to appear in Linear and Nonlinear Analysis, 2017

### Examples of r-convex functions and characterizations of r-convex functions associated with second-order cone

Chien-Hao Huang Department of Mathematics National Taiwan Normal University

Taipei 11677, Taiwan E-mail: qqnick0719@ntnu.edu.tw

Hong-Lin Huang Department of Mathematics National Taiwan Normal University

Taipei 11677, Taiwan E-mail: buddin5678@gmail.com

Jein-Shan Chen ^{1}
Department of Mathematics
National Taiwan Normal University

Taipei 11677, Taiwan E-mail: jschen@math.ntnu.edu.tw

August 23, 2017

Abstract. In this paper, we revisit the concept of r-convex functions which were studied in 1970s. We present several novel examples of r-convex functions that are new to the existing literature. In particular, for any given r, we show examples which are r-convex functions. In addition, we extend the concepts of r-convexity and quasi-convexity to the setting associated with second-order cone. Characterizations about such new func- tions are established. These generalizations will be useful in dealing with optimization problems involved in second-order cones.

Keywords: r-convex function, monotone function, second-order cone, spectral decom- position.

1Corresponding author. The author’s work is supported by Ministry of Science and Technology, Taiwan.

### 1 Introduction

It is known that the concept of convexity plays a central role in many applications includ- ing mathematical economics, engineering, management science, and optimization theory.

Moreover, much attention has been paid to its generalization, to the associated general- ization of the results previously developed for the classical convexity, and to the discovery of necessary and/or sufficient conditions for a function to have generalized convexities.

Some of the known extensions are quasiconvex functions, r-convex functions [1, 24], and so-called SOC-convex functions [7, 8]. Other further extensions can be found in [19, 23].

For a single variable continuous, the midpoint-convex function on R is also a convex func-
tion. This result was generalized in [22] by relaxing continuity to lower-semicontinuity
and replacing the number ^{1}_{2} with an arbitrary parameter α ∈ (0, 1). An analogous con-
sequence was obtained in [18, 23] for quasiconvex functions.

To understand the main idea behind r-convex function, we recall some concepts that were independently defined by Martos [17] and Avriel [2], and has been studied by the latter author. Indeed, this concept relies on the classical definition of convex functions and some well-known results from analysis dealing with weighted means of positive numbers.

Let w = (w_{1}, ..., w_{m}) ∈ R^{m}, q = (q_{1}, ..., q_{m}) ∈ R^{m} be vectors whose components are
positive and nonnegative numbers, respectively, such that Pm

i=1q_{i} = 1. Given the vector
of weights q, the weighted r-mean of the numbers w_{1}, ..., w_{m} is defined as below (see [13]):

Mr(w; q) = Mr(w1, ..., wm; q) :=

_{m}
P

i=1

q_{i}(w_{i})^{r}

1/r

if r 6= 0,

m

Q

i=1

(w_{i})^{q}^{i} if r = 0.

(1)

It is well-known from [13] that for s > r, there holds

M_{s}(w_{1}, ..., w_{m}; q) ≥ M_{r}(w_{1}, ..., w_{m}; q) (2)
for all q1, ..., qm ≥ 0 with Pm

i=1qi = 1. The r-convexity is built based on the aforemen-
tioned weighted r-mean. For a convex set S ⊆ R^{n}, a real-valued function f : S ⊆ R^{n}→ R
is said to be r-convex if, for any x, y ∈ S, λ ∈ [0, 1], q_{2} = λ, q_{1} = 1 − q_{2}, q = (q_{1}, q_{2}),
there has

f (q1x + q2y) ≤ lnMr(e^{f (x)}, e^{f (y)}; q) .
From (1), it can be verified that the above inequality is equivalent to

f ((1 − λ)x + λy) ≤ ln[(1 − λ)e^{rf (x)}+ λe^{rf (y)}]^{1/r} if r 6= 0,

(1 − λ)f (x) + λf (y) if r = 0. (3) Similarly, f is said to be r-concave on S if the inequality (3) is reversed. It is clear from the above definition that a real-valued function is convex (concave) if and only if it is

0-convex (0-concave). Besides, for r < 0 (r > 0), an r-convex (r-concave) function is
called superconvex (superconcave); while for r > 0 (r < 0), it is called subconvex (subcon-
cave). In addition, it can be verified that the r-convexity of f on C with r > 0 (r < 0)
is equivalent to the convexity (concavity) of e^{rf} on S.

A function f : S ⊆ R^{n} → R is said to be quasiconvex on S if, for all x, y ∈ S,
f (λx + (1 − λ)y) ≤ max {f (x), f (y)} , 0 ≤ λ ≤ 1.

Analogously, f is said to be quasiconcave on S if, for all x, y ∈ S, f (λx + (1 − λ)y) ≥ min {f (x), f (y)} , 0 ≤ λ ≤ 1.

From [13], we know that

r→+∞lim M_{r}(w_{1}, ..., w_{m}; q) ≡ M∞(w_{1}, ..., w_{m}) = max{w_{1}, ..., w_{m}},

r→−∞lim Mr(w1, · · · , wm; q) ≡ M−∞(w1, ..., wm) = min{w1, · · · , wm}.

Then, it follows from (2) that M∞(w_{1}, ..., w_{m}) ≥ M_{r}(w_{1}, ..., w_{m}; q) ≥ M−∞(w_{1}, ..., w_{m})
for every real number r. Thus, if f is r-convex on S, it is also (+∞)-convex, that is,
f (λx + (1 − λ)y) ≤ max{f (x), f (y)} for every x, y ∈ S and λ ∈ [0, 1]. Similarly, if f is
r-concave on S, it is also (−∞)-concave, i.e., f (λx + (1 − λ)y) ≥ min{f (x), f (y)}.

The following review some basic properties regarding r-convex function from [1] that will be used in the subsequent analysis.

