2

(

^h

) − μ(

^h

)

^22

≥

⁰

.

Thus, we prove that inequality (7) holds. The proof is complete. 

To close this section, we introduce the regularization of a locally integrable real function. Let

ϕ

^{be a} real function of class C^∞with the following properties:

ϕ ≥

^0,

ϕ

is even, the support supp

ϕ = [−

¹

,

¹

]

^, and

R

ϕ =

1. For each

ε >

^{0, let}

ϕ

_ε

(

^t

) =

¹_ε

ϕ(

_ε^t

)

. Then supp

ϕ

_ε

= [−ε, ε]

^and

ϕ

_εhas all the properties of

ϕ

listed above. If f is a locally integrable real function, we define its regularization of order

ε

^{as the} function

f_ε

(

^s

) :=

^{ f}

(

^s

−

^t

)ϕ

_ε

(

^t

)

^dt

=

^{ f}

(

^s

− ε

^t

)ϕ(

^t

)

^dt

.

⁽⁹⁾ Note that f_εis a C^∞function for each

ε >

^{0, and lim}ε→0f_ε

(

^x

) =

^f

(

^x

)

if f is continuous.

3. Characterizations of SOC-monotone functions

In this section we present some characterizations for SOC-monotone functions, by which the set of continuous SOC-monotone functions is shown to coincide with that of continuous matrix monotone functions of order 2. To this end, we need the following technical lemma.

Lemma 3.1. For any given f

:

J

→ R

with J open, let f^soc

:

S

→ H

be defined by (3).

(a) f^socis continuous on S if and only if f is continuous on J.

(b) f^socis (continuously) differentiable on S iff f is (continuously) differentiable on J. Also, when f is differentiable on J, for any x

=

^xe

+

^x0e

∈

^{S and v}

=

^ve

+

^v0e

∈ H

^,

(

^fsoc

)

^

(

^x

)

^v

=

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

f^

(

x0

)

v if xe

=

0

;

(

^b1

(

^x

) −

^a0

(

^x

))

^xe

,

^ve



^xe

+

^c1

(

^x

)

^v0x_e

+

a0

(

x

)

ve

+

b1

(

x

)

v0e

+

c1

(

x

)

xe

,

ve



e if xe

=

0

,

(10)

where a₀

(

^x

) =

^f(λ_λ²⁽₂^x₍⁾⁾⁻_x)−λ^f(λ₁₍¹_x⁽₎^{x)), b}1

(

^x

) =

^f(λ2(x))+₂^{f(λ1(x)), c}1

(

^x

) =

^{f(λ2(x))−}₂^f(λ1(x)).

(c) If f is differentiable on J, then for any given x

∈

S and all v

∈ H

^,

(

f^soc

)

^

(

x

)

e

= (

f^

)

^soc

(

x

)

and



e

, (

f^soc

)

^

(

x

)

v

 =

^v

, (

f^

)

^soc

(

x

)

^

.

(d) If f^is nonnegative (respectively, positive) on J, then for each x

∈

S,

(

^fsoc

)

^

(

^x

) ≥

⁰

(

respectively

, (

^fsoc

)

^

(

^x

) >

⁰

).

Proof. (a) Suppose that f^socis continuous. Letbe the set composed of those x

=

^{te with t}

∈

^J.

Clearly,

⊆

^{S, and fsoc}is continuous on. Noting that f^soc

(

^x

) =

^f

(

^t

)

e for any x

∈

, it follows that f is continuous on J. Conversely, if f is continuous on J, then f^socis continuous at any x

=

^xe

+

^x0e

∈

^S with x_e

=

^{0 since}

λ

i

(

^x

)

^{and u}i

(

^x

)

^{for i}

=

¹

,

2 are continuous at such points. Next, let x

=

^xe

+

^x0e be an arbitrary element from S with x_e

=

0, and we prove that f^socis continuous at x. Indeed, for any z

=

^ze

+

^z0e

∈

S sufficiently close to x, it is not hard to verify that



f^soc

(

z

) −

f^soc

(

x

) ≤ |

^f

(λ

2

(

^z

)) −

^f

(

^x0

)|

2

+ |

^f

(λ

1

(

^z

)) −

^f

(

^x0

)|

2

+ |

^f

(λ

2

(

^z

)) −

^f

(λ

1

(

^z

))|

2

.

