Some characterizations for SOC-monotone and SOC-convex functions

Transcription

1 J Glob Optim :59 79 DOI 0.007/s z Some characterizations for SOC-monotone and SOC-convex functions Jein-Shan Chen Xin Chen Shaohua Pan Jiawei Zhang Received: 9 June 007 / Accepted: October 008 / Published online: 7 November 008 Springer Science+Business Media, LLC. 008 Abstract We provide some characterizations for SOC-monotone and SOC-convex functions by using differential analysis. From these characterizations, we particularly obtain that a continuously differentiable function defined in an open interval is SOC-monotone SOC-convex of order n 3 if and only if it is -matrix monotone matrix convex, and furthermore, such a function is also SOC-monotone SOC-convex of order n ifitis -matrix monotone matrix convex. In addition, we also prove that Conjecture 4. proposed in Chen Optimization 55: , 006 does not hold in general. Some examples are included to illustrate that these characterizations open convenient ways to verify the SOCmonotonicity and the SOC-convexity of a continuously differentiable function defined on an open interval, which are often involved in the solution methods of the convex second-order cone optimization. Keywords Second-order cone SOC-monotone function SOC-convex function Mathematics Subject Classification 000 6A48 6A5 6B05 90C5 J.-S. Chen B Department of Mathematics, National Taiwan Normal University, Taipei 677, Taiwan jschen@math.ntnu.edu.tw X. Chen Department of Industrial and Enterprise System Engineering, University of Illinois at Urbana Champaign, Urbana 680, IL, USA xinchen@uiuc.edu S. Pan School of Mathematical Sciences, South China University of Technology, Guangzhou 50640, China shhpan@scut.edu.cn J. Zhang Department of Information, Operations and Management Sciences, New York University, New York 00-6, NY, USA jzhang@stern.nyu.edu 3

2 60 J Glob Optim :59 79 Introduction The second-order cone SOC in IR n, also called the Lorentz cone, is a set defined by K n := { x, x IR IR n x x }, where denotes the Euclidean norm, and K denotes the set of nonnegative reals IR +.It is known that K n is a closed convex self-dual cone with nonempty interior intk n.forany x, y IR n, we write x K n y if x y K n ; and write x K n y if x y intk n.inother words, we have x K n 0 if and only if x K n and x K n 0 if and only if x intk n.the relation K n is a partial ordering, but not a linear ordering in K n, i.e., there exist x, y K n such that neither x K n y nor y K n x. To see this, let x =,, y =, 0, and then we have x y = 0, / K, y x = 0, / K. For any x = x, x, y = y, y IR IR n,wedefinetheirjordan product as x y = x, y, y x + x y. we write x to mean x x and write x+y to mean the usual componentwise addition of vectors. Then, +,ande =, 0,...,0 T IR n have the following basic properties see [7,8]: e x = x for all x IR n.x y = y x for all x, y IR n.3x x y = x x y for all x, y IR n.4x + y z = x z + y z for all x, y, z IR n. Note that the Jordan product is not associative. Besides, K n is not closed under Jordan product. We recall from [7,8] that each x = x, x IR IR n admits a spectral factorization, associated with K n, of the form x = λ x u x + λ x u x, 3 where λ x, λ x and u x, u x are the spectral values and the associated spectral vectors of x given by λ i x = x + i x, u i x =, i x for i =,, 4 with x = x x if x = 0 and otherwise x being any vector in IR n such that x =. If x = 0, the factorization is unique. By the spectral factorization, for any f : IR IR, we can define a vector-valued function associated with K n n by f soc x = f λ xu x + f λ xu x, x = x, x IR IR n, 5 and call it the SOC-function induced by f.if f is defined only on a subset of IR, then f soc is defined on the corresponding subset of IR n. The definition is unambiguous whether x = 0 or x = 0. The cases of f soc x = x /, x, expx were discussed in [7]. In fact, the above definition 5 is analogous to one associated with the semidefinite cone; see [9,0]. Recently, the concepts of SOC-monotone and SOC-convex functions are introduced in [5]. Especially, a function f : J IR with J IR is said to be SOC-monotone of order n if x K n y f soc x K n f soc y 6 for any x, y dom f soc IR n, where dom f soc denotes the domain of the function f soc ;and f is said to be SOC-convex of order n if, for any x, y dom f soc, 3 f soc λx + λy K n λf soc x + λ f soc y λ [0, ]. 7