Property 1.1. Let f : S ⊆ R^{n}→ R. Then, the followings hold.

(a) If f is r-convex (r-concave) on S, then f is also s-convex (s-concave) on S for s > r (s < r).

(b) Suppose that f is twice continuously differentiable on S. For any (x, r) ∈ S × R, we define

φ(x, r) = ∇^{2}f (x) + r∇f (x)∇f (x)^{T}.

Then, f is r-convex on S if and only if φ is positive semidefinite for all x ∈ S.

(c) Every r-convex (r-concave) function on a convex set S is also quasiconvex (quasi- concave) on S.

(d) f is r-convex if and only if (−f ) is (−r)-concave.

(e) Let f be r-convex (r-concave), α ∈ R and k > 0. Then f + α is r-convex (r-concave)
and k · f is (_{k}^{r})-convex ((_{k}^{r})-concave).

(f ) Let φ, ψ : S ⊆ R^{n} → R be r-convex (r-concave) and α1, α_{2} > 0. Then, the function
θ defined by

θ(x) = (

lnα1e^{rφ(x)}+ α_{2}e^{rψ(x)}1/r

if r 6= 0,
α_{1}φ(x) + α_{2}ψ(x) if r = 0,
is also r-convex (r-concave).

(g) Let φ : S ⊆ R^{n} → R be r-convex (r-concave) such that r ≤ 0 (r ≥ 0) and let the real
valued function ψ be nondecreasing s-convex (s-concave) on R with s ∈ R. Then,
the composite function θ = ψ ◦ φ is also s-convex (s-concave).

(h) φ : S ⊆ R^{n} → R is r-convex (r-concave) if and only if, for every x, y ∈ S, the
function ψ given by

ψ(λ) = φ ((1 − λ)x + λy) is an r-convex (r-concave) function of λ for 0 ≤ λ ≤ 1.

(i) Let φ be a twice continuously differentiable real quasiconvex function on an open
convex set S ⊆ R^{n}. If there exists a real number r^{∗} satisfying

r^{∗} = sup

x∈S, kzk=1

−z^{T}∇^{2}φ(x)z

[z^{T}∇φ(x)]^{2} (4)

whenever z^{T}∇φ(x) 6= 0, then φ is r-convex for every r ≥ r^{∗}. We obtain the r-
concave analog of the above theorem by replacing supremum in (4) by infimum.

In this paper, we will present new examples of r-convex functions in Section 2. Mean-
while, we extend the r-convexity and quasi-convexity concepts to the setting associated
with second-order cone in Section 4 and Section 5. Applications of r-convexity to opti-
mization theory can be found in [2, 12, 15]. In general, r-convex functions can be viewed
as the functions between convex functions and quasi-convex functions. We believe that
the aforementioned extensions will be beneficial for dealing optimization problems in-
volved second-order constraints. We point out that extending the concepts of r-convex
and quasi-convex functions to the setting associated with second-order cone, which be-
longs to symmetric cones, is not easy and obvious since any two vectors in the Euclidean
Jordan algebra cannot be compared under the partial order K^{n}, see [8]. Nonetheless,
using the projection onto second-order cone pave a way to do such extensions, more de-
tails will be seen in Sections 4 and 5.

To close this section, we recall some background materials regarding second-order
cone. The second-order cone (SOC for short) in R^{n}, also called the Lorentz cone, is
defined by

K^{n}=x = (x_{1}, x_{2}) ∈ R × R^{n−1}| kx_{2}k ≤ x_{1} .

For n = 1, K^{n} denotes the set of nonnegative real number R+. For any x, y in R^{n}, we
write x K^{n} y if x − y ∈ K^{n} and write x K^{n} y if x − y ∈ int(K^{n}). In other words, we
have x K^{n} 0 if and only if x ∈ K^{n} and x K^{n} 0 if and only if x ∈ int(K^{n}). The relation

K^{n} is a partial ordering but not a linear ordering in K^{n}, i.e., there exist x, y ∈ K^{n} such
that neither x K^{n} y nor y K^{n} x. To see this, for n = 2, let x = (1, 1) and y = (1, 0),
we have x − y = (0, 1) /∈ K^{n}, y − x = (0, −1) /∈ K^{n}.

For dealing with second-order cone programs (SOCP) and second-order cone comple- mentarity problems (SOCCP), we need spectral decomposition associated with SOC [9].

More specifically, for any x = (x1, x2) ∈ R × R^{n−1}, the vector x can be decomposed as
x = λ_{1}u^{(1)}_{x} + λ_{2}u^{(2)}_{x} ,

where λ_{1}, λ_{2} and u^{(1)}x , u^{(2)}x are the spectral values and the associated spectral vectors of
x, respectively, given by

λ_{i} = x_{1}+ (−1)^{i}kx_{2}k,
u^{(i)}_{x} =

( 1

2(1, (−1)^{i x}_{kx}^{2}

2k) if x_{2} 6= 0,

1

2(1, (−1)^{i}w) if x_{2} = 0.

for i = 1, 2 with w being any vector in R^{n−1} satisfying kwk = 1. If x_{2} 6= 0, the decompo-
sition is unique.

For any function f : R → R, the following vector-valued function associated with K^{n}
(n ≥ 1) was considered in [7, 8]:

f^{soc}(x) = f (λ_{1})u^{(1)}_{x} + f (λ_{2})u^{(2)}_{x} , ∀x = (x_{1}, x_{2}) ∈ R × R^{n−1}. (5)
If f is defined only on a subset of R, then f^{soc} is defined on the corresponding subset
of R^{n}. The definition (5) is unambiguous whether x_{2} 6= 0 or x_{2} = 0. The cases of
f^{soc}(x) = x^{1/2}, x^{2}, exp(x) are discussed in [10]. In fact, the above definition (5) is analo-
gous to the one associated with positive semidefinite cone S_{+}^{n} [20, 21].

