In particular, for any given r, we show examples which are r-convex functions

Volume 3, Number 3, 2017, 367–384

EXAMPLES OF r-CONVEX FUNCTIONS AND CHARACTERIZATIONS OF r-CONVEX FUNCTIONS

ASSOCIATED WITH SECOND-ORDER CONE

CHIEN-HAO HUANG, HONG-LIN HUANG, AND JEIN-SHAN CHEN^∗

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 functions are established. These generaliza- tions will be useful in dealing with optimization problems involved in second-order cones.

1. Introduction

It is known that the concept of convexity plays a central role in many appli- cations including mathematical economics, engineering, management science, and optimization theory. Moreover, much attention has been paid to its generaliza- tion, to the associated generalization 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 func- tions [7, 8]. Other further extensions can be found in [19, 23]. For a single variable continuous, the midpoint-convex function onR is also a convex function. 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 consequence 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, . . . , wm)∈ R^m, q = (q1, . . . , qm)∈ R^m be vectors

2010 Mathematics Subject Classification. 26A27, 26B05, 90C33.

Key words and phrases. r-Convex function, monotone function, second-order cone, spectral decomposition.

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

whose components are positive and nonnegative numbers, respectively, such that

∑_m

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]):

(1.1) M_r(w; q) = M_r(w₁, . . . , w_m; q) :=







 (∑_m

i=1

qi(wi)^r )_1/r

if r ̸= 0,

∏m i=1

(w_i)^qi if r = 0.

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

(1.2) M_s(w₁, . . . , w_m; q)≥ Mr(w₁, . . . , w_m; q) for all q₁, . . . , q_m ≥ 0 with ∑_m

i=1q_i = 1. The r-convexity is built based on the aforementioned 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)≤ ln{

Mr(e^{f (x)}, e^{f (y)}; q) }

.

From (1.1), it can be verified that the above inequality is equivalent to (1.3) f ((1− λ)x + λy) ≤

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

Similarly, f is said to be r-concave on S if the inequality (1.3) 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 (subconcave). 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 M_r(w₁,· · · , wm; q)≡ M_−∞(w₁, . . . , w_m) = min{w1,· · · , wm}.

Then, it follows from (1.2) that M_∞(w₁, . . . , w_m) ≥ Mr(w₁, . . . , w_m; q) ≥ M_−∞(w1, . . . , wm) 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 (quasiconcave) 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 ((^r_k)-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) = {

ln[

α₁e^rϕ(x)+ α₂e^rψ(x)]1/r

if r ̸= 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) onR 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

(1.4) r^∗ = sup

x∈S, ∥z∥=1

−z^T∇²ϕ(x)z [z^T∇ϕ(x)]²

whenever z^T∇ϕ(x) ̸= 0, then ϕ is r-convex for every r ≥ r^∗. We obtain the r-concave analog of the above theorem by replacing supremum in (1.4) by infimum.

In this paper, we will present new examples of r-convex functions in Section 2.

Meanwhile, 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 optimization 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 deal- ing optimization problems involved second-order constraints. We point out that extending the concepts of r-convex and quasi-convex functions to the setting asso- ciated with second-order cone, which belongs 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 details 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| ∥x2∥ ≤ 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 ∈ Kⁿand 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 com- plementarity 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, λ₂and u⁽¹⁾_x , u⁽²⁾_x are the spectral values and the associated spectral vectors of x, respectively, given by

λ_i = x₁+ (−1)ⁱ∥x2∥, u⁽ⁱ⁾_x =

{ ₁

2(1, (−1)^{i x}_∥x²_2∥) if x₂̸= 0,

1

2(1, (−1)ⁱw) if x₂= 0.

for i = 1, 2 with w being any vector in Rⁿ⁻¹ satisfying ∥w∥ = 1. If x2 ̸= 0, the decomposition is unique.

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

(1.5) f^soc(x) = f (λ₁)u⁽¹⁾_x + f (λ₂)u⁽²⁾_x , ∀x = (x1, x₂)∈ R × Rⁿ⁻¹.

If f is defined only on a subset ofR, then f^socis defined on the corresponding subset of Rⁿ. The definition (1.5) is unambiguous whether x₂̸= 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 (1.5) is analogous to the one associated with positive semidefinite cone S₊ⁿ [20, 21].

Throughout this paper, Rⁿ denotes the space of n-dimensional real column vec- tors, C denotes a convex subset of R, S denotes a convex subset of Rⁿ, and ⟨· , ·⟩

means the Euclidean inner product, whereas∥ · ∥ is the Euclidean norm. The nota- tion “:=” 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 toR 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 examples 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^′(t) = pt^p−1, f^′′(t) = p(p− 1)t^p−2 and sup

s·f^′(t)̸=0,|s|=1

−s · f^′′(t)· s [s· f^′(t)]² = sup

p̸=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 f^′(t) = cos t, f^′′(t) =− sin t, and

−^π₂<t<sup^π₂,|s|=1

−s · f^′′(t)· s

[s· f^′(t)]² = sup

−^π₂<t<^π₂

sin t cos²t =∞.