3 J Glob Optim : The function f is said to be SOC-monotone respectively, SOC-convex if it is SOCmonotone of all order n respectively, SOC-convex of all order n, and f is SOC-convex on J if and only if f is SOC-concave on J. The concepts of SOC-monotone and SOC-convex functions are analogous to matrix monotone and matrix convex functions [,0,,4], and are special cases of operator monotone and operator convex functions [,3,]. For example, the function f is said to be n-matrix convex on J if f λa + λb λf A + λ f B λ [0, ] for arbitrary Hermitian n n matrices A and B with spectra in J. It is clear that the set of SOC-monotone functions and the set of SOC-convex functions are closed under positive linear combinations and pointwise limits. There has been systematic study on matrix monotone and matrix convex functions, and moreover, characterizations for such functions have been explored; see [4,0,,3,4] and the references therein. To the contrast, the study on SOC-monotone and SOC-convex functions just starts its first step. One reason is that they were viewed as special cases of operator monotone and operator convex functions. However, we recently observed that SOC-monotone and SOC-convex functions play an important role in the design of solutions methods for convex second-order cone programs SOCPs; for example, the proximal-like methods in [5] and the augmented Lagrangian method introduced in Sect. 5. On the other hand, we all know that the developments of matrix-valued functions have major contributions in the solution of optimization problems. Thus, we hope similar systematic study on SOC-functions can be exploited so that it can be readily adopted to optimization field. This is the main motivation of the paper. Although some work was done in [5] for SOC-monotone and SOC-convex functions, the focus there is to provide some specific examples by the definition and it seems difficult to exploit the characterizations there to verify whether a given function is SOC-convex or not. In this paper, we employ differential analysis to establish some useful characterizations which will open convenient ways to verify the SOC-monotonicity and the SOC-convexity of a function defined on an open interval. Particularly, from these characterizations, we obtain that a continuously differentiable function defined on an open interval is SOC-monotone SOCconvexofordern 3 if and only if it is -matrix monotone matrix convex, and such a function is also SOC-monotone SOC-convex of order n if it is -matrix monotone matrix convex. Thus, if such functions are -matrix monotone matrix convex, then it must be SOC-monotone SOC-convex. It should be pointed out that the analysis of this paper can not be obtained from those for matrix-valued functions. One of the reasons is that the matrix multiplication is associative whereas the Jordan product is not. Throughout the paper,, denotes the Euclidean inner product, IR n denotes the space of n-dimensional real column vectors, and IR n IR n m is identified with IR n + +n m. Thus, x,...,x m IR n IR n m is viewed as a column vector in IR n + +n m. Also, I represents an identity matrix of suitable dimension; J is a subset of IR; and 0 is a zero matrix or vector of suitable dimension. The notation T means transpose and C i J denotes the family of functions which are defined on J IR to IR and have the i-th continuous derivative. For a function f : IR IR, f i x represents the i-th order derivative of f at x IR, and the first-order and the second-order derivative of f are also written as f and f, respectively. For any f : IR n IR, f x denotes the gradient of f at x IR n and dom f denotes the domain of f. For any differentiable mapping F = F,...,F m T : IR n IR m, Fx =[ F x F m x] is an n m matrix which denotes the transposed Jacobian of F at x. For any symmetric matrices A, B IR n n, we write A B respectively, A B to mean A B is positive semidefinite respectively, positive definite. 3

4 6 J Glob Optim :59 79 Preliminaries In this section, we develop the second-order Taylor s expansion for the vector-valued SOCfunction f soc defined as in 5 which is crucial in our subsequent analysis. To the end, we assume that f C J with J being an open interval in IR and dom f soc IR n. Given any x dom f soc and h = h, h IR IR n,wehavex + th dom f soc for any sufficiently small t > 0. We wish to calculate the Taylor s expansion of the function f soc x + th at x for any sufficiently small t > 0. In particular, we are interested in finding matrices f soc x and A i x for i =,,...,n such that h T A xh f soc x + th = f soc x + t f soc xh + h T A xh t. + ot. 8 h T A n xh For convenience, we omit the variable notion x in λ i x for i =, in the discussions below. It is known that f soc is differentiable respectively, smooth if and only if f is differentiable respectively, smooth; see [6,8]. Moreover, there holds that if x = 0, and otherwise where f soc x = b c x T x c x x a0 I + b a 0 x x T x 9 f soc x = f x I 0 a 0 = f λ f λ, b = f λ + f λ, c = f λ f λ. λ λ Therefore, we only need to derive the formula of A i x for i =,,...,n in 8. We first consider the case where x = 0andx + th = 0. By the definition 5, f soc x + th = f x + th x + th x + th x + th + f x + th + x + th x + th x + th f x + th x + th + f x + th + x + th = f x + th + x + th f x + th x + th 3 := [ ]. x + th x + th