Throughout this paper, R^{n} denotes the space of n-dimensional real column vectors,
C denotes a convex subset of R, S denotes a convex subset of R^{n}, and h· , ·i means the
Euclidean inner product, whereas k · k is the Euclidean norm. The notation “:=” means

“define”. For any f : R^{n} → R, ∇f(x) denotes the gradient of f at x. C^{(i)}(J ) denotes
the family of functions which are defined on J ⊆ R^{n} to R and have the i-th continuous
derivative, while ^{T} means transpose.

### 2 Examples of r-functions

In this section, we try to discover some new r-convex functions which is verified by applying Property 1.1. With these examples, we have a more complete picture about

characterizations of r-convex functions. Moreover, for any given r, we also provide ex- amples which are r-convex functions.

Example 2.1. For any real number p, let f : (0, ∞) → R be defined by f (t) = t^{p}.
(a) If p > 0, then f is convex for p ≥ 1, and (+∞)-convex for 0 < p < 1.

(b) If p < 0, then f is convex.

To see this, we first note that f^{0}(t) = pt^{p−1}, f^{00}(t) = p(p − 1)t^{p−2} and
sup

s·f^{0}(t)6=0,|s|=1

−s · f^{00}(t) · s

[s · f^{0}(t)]^{2} = sup

p6=0

(1 − p)t^{−p}

p = ∞ if 0 < p < 1, 0 if p > 1 or p < 0.

Then, applying Property 1.1 yields the desired result.

Example 2.2. Suppose that f is defined on (−^{π}_{2},^{π}_{2}).

(a) The function f (t) = sin t is ∞-convex.

(b) The function f (t) = tan t is 1-convex.

(c) The function f (t) = ln(sec t) is (−1)-convex.

(d) The function f (t) = ln |sec t + tan t| is 1-convex.

To see (a), we note that f^{0}(t) = cos t, f^{00}(t) = − sin t, and
sup

−^{π}

2<t<^{π}_{2},|s|=1

−s · f^{00}(t) · s

[s · f^{0}(t)]^{2} = sup

−^{π}_{2}<t<^{π}_{2}

sin t

cos^{2}t = ∞.

Hence f (t) = sin t is ∞-convex.

To see (b), we note that f^{0}(t) = sec^{2}t, f^{00}(t) = 2 sec^{2}t · tan t, and
sup

−^{π}_{2}<t<^{π}_{2}

−f^{00}(t)

[f^{0}(t)]^{2} = sup

−^{π}_{2}<t<^{π}_{2}

−2 sec^{2}t · tan t

sec^{4}t = sup

−^{π}_{2}<t<^{π}_{2}

(− sin 2t) = 1.

This says that f (t) = tan t is 1-convex.

To see (c), we note that f^{0}(t) = tan t, f^{00}(t) = sec^{2}t, and
sup

−^{π}

2<t<^{π}_{2}

−f^{00}(t)

[f^{0}(t)]^{2} = sup

−^{π}

2<t<^{π}_{2}

−k sec^{2}t

tan^{2}t = sup

−^{π}

2<t<^{π}_{2}

(− csc^{2}t) = −1.

Then, it is clear to see that f (t) = ln(sec t) is (−1)-convex.

Figure 1: Graphs of r-convex functions with various values of r.

To see (d), we note that f^{0}(t) = sec t, f^{00}(t) = sec t · tan t, and
sup

−^{π}

2<t<^{π}_{2}

−f^{00}(t)

[f^{0}(t)]^{2} = sup

−^{π}

2<t<^{π}_{2}

− sec t · tan t

sec^{2}t = sup

−^{π}

2<t<^{π}_{2}

(− sin t) = 1.

Thus, f (t) = ln |sec t + tan t| is 1-convex.

In light of Example 2.2(b)-(c) and Property 1.1(e), the next example indicates that for any given r ∈ R (no matter positive or negative), we can always construct an r-convex function accordingly. The graphs of various r-convex functions are depicted in Figure 1.

Example 2.3. For any r 6= 0, let f be defined on (−^{π}_{2},^{π}_{2}).

(a) The function f (t) = tan t

r is |r|-convex.

(b) The function f (t) = ln(sec t)

r is (−r)-convex.

(a) First, we compute that f^{0}(t) = sec^{2}t

r , f^{00}(t) = 2 sec^{2}t · tan t

r , and

sup

−^{π}_{2}<t<^{π}_{2}

−f^{00}(t)

[f^{0}(t)]^{2} = sup

−^{π}_{2}<t<^{π}_{2}

(−r sin 2t) = |r|.

This says that f (t) = tan t

r is |r|-convex.

(b) Similarly, from f^{0}(t) = tan t

r , f^{00}(t) = sec^{2}t
r , and
sup

−^{π}_{2}<t<^{π}_{2}

−f^{00}(t)

[f^{0}(t)]^{2} = sup

−^{π}_{2}<t<^{π}_{2}

(−r csc^{2}t) = −r.

Then, it is easy to see that f (t) = ln(sec t)

r is (−r)-convex.

Example 2.4. The function f (x) = ^{1}_{2}ln(kxk^{2}+ 1) defined on R^{2} is 1-convex.

For x = (s, t) ∈ R^{2}, and any real number r 6= 0, we consider the function
φ(x, r) = ∇^{2}f (x) + r∇f (x)∇f (x)^{T}

= 1

(kxk^{2}+ 1)^{2}

t^{2}− s^{2}+ 1 −2st

−2st s^{2}− t^{2}+ 1

+ r

(kxk^{2}+ 1)^{2}

s^{2} st
st t^{2}

= 1

(kxk^{2}+ 1)^{2}

(r − 1)s^{2}+ t^{2}+ 1 (r − 2)st
(r − 2)st s^{2}+ (r − 1)t^{2}+ 1

.

