4 SOC-r-convex functions

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 ¹₂ 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 = (w₁, ..., w_m) ∈ R^m, q = (q₁, ..., 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 w₁, ..., 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)^qi if r = 0.

(1)

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

M_s(w₁, ..., w_m; q) ≥ M_r(w₁, ..., 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 ⊆ Rⁿ, a real-valued function f : S ⊆ Rⁿ→ R is said to be r-convex if, for any x, y ∈ S, λ ∈ [0, 1], q₂ = λ, q₁ = 1 − q₂, q = (q₁, q₂), 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 ⊆ Rⁿ → 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(w₁, ..., w_m; q) ≡ M∞(w₁, ..., w_m) = max{w₁, ..., w_m},

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

Then, it follows from (2) that M∞(w₁, ..., w_m) ≥ M_r(w₁, ..., w_m; q) ≥ M−∞(w₁, ..., 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 ⊆ Rⁿ→ 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) = ∇²f (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 ⊆ Rⁿ → R be r-convex (r-concave) and α1, α₂ > 0. Then, the function θ defined by

θ(x) = (

lnα1e^rφ(x)+ α₂e^rψ(x)1/r

if r 6= 0, α₁φ(x) + α₂ψ(x) if r = 0, is also r-convex (r-concave).

(g) Let φ : S ⊆ Rⁿ → 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 ⊆ Rⁿ → 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 ⊆ Rⁿ. If there exists a real number r^∗ satisfying

r^∗ = sup

x∈S, kzk=1

−z^T∇²φ(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 Kⁿ, 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 Rⁿ, also called the Lorentz cone, is defined by

Kⁿ=x = (x₁, x₂) ∈ R × Rⁿ⁻¹| kx₂k ≤ x₁ .

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

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

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 × Rⁿ⁻¹, the vector x can be decomposed as x = λ₁u⁽¹⁾_x + λ₂u⁽²⁾_x ,

where λ₁, λ₂ and u⁽¹⁾x , u⁽²⁾x are the spectral values and the associated spectral vectors of x, respectively, given by

λ_i = x₁+ (−1)ⁱkx₂k, u⁽ⁱ⁾_x =

( 1

2(1, (−1)^{i x}_kx²

2k) if x₂ 6= 0,

1

2(1, (−1)ⁱw) if x₂ = 0.

for i = 1, 2 with w being any vector in Rⁿ⁻¹ satisfying kwk = 1. If x₂ 6= 0, the decompo- sition is unique.

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

f^soc(x) = f (λ₁)u⁽¹⁾_x + f (λ₂)u⁽²⁾_x , ∀x = (x₁, x₂) ∈ R × Rⁿ⁻¹. (5) If f is defined only on a subset of R, then f^soc is defined on the corresponding subset of Rⁿ. The definition (5) is unambiguous whether x₂ 6= 0 or x₂ = 0. The cases of f^soc(x) = x^1/2, x², 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, Rⁿ denotes the space of n-dimensional real column vectors, C denotes a convex subset of R, S denotes a convex subset of Rⁿ, and h· , ·i means the Euclidean inner product, whereas k · k is the Euclidean norm. The notation “:=” means

“define”. For any f : Rⁿ → R, ∇f(x) denotes the gradient of f at x. C⁽ⁱ⁾(J ) denotes the family of functions which are defined on J ⊆ Rⁿ 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 f⁰(t) = pt^p−1, f⁰⁰(t) = p(p − 1)t^p−2 and sup

s·f⁰(t)6=0,|s|=1

−s · f⁰⁰(t) · s

[s · f⁰(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 (−^π₂,^π₂).

(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 f⁰(t) = cos t, f⁰⁰(t) = − sin t, and sup

−^π

2<t<^π₂,|s|=1

−s · f⁰⁰(t) · s

[s · f⁰(t)]² = sup

−^π₂<t<^π₂

sin t

cos²t = ∞.

Hence f (t) = sin t is ∞-convex.

To see (b), we note that f⁰(t) = sec²t, f⁰⁰(t) = 2 sec²t · tan t, and sup

−^π₂<t<^π₂

−f⁰⁰(t)

[f⁰(t)]² = sup

−^π₂<t<^π₂

−2 sec²t · tan t

sec⁴t = sup

−^π₂<t<^π₂

(− sin 2t) = 1.

