4 SOC-r-convex functions

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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.

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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, ..., wm) ∈ Rm, q = (q1, ..., qm) ∈ Rm be vectors whose components are positive and nonnegative numbers, respectively, such that Pm

i=1qi = 1. Given the vector of weights q, the weighted r-mean of the numbers w1, ..., wm is defined as below (see [13]):

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





m P

i=1

qi(wi)r

1/r

if r 6= 0,

m

Q

i=1

(wi)qi if r = 0.

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

Ms(w1, ..., wm; q) ≥ Mr(w1, ..., wm; 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(ef (x), ef (y); q) . From (1), it can be verified that the above inequality is equivalent to

f ((1 − λ)x + λy) ≤ ln[(1 − λ)erf (x)+ λerf (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

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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 erf 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 Mr(w1, ..., wm; q) ≡ M(w1, ..., wm) = max{w1, ..., wm},

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

Then, it follows from (2) that M(w1, ..., wm) ≥ Mr(w1, ..., wm; 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 (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 (kr)-convex ((kr)-concave).

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(f ) Let φ, ψ : S ⊆ Rn → R be r-convex (r-concave) and α1, α2 > 0. Then, the function θ defined by

θ(x) = (

lnα1erφ(x)+ α2erψ(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

−zT2φ(x)z

[zT∇φ(x)]2 (4)

whenever zT∇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 .

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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

λi = x1+ (−1)ikx2k, u(i)x =

( 1

2(1, (−1)i xkx2

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]:

fsoc(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 fsoc is defined on the corresponding subset of Rn. The definition (5) is unambiguous whether x2 6= 0 or x2 = 0. The cases of fsoc(x) = x1/2, 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

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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) = tp. (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) = ptp−1, f00(t) = p(p − 1)tp−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.

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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|.

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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

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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 = (xTy , 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 xm for the product of m copies of x and xm+n = xm◦ 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 x1/2 such that (x1/2)2 = x1/2◦ x1/2 = x. Indeed,

x1/2 = 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

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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 fsoc defined as in (5) satisfies

x Kn y =⇒ fsoc(x) Kn fsoc(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 fsoc defined as in (5) satisfies

fsoc((1 − λ)x + λy) Kn (1 − λ)fsoc(x) + λfsoc(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(fsoc) ⊆ Rn. Then, the following hold.

(a) f is SOC-monotone of order 2 if and only if f0(τ ) ≥ 0 for any τ ∈ J ;

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(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(fsoc) ⊆ 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) = et and g(t) = ln t. Then, the corresponding SOC-functions of et and ln t are given as below.

(a) For any x = (x1, x2) ∈ R × Rn−1,

fsoc(x) = ex = (

ex1



cosh(kx2k), sinh(kx2k)kxx2

2k



if x2 6= 0,

(ex1, 0) if x2 = 0,

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

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

gsoc(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.

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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) = et 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 ex Kn ey. In particular, if x ∈ Kn, then ex 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 ) = eτ ≥ 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 ex− ey = 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 (t2t)−f (t1)

2−t1 f (t2)−f (t1)

t2−t1 f(1)(t2)

#

= et1+t2 − et2 − et1 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.

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

ex Kn ey

⇐⇒ ex1cosh(kx2k) − ey1cosh(ky2k) ≥

ex1sinh(kx2k) x2

kx2k− ey1sinh(ky2k) y2 ky2k

⇐⇒ [ex1cosh(kx2k) − ey1cosh(ky2k)]2

ex1sinh(kx2k) x2

kx2k− ey1sinh(ky2k) y2 ky2k

2

= e2x1 + e2y1 − 2ex1+y1



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



≥ 0

⇐= e2x1 + e2y1 − 2ex1+y1cosh(kx2k + ky2k) ≥ 0

⇐⇒ cosh(kx2k + ky2k) ≤ e2x1 + e2y1

2ex1+y1 = ex1−y1 + ey1−x1

2 = cosh(x1− y1)

⇐⇒ x1− y1 ≥ kx2k + ky2k.