Since f is continuous on J, and

λ

1

(

z

), λ

2

(

z

) →

x0as z

→

x, it follows that f

(λ

1

(

^z

)) →

^f

(

^x0

)

^{and f}

(λ

2

(

^z

)) →

^f

(

^x0

)

^{as z}

→

^x

.

The last two equations imply that f^socis continuous at x.

(b) When f^socis (continuously) differentiable, using the similar arguments as in part (a) can show that f is (continuously) differentiable. Next assume that f is differentiable. Fix any x

=

xe

+

x0e

∈

S. We first consider the case where x_e

=

^{0. Since}

λ

i

(

^x

)

^{for i}

=

¹

,

^{2 and}_^x_x^e_eare continuously differentiable at such x, it follows that f

(λ

i

(

^x

))

^{and u}i

(

^x

)

are differentiable and continuously differentiable, respectively, at x. Then f^socis differentiable at such x by the definition of f^soc. Also, an elementary computation shows that

[λ

i

(

^x

)]

^v

= 

^v

,

^e

 + (−

¹

)

ⁱ



^xe

,

^v

− 

^v

,

^e



^e





^xe

 =

^v0

+ (−

¹

)

ⁱ



^xe

,

^ve





^xe

 ,

⁽¹¹⁾

 xe



^xe



_

v

=

^v

− 

^v

,

^e



^e



^xe

 − 

^xe

,

^v

− 

^v

,

^e



^e



^xe



^xe



³

=

^ve



^xe

 − 

^xe

,

^ve



^xe



^xe



^{3 (12)}

for any v

=

^ve

+

^v0e

∈ H

, and consequently, [f

(λ

i

(

^x

))

^]v

=

^f

(λ

i

(

^x

))



v₀

+ (−

¹

)

ⁱ



^xe

,

^ve





^xe





,

[u_i

(

^x

)

^]v

=

¹

2

(−

¹

)

ⁱ

 ve



^xe

 − 

xe

,

ve



xe



^xe



³



.

Together with the definition of f^soc, we calculate that

(

f^soc

)

^

(

x

)

v is equal to f^

(λ

1

(

x

))

2



v₀

− 

xe

,

ve





xe



 

e

− 

^xx^ee





−

^f

(λ

1

(

x

))

2

 ve



xe

 − 

xe

,

ve



xe



xe



³



+

^f

(λ

2

(

^x

))

2



v0

+ 

^xe

,

^ve





^xe



 

e

+ 

^xxee





+

^f

(λ

2

(

^x

))

2

 v_e



^xe

 − 

^xe

,

^ve



^xe



^xe



³



=

^b1

(

^x

)

^v0e

+

^c1

(

^x

) 

^xe

,

^ve



^e

+

^c1

(

^x

)

^v0x_e

+

^b1

(

^x

)

^xe

,

^ve



^xe

+

^a0

(

^x

)

^ve

−

^a0

(

^x

)

^xe

,

^ve



^xe

,

where

λ

2

(

^x

) − λ

1

(

^x

) =

²



^xe



is used for the last equality. Thus, we get (10) for x_e

=

^{0. We next} consider the case where xe

=

0. Under this case, for any v

=

^ve

+

^v0e

∈ H

^,

f^soc

(

^x

+

^v

) −

^fsoc

(

^x

) =

^f

(

^x0

+

^v0

− 

^ve

)

2

(

^e

−

^ve

) +

^f

(

^x0

+

^v0

+ 

^ve

)

2

(

^e

+

^ve

) −

^f

(

^x0

)

^e

=

^f

(

^x0

)(

^v0

− 

^ve

)

2 e

+

^f

(

^x0

)(

^v0

+ 

^ve

)

2 e

+

^f

(

^x0

)(

^v0

+ 

^ve

)

2 ve

−

^f

(

^x0

)(

^v0

− 

^ve

)

2 ve

+

^o

(

^v

)

=

f^

(

x0

)(

v0e

+ 

ve



ve

) +

o

(

v

),

where v_e

=

_^v_v^e_e^{if v}e

=

0, and otherwise v_eis an arbitrary unit vector from



^e



^{⊥. Hence,}



^fsoc

(

^x

+

^v

) −

^fsoc

(

^x

) −

^f

(

^x0

)

^v

 =

^o

(

^v

).