Applying Property 1.1(b), we know that f is r-convex if and only if φ is positive semidef- inite, which is equivalent to

(r − 1)s^{2}+ t^{2} + 1 ≥ 0 (6)

(r − 1)s^{2}+ t^{2}+ 1 (r − 2)st
(r − 2)st s^{2} + (r − 1)t^{2}+ 1

≥ 0. (7)

It is easy to verify the inequality (6) holds for all x ∈ R^{2} if and only if r ≥ 1. Moreover,
we note that

(r − 1)s^{2}+ t^{2}+ 1 (r − 2)st
(r − 2)st s^{2}+ (r − 1)t^{2}+ 1

≥ 0

⇐⇒ s^{2}t^{2}+ s^{2}+ t^{2}+ 1 + (r − 1)^{2}s^{2}t^{2}+ (r − 1)(s^{4}+ s^{2}+ t^{4}+ t^{2}) − (r − 2)^{2}s^{2}t^{2} ≥ 0

⇐⇒ s^{2} + t^{2}+ 1 + (2r − 2)s^{2}t^{2}+ (r − 1)(s^{4} + s^{2} + t^{4}+ t^{2}) ≥ 0,

and hence the inequality (7) holds for all x ∈ R^{2} whenever r ≥ 1. Thus, we conclude by
Property 1.1(b) that f is 1-convex on R^{2}.

### 3 Properties of SOC-functions

As mentioned in Section 1, another contribution of this paper is extending the concept of r-convexity to the setting associated with second-order cone. To this end, we recall what

10 5

0 -5 -10 -10

-5 0 5 0 1 2 3

10

Figure 2: Graphs of 1-convex functions f (x) = ^{1}_{2}ln(kxk^{2}+ 1).

SOC-convex function means. For any x = (x_{1}, x_{2}) ∈ R×R^{n−1}and y = (y_{1}, y_{2}) ∈ R×R^{n−1},
we define their Jordan product as

x ◦ y = (x^{T}y , y_{1}x_{2}+ x_{1}y_{2}).

We write x^{2} to mean x ◦ x and write x + y to mean the usual componentwise addition
of vectors. Then, ◦, +, together with e^{0} = (1, 0, . . . , 0)^{T} ∈ R^{n} and for any x, y, z ∈ R^{n},
the following basic properties [10, 11] hold: (1) e^{0} ◦ x = x, (2) x ◦ y = y ◦ x, (3)
x ◦ (x^{2}◦ y) = x^{2}◦ (x ◦ y), (4) (x + y) ◦ z = x ◦ z + y ◦ z. Notice that the Jordan product is
not associative in general. However, it is power associative, i.e., x ◦ (x ◦ x) = (x ◦ x) ◦ x
for all x ∈ R^{n}. Thus, we may, without loss of ambiguity, write x^{m} for the product of m
copies of x and x^{m+n} = x^{m}◦ x^{n} for all positive integers m and n. Here, we set x^{0} = e^{0}.
Besides, K^{n} is not closed under Jordan product.

For any x ∈ K^{n}, it is known that there exists a unique vector in K^{n} denoted by x^{1/2}
such that (x^{1/2})^{2} = x^{1/2}◦ x^{1/2} = x. Indeed,

x^{1/2} =
s,x_{2}

2s

, where s = s

1 2

x_{1}+

q

x^{2}_{1}− kx_{2}k^{2}

.

In the above formula, the term x_{2}/s is defined to be the zero vector if x_{2} = 0 and
s = 0, i.e., x = 0. For any x ∈ R^{n}, we always have x^{2} ∈ K^{n}, i.e., x^{2} K^{n} 0. Hence,
there exists a unique vector (x^{2})^{1/2} ∈ K^{n} denoted by |x|. It is easy to verify that

|x| K^{n} 0 and x^{2} = |x|^{2} for any x ∈ R^{n}. It is also known that |x| K^{n} x. For any
x ∈ R^{n}, we define [x]_{+} to be the nearest point projection of x onto K^{n}, which is the same

definition as in R^{n}+. In other words, [x]_{+} is the optimal solution of the parametric SOCP:

[x]+= arg min{kx − yk | y ∈ K^{n}}. In addition, it can be verified that [x]+ = (x + |x|)/2;

see [10, 11].

Property 3.1. ([11, Proposition 3.3]) For any x = (x_{1}, x_{2}) ∈ R × R^{n−1}, we have
(a) |x| = (x^{2})^{1/2}= |λ1|u^{(1)}x + |λ2|u^{(2)}x .

(b) [x]_{+} = [λ_{1}]_{+}u^{(1)}x + [λ_{2}]_{+}u^{(2)}x = ^{1}_{2}(x + |x|).

Next, we review the concepts of SOC-monotone and SOC-convex functions which are introduced in [7].

Definition 3.1. For a real valued function f : R → R,

(a) f is said to be SOC-monotone of order n if its corresponding vector-valued function
f^{soc} defined as in (5) satisfies

x K^{n} y =⇒ f^{soc}(x) K^{n} f^{soc}(y).

The function f is said to be SOC-monotone if f is SOC-monotone of all order n.

(b) f is said to be SOC-convex of order n if its corresponding vector-valued function f^{soc}
defined as in (5) satisfies

f^{soc}((1 − λ)x + λy) _{K}^{n} (1 − λ)f^{soc}(x) + λf^{soc}(y) (8)
for all x, y ∈ R^{n} and 0 ≤ λ ≤ 1. Similarly, f is said to be SOC-concave of order
n on C if the inequality (8) is reversed. The function f is said to be SOC-convex
(respectively, SOC-concave) if f is SOC-convex of all order n (respectively, SOC-
concave of all order n).

The concepts of SOC-monotone and SOC-convex functions are analogous to matrix monotone and matrix convex functions [5, 14], and are special cases of operator monotone and operator convex functions [3, 6, 16]. Examples of SOC-monotone and SOC-convex functions are given in [7]. It is clear that the set of SOC-monotone functions and the set of SOC-convex functions are both closed under linear combinations and under pointwise limits.