5 J Glob Optim : To derive the Taylor s expansion of f soc x + th at x with x = 0, we first write out and expand x + th.noticethat x + th = x + tx T h + t h = x + t x T h x + h t x. Therefore, using the fact that + ɛ = + ɛ 8 ɛ + oɛ, we may obtain where with x + th = x + t α x + β t x + ot, 3 α = x T h x, β = h x T h x = h α = h T M x h, M x = I x x T x. Furthermore, from 3 and the fact that + ɛ = ɛ + ɛ + oɛ, it follows that x + th = x α t x + t α x β x + ot. 4 Combining Eqs. 3and4 then yields that x + th x + th = x x + t h x α x x x + t α x β x x x h α + ot x x = x x + tm h x x + t 3 ht x x T h x x 4 x h x x x h h T x x + ot. 5 x In addition, from 3, we have the following equalities f x + th x + th α = f x + th x + t x + β t x + ot = f λ + th α β t x + ot = f λ + tf λ h α + t f β λ x + f λ h α + ot 6 3

6 64 J Glob Optim :59 79 and f x + th + x + th = f λ + th + α + β t x + ot = f λ + tf λ h + α + t f β λ x + f λ h + α + ot. 7 For i = 0,,, we define a i = f i λ f i λ, b i = f i λ + f i λ, c i = f i λ f i λ, λ λ 8 where f i means the i-th derivative of f and f 0 is the same as the original f. Then, by the Eqs. 6 8, it can be verified that = f x + th + x + th + f x + th x + th = b 0 + t b h + c α + t a β + b h + α + c h α + ot = b 0 + t b h + c h T x + x t h T A xh + ot, where A x = b c x T x c x x a I + b a x x T x. 9 Note that in the above expression for, b 0 is exactly the first component of f soc x and b h + c h T x x is the first component of f soc xh. Using the same techniques again, f x + th + x + th f x + th x + th = c 0 + t c h + b α + t b β x + c h + α + b h α + ot = c 0 + t c h + b α + t h T Bxh + ot, 0 where Bx = c b x T x b x x c I + b x c M x. Using Eqs. 5and0, we obtain that = x + th f x + th + x + th f x + th x + th x + th = c 0 x x + t x x c h + b α + c 0 h M x + x t W + ot, 3

7 J Glob Optim : where W = x h x ht Bxh + M x c h + b α x +c 0 3 ht x x T h x x 4 x h x x x h h T x x. x Now we denote where d := b a 0 x V := c h + b α x Cx := Then U can be further recast as = b a 0 λ λ, U := h T Cxh c 0 x T h x 3 = a h + d x T h x, c b a x T x b a x x di + c 3d x x T x. U = h T Bxh + c 0 3 ht x x T h x 4 c 0 h x x T h x c h + b α. Consequently, W = x x U + h V. We next consider the case where x = 0andx + th = 0. By definition 5, f soc x + th = f x + th h h + f x + th + h h f x + th h + f x + th + h = f x + th + h f x + th h h h Using the Taylor expansion of f at x, we can obtain that [ ] f x + th h + f x + th + h = f x + tf x h + t f x h T h + ot, [ ] f x + th h f x + th + h = tf x h + t f x h h + ot. h h. 3 Therefore, f soc x + th = f soc x + tf x h + [ t f h x T ] h. 4 h h 3

8 66 J Glob Optim :59 79 Thus, under this case, we have that A x = f x I, A i x = f x 0 ē T i ē i 0 i =,...,n, 5 where e j IR n is the vector whose jth component is and the others are 0. Summing up the above discussions, we may obtain the following conclusion. Proposition. Let f C J with J being an open interval in IR and dom f soc IR n. Then, for given x dom f soc,h IR n and any sufficiently small t > 0, h T A xh f soc x + th = f soc x + t f soc xh + h T A xh t. + ot, h T A n xh where f soc x and A i x, i =,,...,n are given by 0 and 5 ifx = 0; and otherwise f soc x and A x are given by 9 and 9, respectively, and for i, where A i x = Cx x i x + B i x [ B i x = vei T + e i v T, v = a d x ] T T. x From Proposition 4.3 of [5] and Proposition., we readily have the following result. Proposition. Let f C J with J being an open interval in IR and dom f soc IR n. Then, f is SOC-convex if and only if for any x dom f soc and h IR n, the vector h T A xh h T A xh. Kn. h T A n xh 3 Characterizations of SOC-monotone functions Now we are ready to show our main result concerning the characterization of SOC-monotone functions. We need the following technical lemmas for the proof. The first one is so-called S-Lemma whose proof can be found in [6,8]. Lemma 3. Let A, B be symmetric matrices and y T Ay > 0 for some y. Then, the implication [ z T Az 0 z T Bz 0 ] is valid if and only if B λa forsomeλ 0. Lemma 3. Given θ IR,a IR n, and a symmetric matrix A IR n n.letb n := {z IR n z }. Then, the following results hold: [ ] a For any h K n,ah K n is equivalent to A K z n for any z B n. b For any z B n, θ + a T z 0 is equivalent to θ a. 3