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

To see (c), we note that f⁰(t) = tan t, f⁰⁰(t) = sec²t, and sup

−^π

2<t<^π₂

−f⁰⁰(t)

[f⁰(t)]² = sup

−^π

2<t<^π₂

−k sec²t

tan²t = sup

−^π

2<t<^π₂

(− csc²t) = −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 f⁰(t) = sec t, f⁰⁰(t) = sec t · tan t, and sup

−^π

2<t<^π₂

−f⁰⁰(t)

[f⁰(t)]² = sup

−^π

2<t<^π₂

− sec t · tan t

sec²t = sup

−^π

2<t<^π₂

(− 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 (−^π₂,^π₂).

(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 f⁰(t) = sec²t

r , f⁰⁰(t) = 2 sec²t · tan t

r , and

sup

−^π₂<t<^π₂

−f⁰⁰(t)

[f⁰(t)]² = sup

−^π₂<t<^π₂

(−r sin 2t) = |r|.

This says that f (t) = tan t

r is |r|-convex.

(b) Similarly, from f⁰(t) = tan t

r , f⁰⁰(t) = sec²t r , and sup

−^π₂<t<^π₂

−f⁰⁰(t)

[f⁰(t)]² = sup

−^π₂<t<^π₂

(−r csc²t) = −r.

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

r is (−r)-convex.

Example 2.4. The function f (x) = ¹₂ln(kxk²+ 1) defined on R² is 1-convex.

For x = (s, t) ∈ R², and any real number r 6= 0, we consider the function φ(x, r) = ∇²f (x) + r∇f (x)∇f (x)^T

= 1

(kxk²+ 1)²

t²− s²+ 1 −2st

−2st s²− t²+ 1



+ r

(kxk²+ 1)²

s² st st t²



= 1

(kxk²+ 1)²

(r − 1)s²+ t²+ 1 (r − 2)st (r − 2)st s²+ (r − 1)t²+ 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)s²+ t² + 1 ≥ 0 (6)

(r − 1)s²+ t²+ 1 (r − 2)st (r − 2)st s² + (r − 1)t²+ 1

≥ 0. (7)

It is easy to verify the inequality (6) holds for all x ∈ R² if and only if r ≥ 1. Moreover, we note that

(r − 1)s²+ t²+ 1 (r − 2)st (r − 2)st s²+ (r − 1)t²+ 1

≥ 0

⇐⇒ s²t²+ s²+ t²+ 1 + (r − 1)²s²t²+ (r − 1)(s⁴+ s²+ t⁴+ t²) − (r − 2)²s²t² ≥ 0

⇐⇒ s² + t²+ 1 + (2r − 2)s²t²+ (r − 1)(s⁴ + s² + t⁴+ t²) ≥ 0,

and hence the inequality (7) holds for all x ∈ R² whenever r ≥ 1. Thus, we conclude by Property 1.1(b) that f is 1-convex on R².

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) = ¹₂ln(kxk²+ 1).

SOC-convex function means. For any x = (x₁, x₂) ∈ R×Rⁿ⁻¹and y = (y₁, y₂) ∈ R×Rⁿ⁻¹, we define their Jordan product as

x ◦ y = (x^Ty , y₁x₂+ x₁y₂).

We write x² to mean x ◦ x and write x + y to mean the usual componentwise addition of vectors. Then, ◦, +, together with e⁰ = (1, 0, . . . , 0)^T ∈ Rⁿ and for any x, y, z ∈ Rⁿ, the following basic properties [10, 11] hold: (1) e⁰ ◦ x = x, (2) x ◦ y = y ◦ x, (3) x ◦ (x²◦ y) = x²◦ (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 ∈ Rⁿ. Thus, we may, without loss of ambiguity, write x^m for the product of m copies of x and x^m+n = x^m◦ xⁿ for all positive integers m and n. Here, we set x⁰ = e⁰. Besides, Kⁿ is not closed under Jordan product.

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

x^1/2 = s,x₂

2s



, where s = s

1 2

 x₁+

q

x²₁− kx₂k²

 .

In the above formula, the term x₂/s is defined to be the zero vector if x₂ = 0 and s = 0, i.e., x = 0. For any x ∈ Rⁿ, we always have x² ∈ Kⁿ, i.e., x² Kⁿ 0. Hence, there exists a unique vector (x²)^1/2 ∈ Kⁿ denoted by |x|. It is easy to verify that

|x| Kⁿ 0 and x² = |x|² for any x ∈ Rⁿ. It is also known that |x| Kⁿ x. For any x ∈ Rⁿ, we define [x]₊ to be the nearest point projection of x onto Kⁿ, which is the same

definition as in Rⁿ+. In other words, [x]₊ is the optimal solution of the parametric SOCP:

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

see [10, 11].