Property 3.2. ([8, Theorem 3.1]) Let f ∈ C^{(1)}(J ) with J being an open interval and
dom(f^{soc}) ⊆ R^{n}. Then, the following hold.

(a) f is SOC-monotone of order 2 if and only if f^{0}(τ ) ≥ 0 for any τ ∈ J ;

(b) f is SOC-monotone of order n ≥ 3 if and only if the 2 × 2 matrix

f^{(1)}(t_{1}) f (t_{2}) − f (t_{1})
t2− t1

f (t_{2}) − f (t_{1})

t_{2}− t_{1} f^{(1)}(t_{2})

O for all t_{1}, t_{2} ∈ J and t_{1} 6= t_{2}.

Property 3.3. ([8, Theorem 4.1]) Let f ∈ C^{(2)}(J ) with J being an open interval in R
and dom(f^{soc}) ⊆ R^{n}. Then, the following hold.

(a) f is SOC-convex of order 2 if and only if f is convex;

(b) f is SOC-convex of order n ≥ 3 if and only if f is convex and the inequality 1

2f^{(2)}(t_{0})[f (t0) − f (t) − f^{(1)}(t)(t0− t)]

(t_{0}− t)^{2} ≥ [f (t) − f (t0) − f^{(1)}(t0)(t − t0)]

(t_{0}− t)^{4} (9)

holds for any t_{0}, t ∈ J and t_{0} 6= t.

Property 3.4. ([4, Theorem 3.3.7]) Let f : S → R where S is a nonempty open convex
set in R^{n}. Suppose f ∈ C^{2}(S). Then, f is convex if and only if ∇^{2}f (x) O, for all
x ∈ S.

Property 3.5. ([7, Proposition 4.1]) Let f : [0, ∞] → [0, ∞] be continuous. If f is SOC-concave, then f is SOC-monotone.

Property 3.6. ([11, Proposition 3.2]) Suppose that f (t) = e^{t} and g(t) = ln t. Then, the
corresponding SOC-functions of e^{t} and ln t are given as below.

(a) For any x = (x_{1}, x_{2}) ∈ R × R^{n−1},

f^{soc}(x) = e^{x} =
(

e^{x}^{1}

cosh(kx2k), sinh(kx_{2}k)_{kx}^{x}^{2}

2k

if x2 6= 0,

(e^{x}^{1}, 0) if x_{2} = 0,

where cosh(α) = (e^{α}+ e^{−α})/2 and sinh(α) = (e^{α}− e^{−α})/2 for α ∈ R.

(b) For any x = (x_{1}, x_{2}) ∈ int(K^{n}), ln x is well-defined and

g^{soc}(x) = ln x =
( 1

2

ln(x^{2}_{1}− kx_{2}k^{2}), ln

x1+kx2k x1−kx2k

x2

kx2k

if x_{2} 6= 0,

(ln x_{1}, 0) if x_{2} = 0.

With these, we have the following technical lemmas that will be used in the subsequent analysis.

Lemma 3.1. Let f : R → R be f (t) = e^{t} and x = (x_{1}, x_{2}) ∈ R × R^{n−1}, y = (y_{1}, y_{2}) ∈
R × R^{n−1}. Then, the following hold.

(a) f is SOC-monotone of order 2 on R.

(b) f is not SOC-monotone of order n ≥ 3 on R.

(c) If x_{1}− y_{1} ≥ kx_{2}k + ky_{2}k, then e^{x} K^{n} e^{y}. In particular, if x ∈ K^{n}, then e^{x} K^{n} e^{(0,0)}.
Proof. (a) By applying Property 3.2(a), it is clear that f is SOC-monotone of order 2
since f^{0}(τ ) = e^{τ} ≥ 0 for all τ ∈ R.

(b) Take x = (2, 1.2, −1.6), y = (−1, 0, −4), then we have x − y = (3, 1.2, 2.4) K^{n} 0.

But, we compute that
e^{x}− e^{y} = e^{2}

cosh(2), sinh(2)(1.2, −1.6) 2

− e^{−1}

cosh(4), sinh(4)(0, −4) 4

= 1

2(e^{4}+ 1, .6(e^{4}− 1), −.8(e^{4}− 1)) − (e^{3}+ e^{−5}, 0, −e^{3}+ e^{−5})

= (17.7529, 16.0794, −11.3999) ^{K}^{n} 0.

The last inequality is because k(16.0794, −11.3999)k = 19.7105 > 17.7529.

We also present an alternative argument for part(b) here. First, we observe that

det

"

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

2−t_{1}
f (t2)−f (t1)

t2−t_{1} f^{(1)}(t_{2})

#

= e^{t}^{1}^{+t}^{2} − e^{t}^{2} − e^{t}^{1}
t_{2}− t_{1}

2

≥ 0 (10)

if and only if 1 ≥ e^{(t}^{2}^{−t}^{1}^{)/2}− e^{(t}^{1}^{−t}^{2}^{)/2}
t2− t1

^{2}

. Denote s := (t_{2} − t_{1})/2, then the above
inequality holds if and only if 1 ≥ (sinh(s)/s)^{2}. In light of Taylor Theorem, we know
sinh(s)/s = 1 + s^{2}/6 + s^{4}/120 + · · · > 1 for s 6= 0. Hence, (10) does not hold. Then,
applying Property 3.2(b) says f is not SOC-monotone of order n ≥ 3 on R.