This shows that f^socis differentiable at such x with

(

^fsoc

)

^

(

^x

)

^v

=

^f

(

^x0

)

^v.

Assume that f is continuously differentiable. From (10), it is easy to see that

(

^fsoc

)

^

(

^x

)

is continuous at every x with xe

=

0. We next argue that

(

^fsoc

)

^

(

^x

)

is continuous at every x with xe

=

^{0. Fix any} x

=

x0e with x0

∈

J. For any z

=

ze

+

z0e with ze

=

0, we have

(

^fsoc

)

^

(

^z

)

^v

− (

^fsoc

)

^

(

^x

)

^v

 ≤ |

^b1

(

^z

) −

^a0

(

^z

)|

^ve

 + |

^b1

(

^z

) −

^f

(

^x0

)||

^v0

|

+|

^a0

(

^z

) −

^f

(

^x0

)|

^ve

 + |

^c1

(

^z

)|(|

^v0

| + 

^ve

).

⁽¹³⁾ Since f is continuously differentiable on J and

λ

2

(

^z

) →

^x0

, λ

1

(

^z

) →

^x0as z

→

^{x, we have}

a₀

(

^z

) →

^f

(

^x0

),

^b1

(

^z

) →

^f

(

^x0

)

^{and c}1

(

^z

) →

⁰

.

Together with Eq. (13), we obtain that

(

f^soc

)

^

(

z

) → (

f^soc

)

^

(

x

)

as z

→

x.

(c) The result is direct by the definition of

(

^f

)

^socand a simple computation from (10).

(d) Suppose that f^

(

^t

) ≥

0 for all t

∈

J. Fix any x

=

^xe

+

^x0e

∈

^{S. If x}e

=

0, the result is direct. It remains to consider the case xe

=

^{0. Since f}

(

^t

) ≥

0 for all t

∈

J, we have b1

(

^x

) ≥

^{0, b}1

(

^x

) −

^c1

(

^x

) =

f^

(λ

1

(

^x

)) ≥

^{0, b}1

(

^x

) +

^c1

(

^x

) =

^f

(λ

2

(

^x

)) ≥

^{0 and a}0

(

^x

) ≥

0. From part (b) and the definitions of b₁

(

^x

)

^{and c}1

(

^x

)

, it follows that for any h

=

^he

+

^h0e

∈ H

^,



h

, (

f^soc

)

^

(

x

)

h

 = (

b1

(

x

) −

a0

(

x

))

xe

,

he



²

+

2c1

(

x

)

h0



xe

,

he

 +

b1

(

x

)

h²₀

+

a0

(

x

)

he



²

=

^a0

(

^x

)



^he



²

− 

^xe

,

^he



^2

+

¹

2

(

^b1

(

^x

) −

^c1

(

^x

))

^[h0

− 

^xe

,

^he



^]2

+

¹

2

(

b1

(

x

) +

c1

(

x

))

^[h0

+ 

xe

,

he



^]2

≥

0

.

This implies that the operator

(

f^soc

)

^

(

x

)

is positive semidefinite. Particularly, if f^

(

t

) >

0 for all t

∈

J, we have that



^h

, (

^fsoc

)

^

(

^x

)

^h

 >

0 for all h

=

0. The proof is complete. 

Lemma3.1(d) shows that the differential operator

(

^fsoc

)

^

(

^x

)

corresponding to a differentiable non- decreasing f is positive semidefinite. So, the differential operator

(

f^soc

)

^

(

x

)

associated with a differen- tiable SOC-monotone function is also positive semidefinite.

Proposition 3.1. Assume that f

∈

C¹

(

J

)

with J open. Then f is SOC-monotone if and only if

(

f^soc

)

^

(

x

)

h

∈

K for any x

∈

^{S and h}

∈

^K.

Proof. If f is SOC-monotone, then for any x

∈

S, h

∈

K and t

>

0, we have f^soc

(

^x

+

^th

) −

^fsoc

(

^x

)

K 0

,

which, by the continuous differentiability of f^socand the closedness of K, implies that

(

f^soc

)

^

(

x

)

h

K 0

.