Hence f (t) = sin t is∞-convex.

To see (b), we note that f^′(t) = sec²t, f^′′(t) = 2 sec²t· tan t, and sup

−^π₂<t<^π₂

−f^′′(t)

[f^′(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 f^′(t) = tan t, f^′′(t) = sec²t, and sup

−^π2<t<^π₂

−f^′′(t)

[f^′(t)]² = sup

−^π2<t<^π₂

−k sec²t

tan²t = sup

−^π2<t<^π₂

(− csc²t) =−1.

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

To see (d), we note that f^′(t) = sec t, f^′′(t) = sec t· tan t, and sup

−^π₂<t<^π₂

−f^′′(t)

[f^′(t)]² = sup

−^π₂<t<^π₂

− sec t · tan t

sec²t = sup

−^π₂<t<^π₂

(− sin t) = 1.

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

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

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 r̸= 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 f^′(t) = sec²t

r , f^′′(t) = 2 sec²t· tan t

r , and

−^π₂sup<t<^π₂

−f^′′(t)

[f^′(t)]² = sup

−^π₂<t<^π₂

(−r sin 2t) = |r|.

This says that f (t) = tan t

r is |r|-convex.

(b) Similarly, from f^′(t) = tan t

r , f^′′(t) = sec²t r , and sup

−^π₂<t<^π₂

−f^′′(t)

[f^′(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(∥x∥²+ 1) defined onR² is 1-convex.

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

= 1

(∥x∥²+ 1)²

[t²− s²+ 1 −2st

−2st s²− t²+ 1 ]

+ r

(∥x∥²+ 1)²

[s² st st t² ]

= 1

(∥x∥²+ 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 semidefinite, which is equivalent to

(r− 1)s²+ t²+ 1≥ 0 (2.1)

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

 ≥ 0.

(2.2)

It is easy to verify the inequality (2.1) 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 (2.2) holds for all x ∈ R² whenever r ≥ 1. Thus, we conclude by Property 1.1(b) that f is 1-convex onR².

Figure 2. Graphs of 1-convex functions f (x) = ¹₂ln(∥x∥²+ 1).

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 SOC-convex function means. For any x = (x1, x2)∈ R × Rⁿ⁻¹ and 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 e^′ = (1, 0, . . . , 0)^T ∈ Rⁿ and for any x, y, z ∈ Rⁿ, the following basic properties [10, 11] hold: (1) e^′ ◦ 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 = e^′. 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 =

√ 1 2

( x₁+

√

x²₁− ∥x2∥² )

.

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{∥x − y∥ | 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 3.2. 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 (1.5) 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 (1.5) satisfies

(3.1) f^soc((1− λ)x + λy) ≼_Kⁿ (1− λ)f^soc(x) + λf^soc(y)

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 (3.1) 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 ma- trix 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 combi- nations and under pointwise limits.

Property 3.3 ([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 f^′(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 ̸= t2.

Property 3.4 ([8, Theorem 4.1]). Let f ∈ C⁽²⁾(J ) with J being an open interval inR 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 (3.2) 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 holds for any t0, t∈ J and t0 ̸= t.

Property 3.5 ([4, Theorem 3.3.7]). Let f : S → R where S is a nonempty open convex set inRⁿ. Suppose f ∈ C²(S). Then, f is convex if and only if∇²f (x)≽ O, for all x∈ S.

Property 3.6 ([7, Proposition 4.1]). Let f : [0,∞] → [0, ∞] be continuous. If f is SOC-concave, then f is SOC-monotone.

Property 3.7 ( [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(∥x2∥), sinh(∥x2∥)_∥x^x2_2∥)

if x2 ̸= 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− ∥x2∥²), ln

(x1+∥x²∥ x1−∥x2∥

) x2

∥x2∥

)

if x2̸= 0,

(ln x1, 0) if x2= 0.

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

Lemma 3.8. 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 ≥ ∥x2∥ + ∥y2∥, then e^x ≽_Kⁿ e^y. In particular, if x ∈ Kⁿ, then e^x ≽Kⁿ e^(0,0).

Proof. (a) By applying Property 3.3(a), it is clear that f is SOC-monotone of order 2 since f^′(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 ∥(16.0794, −11.3999)∥ = 19.7105 > 17.7529.

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

(3.3) 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

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 ̸= 0. Hence, (3.3) does not hold.

Then, applying Property 3.3(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(∥x2∥) − e^y1cosh(∥y2∥)

≥

e^x1sinh(∥x2∥) x2

∥x2∥ − e^y1sinh(∥y2∥) y2

∥y2∥

⇐⇒ [e^x1cosh(∥x2∥) − e^y1cosh(∥y2∥)]²

−

e^x1sinh(∥x2∥) x2

∥x2∥− e^y1sinh(∥y2∥) y2

∥y2∥ ²

= e^2x1 + e^2y1

−2e^x1+y1 [

cosh(∥x2∥) cosh(∥y2∥) − sinh(∥x2∥) sinh(∥y2∥) ⟨x2, y2⟩

∥x2∥∥y2∥ ]

≥ 0

⇐= e^2x1+ e^2y1 − 2e^x1+y1cosh(∥x2∥ + ∥y2∥) ≥ 0

⇐⇒ cosh(∥x2∥ + ∥y2∥) ≤ e^2x1 + e^2y1

2e^x1+y1 = e^x1−y1 + e^y1−x1

2 = cosh(x1− y1)

⇐⇒ x1− y1 ≥ ∥x2∥ + ∥y2∥.