(c) The desired result follows by the following implication:

e^{x} K^{n} e^{y}

⇐⇒ e^{x}^{1}cosh(kx_{2}k) − e^{y}^{1}cosh(ky_{2}k) ≥

e^{x}^{1}sinh(kx_{2}k) x_{2}

kx_{2}k− e^{y}^{1}sinh(ky_{2}k) y_{2}
ky_{2}k

⇐⇒ [e^{x}^{1}cosh(kx_{2}k) − e^{y}^{1}cosh(ky_{2}k)]^{2}−

e^{x}^{1}sinh(kx_{2}k) x_{2}

kx_{2}k− e^{y}^{1}sinh(ky_{2}k) y_{2}
ky_{2}k

2

= e^{2x}^{1} + e^{2y}^{1} − 2e^{x}^{1}^{+y}^{1}

cosh(kx_{2}k) cosh(ky_{2}k) − sinh(kx_{2}k) sinh(ky_{2}k) hx_{2}, y_{2}i
kx_{2}kky_{2}k

≥ 0

⇐= e^{2x}^{1} + e^{2y}^{1} − 2e^{x}^{1}^{+y}^{1}cosh(kx_{2}k + ky_{2}k) ≥ 0

⇐⇒ cosh(kx_{2}k + ky_{2}k) ≤ e^{2x}^{1} + e^{2y}^{1}

2e^{x}^{1}^{+y}^{1} = e^{x}^{1}^{−y}^{1} + e^{y}^{1}^{−x}^{1}

2 = cosh(x_{1}− y_{1})

⇐⇒ x_{1}− y_{1} ≥ kx_{2}k + ky_{2}k.

2

Lemma 3.2. Let f (t) = e^{t} be defined on R, then f is SOC-convex of order 2. However,
f is not SOC-convex of order n ≥ 3.

Proof. (a) By applying Property 3.3 (a), it is clear that f is SOC-convex since expo- nential function is a convex function on R.

(b) As below, it is a counterexample which shows f (t) = e^{t} is not SOC-convex of order
n ≥ 3. To see this, we compute that

e[(2,0,−1)+(6,−4,−3)]/2 = e^{(4,−2,−2)}

= e^{4}

cosh(2√

2) , sinh(2√

2) · (−2, −2)/(2√ 2) + (463.48, −325.45, −325.45)

and

1

2 e^{(2,0,−1)}+ e^{(6,−4,−3)}

= 1

2e^{2}(cosh(1), 0, − sinh(1)) + e^{6}(cosh(5), sinh(5) · (−4, −3)/5)

= (14975, −11974, −8985).

We see that 14975 − 463.48 = 14511.52, but

k(−11974, −8985) − (−325.4493, −325.4493)k = 14515 > 14511.52 which is a contradiction. 2

Lemma 3.3. ([8, Proposition 5.1]) The function g(t) = ln t is SOC-monotone of order n ≥ 2 on (0, ∞).

In general, to verify the SOC-convexity of e^{t}(as shown in Proposition 3.1), we observe
that the following fact

0 ≺K^{n} e^{rf}^{soc}(λx+(1−λ)y)K^{n} w =⇒ rf^{soc}(λx + (1 − λ)y) K^{n} ln(w)

is important and often needed. Note for x_{2} 6= 0, we also have some observations as below.

(a) e^{x} K^{n} 0 ⇐⇒ cosh(kx2k) ≥ | sinh(kx2k)| ⇐⇒ e^{−kx}^{2}^{k} > 0 .
(b) 0 ≺_{K}^{n} ln(x) ⇐⇒ ln(x^{2}_{1} − kx_{2}k^{2}) >

ln

x1+kx2k x1−kx2k

⇐⇒ ln(x_{1} − kx_{2}k) > 0 ⇐⇒

x_{1}− kx_{2}k > 1. Hence (1, 0) ≺K^{n} x implies 0 ≺K^{n} ln(x).

(c) ln(1, 0) = (0, 0) and e^{(0,0)} = (1, 0).

### 4 SOC-r-convex functions

In this section, we define the so-called SOC-r-convex functions which is viewed as the natural extension of r-convex functions to the setting associated with second-order cone.

Definition 4.1. Suppose that r ∈ R and f : C ⊆ R → R where C is a convex subset of
R. Let f^{soc} : S ⊆ R^{n} → R^{n} be its corresponding SOC-function defined as in (5). The
function f is said to be SOC-r-convex of order n on C if, for x, y ∈ S and λ ∈ [0, 1],
there holds

f^{soc}(λx + (1 − λ)y) _{K}^{n}
(1

rln λe^{rf}^{soc}^{(x)}+ (1 − λ)e^{rf}^{soc}^{(y)}

r 6= 0,

λf^{soc}(x) + (1 − λ)f^{soc}(y) r = 0. (11)
Similarly, f is said to be SOC-r-concave of order n on C if the inequality (11) is reversed.

We say f is SOC-r-convex (respectively, SOC-r-concave) on C if f is SOC-r-convex of all order n (respectively, SOC-r-concave of all order n) on C.

It is clear from the above definition that a real function is SOC-convex (SOC-concave)
if and only if it is SOC-0-convex (SOC-0-concave). In addition, a function f is SOC-r-
convex if and only if −f is SOC-(−r)-concave. From [1, Theorem 4.1], it is shown that
φ : R → R is r-convex with r 6= 0 if and only if e^{rφ} is convex whenever r > 0 and concave
whenever r < 0. However, we observe that the exponential function e^{t}is not SOC-convex
for n ≥ 3 by Lemma 3.2. This is a hurdle to build parallel result for general n in the
setting of SOC case. As seen in Proposition 4.3, the parallel result is true only for n = 2.

Indeed, for n ≥ 3, only one direction holds which can be viewed as a weaker version of [1, Theorem 4.1].

Proposition 4.1. Let f : [0, ∞) → [0, ∞) be continuous. If f is SOC-r-concave with r ≥ 0, then f is SOC-monotone.