Property 3.1. ([11, Proposition 3.3]) For any x = (x₁, x₂) ∈ R × Rⁿ⁻¹, we have (a) |x| = (x²)^1/2= |λ1|u⁽¹⁾x + |λ2|u⁽²⁾x .

(b) [x]₊ = [λ₁]₊u⁽¹⁾x + [λ₂]₊u⁽²⁾x = ¹₂(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 Kⁿ y =⇒ f^soc(x) Kⁿ 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) _Kⁿ (1 − λ)f^soc(x) + λf^soc(y) (8) for all x, y ∈ Rⁿ 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) ⊆ Rⁿ. Then, the following hold.

(a) f is SOC-monotone of order 2 if and only if f⁰(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⁾(t₁) f (t₂) − f (t₁) t2− t1

f (t₂) − f (t₁)

t₂− t₁ f⁽¹⁾(t₂)





 O for all t₁, t₂ ∈ J and t₁ 6= t₂.

Property 3.3. ([8, Theorem 4.1]) Let f ∈ C⁽²⁾(J ) with J being an open interval in R and dom(f^soc) ⊆ Rⁿ. 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⁾(t₀)[f (t0) − f (t) − f⁽¹⁾(t)(t0− t)]

(t₀− t)² ≥ [f (t) − f (t0) − f⁽¹⁾(t0)(t − t0)]

(t₀− t)⁴ (9)

holds for any t₀, t ∈ J and t₀ 6= t.

Property 3.4. ([4, Theorem 3.3.7]) Let f : S → R where S is a nonempty open convex set in Rⁿ. Suppose f ∈ C²(S). Then, f is convex if and only if ∇²f (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 = (x₁, x₂) ∈ R × Rⁿ⁻¹,

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

e^x1



cosh(kx2k), sinh(kx₂k)_kx^x2

2k



if x2 6= 0,

(e^x1, 0) if x₂ = 0,

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

(b) For any x = (x₁, x₂) ∈ int(Kⁿ), ln x is well-defined and

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

2



ln(x²₁− kx₂k²), ln

x1+kx2k x1−kx2k

 x2

kx2k



if x₂ 6= 0,

(ln x₁, 0) if x₂ = 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 = (x₁, x₂) ∈ R × Rⁿ⁻¹, y = (y₁, y₂) ∈ R × Rⁿ⁻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 x₁− y₁ ≥ kx₂k + ky₂k, then e^x Kⁿ e^y. In particular, if x ∈ Kⁿ, then e^x Kⁿ e^(0,0). Proof. (a) By applying Property 3.2(a), it is clear that f is SOC-monotone of order 2 since f⁰(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) Kⁿ 0.

But, we compute that e^x− e^y = e²



cosh(2), sinh(2)(1.2, −1.6) 2



− e⁻¹



cosh(4), sinh(4)(0, −4) 4



= 1

2(e⁴+ 1, .6(e⁴− 1), −.8(e⁴− 1)) − (e³+ e⁻⁵, 0, −e³+ e⁻⁵)

= (17.7529, 16.0794, −11.3999) ^Kn 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⁾(t₁) ^{f (t2}_t^{)−f (t1)}

2−t₁ f (t2)−f (t1)

t2−t₁ f⁽¹⁾(t₂)

#

= e^t1+t2 − e^t2 − e^t1 t₂− t₁

2

≥ 0 (10)

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

²

. Denote s := (t₂ − t₁)/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 + s²/6 + s⁴/120 + · · · > 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 Kⁿ e^y

⇐⇒ e^x1cosh(kx₂k) − e^y1cosh(ky₂k) ≥

e^x1sinh(kx₂k) x₂

kx₂k− e^y1sinh(ky₂k) y₂ ky₂k

⇐⇒ [e^x1cosh(kx₂k) − e^y1cosh(ky₂k)]²−

e^x1sinh(kx₂k) x₂

kx₂k− e^y1sinh(ky₂k) y₂ ky₂k

2

= e^2x1 + e^2y1 − 2e^x1+y1



cosh(kx₂k) cosh(ky₂k) − sinh(kx₂k) sinh(ky₂k) hx₂, y₂i kx₂kky₂k



≥ 0

⇐= e^2x1 + e^2y1 − 2e^x1+y1cosh(kx₂k + ky₂k) ≥ 0

⇐⇒ cosh(kx₂k + ky₂k) ≤ e^2x1 + e^2y1

2e^x1+y1 = e^x1−y1 + e^y1−x1

2 = cosh(x₁− y₁)

⇐⇒ x₁− y₁ ≥ kx₂k + ky₂k.