Lemma 3.9. 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 3.4 (a), it is clear that f is SOC-convex since exponential function is a convex function onR.

(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

2

[e²(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

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

Lemma 3.10 ([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)}≼_Kⁿw =⇒ rf^soc(λx + (1− λ)y) ≼_Kⁿ ln(w) is important and often needed. Note for x2 ̸= 0, we also have some observations as below.

(a) e^x ≻_Kⁿ 0 ⇐⇒ cosh(∥x2∥) ≥ | sinh(∥x2∥)| ⇐⇒ e^−∥x2∥ > 0 . (b) 0 ≺Kⁿ ln(x) ⇐⇒ ln(x²1− ∥x2∥²) > ln(

x1+∥x2∥

x1−∥x²∥) ⇐⇒ ln(x1 − ∥x2∥) >

0 ⇐⇒ x1− ∥x2∥ > 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→ Rⁿbe its corresponding SOC-function defined as in (1.5). 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

(4.1) f^soc(λx + (1− λ)y) ≼_Kⁿ {1

rln(

λe^rfsoc(x)+ (1− λ)e^rfsoc(y))

r ̸= 0, λf^soc(x) + (1− λ)f^soc(y) r = 0.

Similarly, f is said to be SOC-r-concave of order n on C if the inequality (4.1) 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 ̸= 0 if and only if e^rϕis 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 3.9. This is a hurdle to build parallel result for general n in the setting of SOC case. As seen in Proposition 4.5, 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 4.2. 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₍₁^−λ)λ_−λ) (x− y). If r = 0, then f is SOC-concave and SOC-monotone by Property 3.6. 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 3.8-3.10. Letting λ→ 1, we obtain that f^soc(x) ≽_Kⁿ f^soc(y), which says that f is SOC-monotone.



In fact, in light of Lemma 3.8-3.10, we have the following Lemma which is useful for subsequent analysis.

Lemma 4.3. 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 4.4. 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

r ln (λe^rx+ (1− λ)e^ry)

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

⇐⇒ e^{λrx+(1−λ)ry}≼_Kⁿ λe^rx+ (1− λ)e^ry,

where the first “⇐⇒” is true due to Lemma 4.3, whereas the second “⇐⇒” holds because e^t and ln t are SOC-monotone of order 2 by Lemma 3.8 and Lemma 3.10.

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. 

Proposition 4.5. 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 3.10) yields

e^{rfsoc(λx+(1−λ)y)}≼_Kⁿλe^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. 

Combining with Property 3.4, we can characterize the SOC-r-convexity as follows.

Proposition 4.6. Let f ∈ C⁽²⁾(J ) with J being an open interval inR 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 (3.2).

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 4.7. For n = 2, the following hold.

(a) The function f (t) = t² is SOC-r-convex onR 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 h^′(t) = 2rte^rt2 and h^′′(t) = (1 + 2rt²)2re^rt2. From Property 3.5, we know h is convex if and only if h^′′(t)≥ 0.

Thus, the desired result holds by applying Property 3.4 and Proposition 4.5. The arguments for other cases are similar and we omit them. 

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∈ Rⁿ, 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 5.2. 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)≼Kⁿf^soc(x) ⇐⇒ (1 − λ)f^soc(y)≼Kⁿ (1− λ)f^soc(x)

⇐⇒ λf^soc(x) + (1− λ)f^soc(y)≼_Kⁿf^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

λf^soc(x) + (1− λ)f^soc(y)≼_Kⁿ 1

2(f^soc(x) + f^soc(y) +|f^soc(x)− f^soc(y)|) .

To this end, we note that

|f^soc(x)− f^soc(y)| ≽_Kⁿf^soc(x)− f^soc(y) and

|f^soc(x)− f^soc(y)| ≽_Kⁿ f^soc(y)− f^soc(x), which respectively implies

1

2(f^soc(x) + f^soc(y) +|f^soc(x)− f^soc(y)|) ≽_Kⁿ x, (5.1)

1

2(f^soc(x) + f^soc(y) +|f^soc(x)− f^soc(y)|) ≽Kⁿ y.

(5.2)

Then, adding up (5.1) ×λ and (5.2) ×(1 − λ) yields the desired result.  Proposition 5.3. 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 5.2. Thus, the desired result follows. 

From Proposition 5.3, 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 associated with SOC which will be helpful in dealing with optimization problems involved in second-order cones. In particular, we obtain some characteri- zations 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.

Manuscript received revised