Proof. For any 0 < λ < 1, we can write λx = λy + ^{(1−λ)λ}_{(1−λ)}(x − y). If r = 0, then f is
SOC-concave and SOC-monotone by Property 3.5. If r > 0, then

f^{soc}(λx) K^{n}

1 r ln

λe^{rf}^{soc}^{(y)}+ (1 − λ)e^{rf}^{soc}^{(}^{1−λ}^{λ} ^{(x−y))}

K^{n}

1

r ln λe^{r(0,0)}+ (1 − λ)e^{r(0,0)}

= 1

r ln (λ(1, 0) + (1 − λ)(1, 0))

= 0,

where the second inequality is due to x − y K^{n} 0 and Lemmas 3.1-3.3. Letting λ → 1,
we obtain that f^{soc}(x) K^{n} f^{soc}(y), which says that f is SOC-monotone. 2

In fact, in light of Lemma 3.1-3.3, we have the following Lemma which is useful for subsequent analysis.

Lemma 4.1. Let z ∈ R^{n} and w ∈ int(K^{n}). Then, the following hold.

(a) For n = 2 and r > 0, z K^{n} ln(w)/r ⇐⇒ rz K^{n} ln(w) ⇐⇒ e^{rz} K^{n} w.

(b) For n = 2 and r > 0, z K^{n} ln(w)/r ⇐⇒ rz K^{n} ln(w) ⇐⇒ e^{rz} K^{n} w.

(c) For n ≥ 2, if e^{rz} K^{n} w, then rz K^{n} ln(w).

Proposition 4.2. For n = 2 and let f : R → R. Then, the following hold.

(a) The function f (t) = t is SOC-r-convex (SOC-r-concave) on R for r > 0 (r < 0).

(b) If f is SOC-convex, then f is SOC-r-convex (SOC-r-concave) for r > 0 (r < 0).

Proof. (a) For r > 0, x, y ∈ R^{n} and λ ∈ [0, 1], we note that the corresponding vector-
valued SOC-function of f (t) = t is f^{soc}(x) = x. Therefore, to prove the desired result,
we need to verify that

f^{soc}(λx + (1 − λ)y) K^{n}

1

rln λe^{rf}^{soc}^{(x)}+ (1 − λ)e^{rf}^{soc}^{(y)} .
To this end, we see that

λx + (1 − λ)y K^{n}

1

rln (λe^{rx}+ (1 − λ)e^{ry})

⇐⇒ λrx + (1 − λ)ry K^{n} ln (λe^{rx}+ (1 − λ)e^{ry})

⇐⇒ eλrx+(1−λ)ry K^{n} λe^{rx}+ (1 − λ)e^{ry},

where the first “⇐⇒” is true due to Lemma 4.1, whereas the second “⇐⇒” holds because
e^{t} and ln t are SOC-monotone of order 2 by Lemma 3.1 and Lemma 3.3. Then, using the
fact that e^{t} is SOC-convex of order 2 gives the desired result.

(b) For any x, y ∈ R^{n} and 0 ≤ λ ≤ 1, it can be verified that
f^{soc}(λx + (1 − λ)y) K^{n} λf^{soc}(x) + (1 − λ)f^{soc}(y)

_{K}^{n} 1

rln λe^{rf}^{soc}^{(x)}+ (1 − λ)e^{rf}^{soc}^{(y)} ,

where the second inequality holds according to the proof of (a). Thus, the desired result follows. 2

Proposition 4.3. Let f : R → R. Then f is SOC-r-convex if e^{rf} is SOC-convex (SOC-
concave) for n ≥ 2 and r > 0 (r < 0). For n = 2, we can replace “if” by “if and only
if”.

Proof. Suppose that e^{rf} is SOC-convex. For any x, y ∈ R^{n} and 0 ≤ λ ≤ 1, using that
fact that ln t is SOC-monotone (Lemma 3.3) yields

e^{rf}^{soc}(λx+(1−λ)y) _{K}^{n} λe^{rf}^{soc}^{(x)}+ (1 − λ)e^{rf}^{soc}^{(y)}

=⇒ rf^{soc}(λx + (1 − λ)y) K^{n} ln λe^{rf}^{soc}^{(x)}+ (1 − λ)e^{rf}^{soc}^{(y)}

⇐⇒ f^{soc}(λx + (1 − λ)y) K^{n}

1

rln λe^{rf}^{soc}^{(x)}+ (1 − λ)e^{rf}^{soc}^{(y)} .

When n = 2, e^{t} is SOC-monotone as well, which implies that the “=⇒” can be replaced
by “⇐⇒”. Thus, the proof is complete. 2

Combining with Property 3.3, we can characterize the SOC-r-convexity as follows.

Proposition 4.4. Let f ∈ C^{(2)}(J ) with J being an open interval in R and dom(f^{soc}) ⊆
R^{n}. Then, for r > 0, the followings hold.

(a) f is SOC-r-convex of order 2 if and only if e^{rf} is convex;

(b) f is SOC-r-convex of order n ≥ 3 if e^{rf} is convex and satisfies the inequality (9).

Next, we present several examples of SOC-r-convex and SOC-r-concave functions of order 2. For examples of SOC-r-convex and SOC-r-concave functions (of order n), we are still unable to discover them.

Example 4.1. For n = 2, the following hold.

(a) The function f (t) = t^{2} is SOC-r-convex on R for r ≥ 0.

(b) The function f (t) = t^{3} is SOC-r-convex on [0, ∞) for r > 0, while it is SOC-r-
concave on (−∞, 0] for r < 0.

(c) The function f (t) = 1/t is SOC-r-convex on [−r/2, 0) or (0, ∞) for r > 0, while it is SOC-r-concave on (−∞, 0) or (0, −r/2] for r < 0.

(d) The function f (t) =√

t is SOC-r-convex on [1/r^{2}, ∞) for r > 0, while it is SOC-r-
concave on [0, ∞) for r < 0.

(e) The function f (t) = ln t is SOC-r-convex (SOC-r-concave) on (0, ∞) for r > 0 (r < 0).