2

Lemma 3.2. Let f (t) = et 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) = et 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

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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 et(as shown in Proposition 3.1), we observe that the following fact

0 ≺Kn erfsoc(λx+(1−λ)y)Kn w =⇒ rfsoc(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) ex 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 fsoc : 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

fsoc(λx + (1 − λ)y) Kn (1

rln λerfsoc(x)+ (1 − λ)erfsoc(y)

r 6= 0,

λfsoc(x) + (1 − λ)fsoc(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 is convex whenever r > 0 and concave whenever r < 0. However, we observe that the exponential function etis 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].

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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

fsoc(λx) Kn

1 r ln

λerfsoc(y)+ (1 − λ)erfsoc(1−λλ (x−y))

Kn

1

r ln λer(0,0)+ (1 − λ)er(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 fsoc(x) Kn fsoc(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) ⇐⇒ erz Kn w.

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

(c) For n ≥ 2, if erz 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 fsoc(x) = x. Therefore, to prove the desired result, we need to verify that

fsoc(λx + (1 − λ)y) Kn

1

rln λerfsoc(x)+ (1 − λ)erfsoc(y) . To this end, we see that

λx + (1 − λ)y Kn

1

rln (λerx+ (1 − λ)ery)

⇐⇒ λrx + (1 − λ)ry Kn ln (λerx+ (1 − λ)ery)

⇐⇒ eλrx+(1−λ)ry Kn λerx+ (1 − λ)ery,

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where the first “⇐⇒” is true due to Lemma 4.1, whereas the second “⇐⇒” holds because et and ln t are SOC-monotone of order 2 by Lemma 3.1 and Lemma 3.3. Then, using the fact that et 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 fsoc(λx + (1 − λ)y) Kn λfsoc(x) + (1 − λ)fsoc(y)

Kn 1

rln λerfsoc(x)+ (1 − λ)erfsoc(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 erf 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 erf 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

erfsoc(λx+(1−λ)y) Kn λerfsoc(x)+ (1 − λ)erfsoc(y)

=⇒ rfsoc(λx + (1 − λ)y) Kn ln λerfsoc(x)+ (1 − λ)erfsoc(y)

⇐⇒ fsoc(λx + (1 − λ)y) Kn

1

rln λerfsoc(x)+ (1 − λ)erfsoc(y) .

When n = 2, et 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(fsoc) ⊆ Rn. Then, for r > 0, the followings hold.

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

(b) f is SOC-r-convex of order n ≥ 3 if erf 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.

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(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) := ert2. Then, we have h0(t) = 2rtert2 and h00(t) = (1 + 2rt2)2rert2. 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

fsoc(λx + (1 − λ)y) Kn fsoc(y) + [fsoc(x) − fsoc(y)]+

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where

fsoc(y) + [fsoc(x) − fsoc(y)]+

=

fsoc(x) if fsoc(x) Kn fsoc(y), fsoc(y) if fsoc(x) ≺Kn fsoc(y),

1

2(fsoc(x) + fsoc(y) + |fsoc(x) − fsoc(y)|) if fsoc(x) − fsoc(y) /∈ Kn∪ (−Kn).

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

fsoc(λx + (1 − λ)y) Kn fsoc(x) − [fsoc(x) − fsoc(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

fsoc(y) Kn fsoc(x) ⇐⇒ (1 − λ)fsoc(y) Kn (1 − λ)fsoc(x)

⇐⇒ λfsoc(x) + (1 − λ)fsoc(y) Kn fsoc(x).

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

Hence, it remains to consider the case of fsoc(x) − fsoc(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.

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Proof. For any x, y ∈ Rn and 0 ≤ λ ≤ 1, it can be verified that

fsoc(λx + (1 − λ)y) Kn λfsoc(x) + (1 − λ)fsoc(y) Kn fsoc(y) + [fsoc(x) − fsoc(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, 1t, t1/2 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 et. However, for n > 2, the expressions of ex 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 etmake things much easier than the general case n ≥ 3. To conquer this difficulty, we believe that we have to derive more properties of ex. 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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