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)}

= e⁴

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

2e²(cosh(1), 0, − sinh(1)) + e⁶(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 ≺Kⁿ e^rfsoc(λx+(1−λ)y)Kⁿ w =⇒ rf^soc(λx + (1 − λ)y) Kⁿ ln(w)

is important and often needed. Note for x₂ 6= 0, we also have some observations as below.

(a) e^x Kⁿ 0 ⇐⇒ cosh(kx2k) ≥ | sinh(kx2k)| ⇐⇒ e^−kx2k > 0 . (b) 0 ≺_Kⁿ ln(x) ⇐⇒ ln(x²₁ − kx₂k²) >

  ln

x1+kx2k x1−kx2k

 

 ⇐⇒ ln(x₁ − kx₂k) > 0 ⇐⇒

x₁− kx₂k > 1. Hence (1, 0) ≺Kⁿ x implies 0 ≺Kⁿ 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 ⊆ Rⁿ → Rⁿ 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) _Kⁿ (1

rln λe^rfsoc(x)+ (1 − λ)e^rfsoc(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^tis 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) Kⁿ

1 r ln

λe^rfsoc(y)+ (1 − λ)e^{rfsoc(1−λλ (x−y))}

Kⁿ

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 Kⁿ 0 and Lemmas 3.1-3.3. Letting λ → 1, we obtain that f^soc(x) Kⁿ 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 ∈ Rⁿ and w ∈ int(Kⁿ). Then, the following hold.

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

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

(c) For n ≥ 2, if e^rz Kⁿ w, then rz Kⁿ 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 ∈ Rⁿ 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) Kⁿ

1

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

λx + (1 − λ)y Kⁿ

1

rln (λe^rx+ (1 − λ)e^ry)

⇐⇒ λrx + (1 − λ)ry Kⁿ ln (λe^rx+ (1 − λ)e^ry)

⇐⇒ eλrx+(1−λ)ry Kⁿ 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 ∈ Rⁿ and 0 ≤ λ ≤ 1, it can be verified that f^soc(λx + (1 − λ)y) Kⁿ λf^soc(x) + (1 − λ)f^soc(y)

_Kⁿ 1

rln λe^rfsoc(x)+ (1 − λ)e^rfsoc(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 ∈ Rⁿ and 0 ≤ λ ≤ 1, using that fact that ln t is SOC-monotone (Lemma 3.3) yields

e^rfsoc(λx+(1−λ)y) _Kⁿ λe^rfsoc(x)+ (1 − λ)e^rfsoc(y)

=⇒ rf^soc(λx + (1 − λ)y) Kⁿ ln λe^rfsoc(x)+ (1 − λ)e^rfsoc(y)

⇐⇒ f^soc(λx + (1 − λ)y) Kⁿ

1

rln λe^rfsoc(x)+ (1 − λ)e^rfsoc(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) ⊆ Rⁿ. 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) = t² is SOC-r-convex on R for r ≥ 0.

(b) The function f (t) = t³ 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/r², ∞) 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^rt2. Then, we have h⁰(t) = 2rte^rt2 and h⁰⁰(t) = (1 + 2rt²)2re^rt2. From Property 3.4, we know h is convex if and only if h⁰⁰(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 ⊆ Rⁿ → 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 _Kⁿ is not a linear ordering. Hence, it is not possible to compare any two vectors (elements) via Kⁿ. 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) Kⁿ 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) Kⁿ f^soc(y), f^soc(y) if f^soc(x) ≺Kⁿ f^soc(y),

1

2(f^soc(x) + f^soc(y) + |f^soc(x) − f^soc(y)|) if f^soc(x) − f^soc(y) /∈ Kⁿ∪ (−Kⁿ).

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

f^soc(λx + (1 − λ)y) Kⁿ 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 = (x₁, x₂) ∈ R × Rⁿ⁻¹, y = (y₁, y₂) ∈ R × Rⁿ⁻¹, and 0 ≤ λ ≤ 1, we have

f^soc(y) Kⁿ f^soc(x) ⇐⇒ (1 − λ)f^soc(y) Kⁿ (1 − λ)f^soc(x)

⇐⇒ λf^soc(x) + (1 − λ)f^soc(y) Kⁿ f^soc(x).

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

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

1

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

|x − y| Kⁿ x − y and |x − y| Kⁿ y − x, which respectively implies

1

2(x + y + |x − y|) Kⁿ x, (12)

1

2(x + y + |x − y|) Kⁿ 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 ∈ Rⁿ and 0 ≤ λ ≤ 1, it can be verified that

f^soc(λx + (1 − λ)y) _Kⁿ λf^soc(x) + (1 − λ)f^soc(y) _Kⁿ 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, t² on R, and t³, 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^tmake 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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