Proof. (a) First, we denote h(t) := e^{rt}^{2}. Then, we have h^{0}(t) = 2rte^{rt}^{2} and h^{00}(t) =
(1 + 2rt^{2})2re^{rt}^{2}. From Property 3.4, we know h is convex if and only if h^{00}(t) ≥ 0. Thus,
the desired result holds by applying Property 3.3 and Proposition 4.3. The arguments
for other cases are similar and we omit them. 2

### 5 SOC-quasiconvex Functions

In this section, we define the so-called SOC-quasiconvex functions which is a natural extension of quasiconvex functions to the setting associated with second-order cone.

Recall that a function f : S ⊆ R^{n} → R is said to be quasiconvex on S if, for any
x, y ∈ S and 0 ≤ λ ≤ 1, there has

f (λx + (1 − λ)y) ≤ max {f (x), f (y)} .

We point out that the relation _{K}^{n} is not a linear ordering. Hence, it is not possible to
compare any two vectors (elements) via K^{n}. Nonetheless, we note that

max{a, b} = b + [a − b]_{+} = 1

2(a + b + |a − b|), for any a, b ∈ R.

This motivates us to define SOC-quasiconvex functions in the setting of second-order cone.

Definition 5.1. Let f : C ⊆ R → R and 0 ≤ λ ≤ 1. The function f is said to be SOC-quasiconvex of order n on C if, for any x, y ∈ C, there has

f^{soc}(λx + (1 − λ)y) K^{n} f^{soc}(y) + [f^{soc}(x) − f^{soc}(y)]_{+}

where

f^{soc}(y) + [f^{soc}(x) − f^{soc}(y)]_{+}

=

f^{soc}(x) if f^{soc}(x) K^{n} f^{soc}(y),
f^{soc}(y) if f^{soc}(x) ≺K^{n} f^{soc}(y),

1

2(f^{soc}(x) + f^{soc}(y) + |f^{soc}(x) − f^{soc}(y)|) if f^{soc}(x) − f^{soc}(y) /∈ K^{n}∪ (−K^{n}).

Similarly, f is said to be SOC-quasiconcave of order n if

f^{soc}(λx + (1 − λ)y) K^{n} f^{soc}(x) − [f^{soc}(x) − f^{soc}(y)]_{+}.

The function f is called SOC-quasiconvex (SOC-quasiconcave) if it is SOC-quasiconvex of all order n (SOC-quasiconcave of all order n).

Proposition 5.1. Let f : R → R be f (t) = t. Then, f is SOC-quasiconvex on R.

Proof. First, for any x = (x_{1}, x_{2}) ∈ R × R^{n−1}, y = (y_{1}, y_{2}) ∈ R × R^{n−1}, and 0 ≤ λ ≤ 1,
we have

f^{soc}(y) K^{n} f^{soc}(x) ⇐⇒ (1 − λ)f^{soc}(y) K^{n} (1 − λ)f^{soc}(x)

⇐⇒ λf^{soc}(x) + (1 − λ)f^{soc}(y) K^{n} f^{soc}(x).

Recall that the corresponding SOC-function of f (t) = t is f^{soc}(x) = x. Thus, for all
x ∈ R^{n}, this implies f^{soc}(λx + (1 − λ)y) = λf^{soc}(x) + (1 − λ)f^{soc}(y) K^{n} f^{soc}(x) under
this case: f^{soc}(y) _{K}^{n} f^{soc}(x). The argument is similar to the case of f^{soc}(x) _{K}^{n} f^{soc}(y).

Hence, it remains to consider the case of f^{soc}(x) − f^{soc}(y) /∈ K^{n}∪ (−K^{n}), i.e., it suffices
to show that λx + (1 − λ)y K^{n}

1

2(x + y + |x − y|). To this end, we note that

|x − y| K^{n} x − y and |x − y| K^{n} y − x,
which respectively implies

1

2(x + y + |x − y|) K^{n} x, (12)

1

2(x + y + |x − y|) K^{n} y. (13)

Then, adding up (12) ×λ and (13) ×(1 − λ) yields the desired result. 2

Proposition 5.2. If f : C ⊆ R → R is SOC-convex on C, then f is also SOC- quasiconvex on C.

Proof. For any x, y ∈ R^{n} and 0 ≤ λ ≤ 1, it can be verified that

f^{soc}(λx + (1 − λ)y) _{K}^{n} λf^{soc}(x) + (1 − λ)f^{soc}(y) _{K}^{n} f^{soc}(y) + [f^{soc}(x) − f^{soc}(y)]_{+},
where the second inequality holds according to the proof of Proposition 5.1. Thus, the
desired result follows. 2

From Proposition 5.2, we can easily construct examples of SOC-quasiconvex func-
tions. More specifically, all the SOC-convex functions which were verified in [7] are
SOC-quasiconvex functions, for instances, t^{2} on R, and t^{3}, ^{1}_{t}, t^{1/2} on (0, ∞).

### 6 Final Remarks

In this paper, we revisit the concept of r-convex functions and provide a way to construct r-convex functions for any given r ∈ R. We also extend such concept to the setting asso- ciated with SOC which will be helpful in dealing with optimization problems involved in second-order cones. In particular, we obtain some characterizations for SOC-r-convexity and SOC-quasiconvexity.

Indeed, this is just the first step and there still have many things to clarify. For
example, in Section 4, we conclude that SOC-convexity implies SOC-r-convexity for
n = 2 only. The key role therein relies particularly on the SOC-convexity and SOC-
monotonicity of e^{t}. However, for n > 2, the expressions of e^{x} and ln(x) associated with
second-order cone are very complicated so that it is hard to compare any two elements.

In other words, when n = 2, the SOC-convexity and SOC-monotonicity of e^{t}make things
much easier than the general case n ≥ 3. To conquer this difficulty, we believe that we
have to derive more properties of e^{x}. In particular, “Does SOC-r-convex function have
similar results as shown in Property 1.1?” is an important future direction.

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