9 J Glob Optim : [ ] θ a T c If A = with H being an n n symmetric matrix, then for any a H h K n,ah K n is equivalent to θ a and there exists λ 0 such that the matrix [ θ a λ θa T a T ] H θa H T a aa T H T O. H + λi [ ] Proof a For any h K n, suppose that Ah K n.leth = where z B z n.then h K n and the desired result follows. For the other direction, if h = 0, the conclusion is obvious. Now let h := h, h be any nonzero vector in K n. Then, h > 0and h h. Consequently, h B n and A h K n.sincek n isacone,we h h have h A h = Ah K n. h b For z B n, suppose θ + a T z 0. If a = 0, then the result is clear since θ 0. If a = 0, let z := a/ a. Clearly, z B n and hence θ + at a 0 which gives a θ a 0. For the other direction, the result follows from the Cauchy Schwarz inequality: c and θ + a T z θ a z θ a 0. [ ] From part a, Ah K n for any h K n is equivalent to A K z n for any z B n. Notice that [ ] [ ][ ] [ θ a T θ + a A = = T ] z. z a H z a + Hz Then, Ah K n for any h K n is equivalent to the following two things: θ + a T z 0, for any z B n 6 a + Hz T a + Hz θ + a T z, for any z B n. 7 By part b, 6 is equivalent to θ a. Now, we write the expression of 7asbelow: z T aa T H T Hz + θa T a T Hz + θ a T a 0, for any z B n, which can be further simplified as [ z T ] [ θ a θa T a T H θa H T aaa T H T H ][ ] 0, for any z B n. z Observe that z B n is the same as [ z T ] [ 0 0 I ][ ] 0. z 3

10 68 J Glob Optim :59 79 Thus, by applying the S-Lemma Lemma 3., there exists λ 0 such that [ θ a θa T a T ] [ ] H 0 θa H T aaa T H T λ O H 0 I This completes the proof of part c. Theorem 3. Let f C J with J being an open interval and dom f soc IR n. Then, i when n =, f is SOC-monotone if and only if f τ 0 for any τ J; ii when n 3, f is SOC-monotone if and only if the matrix f f t f t t t t f t f t O for all t, t J. f t t t Proof By the definition of SOC-monotonicity, f is SOC-monotone if and only if f soc x + h f soc x K n 8 for any x dom f soc and h K n such that x + h dom f soc. By the first-order Taylor expansion of f soc, i.e., f soc x + h = f soc x + f soc x + thh for some t 0,, it is clear that 8 is equivalent to f soc x + thh K n for any x dom f soc and h K n such that x + h dom f soc,andsomet 0,. Lety := x + th = µ v + µ v for such x, h and t. We next proceed the arguments by the two cases of y = 0andy = 0. Case : y = 0. Under this case, we notice that [ ] f soc θ a T y =, a H where with θ = b, a = c y y, and H =ã0 I + b ã 0 y y T y, ã 0 = f µ f µ, b = f µ + f µ, c = f µ f µ. 9 µ µ In addition, we also observe that θ a = b c, θa T a T H = 0 and aa T H T H = ã 0 I + c b + ã 0 y y T y. Thus, by Lemma 3., f is SOC-monotone if and only if a b c ; 3

11 J Glob Optim : b and there exists λ 0 such that the matrix b c λ 0 0 λ ã 0 I + c b + ã 0 y y T O. y When n =, a together with b is equivalent to saying that f µ 0and f µ 0. Then we conclude that f is SOC-monotone if and only if f τ 0foranyτ J. When n 3, b is equivalent to saying that b c = λ 0andλ ã 0 0, i.e., b c ã 0. Therefore, a together with b is equivalent to f µ f µ f µ µ µ f µ f µ µ µ f µ O for any x IR n, h K n such that x + h dom f soc,andsomet 0,. Thus, we conclude that f is SOC-monotone if and only if f f t f t t t t f t f t O for all t, t J. f t t t Case : y = 0. Now we have µ = µ and f soc y = f µ I = f µ I. Hence, f is SOC-monotone is equivalent to f µ 0, which is also equivalent to f f µ f µ µ µ µ f µ f µ O f µ µ µ since f µ = f µ and f µ f µ µ µ = f µ = f µ by the Taylor formula and µ = µ. Thus, similar to Case, the conclusion also holds under this case. From Theorem 3. and [, Theorem ], we immediately have the following results. Corollary 3. Let f C J with J being an open interval in IR. Then, a f is SOC-monotone of order n 3 if and only if it is -matrix monotone, and f is SOC-monotone of order n if it is -matrix monotone. b Suppose that n 3 and f is SOC-monotone of order n. Then, f t 0 = 0 for some t 0 J if and only if f is a constant function on J. Note that the SOC-monotonicity of order does not imply the -matrix monotonicity. For example, f t = t is SOC-monotone of order on 0, + by Example 3. a in [5], but by [, Theorem ] we can verify that it is not -matrix monotone. Corollary 3. a implies that a continuously differentiable function defined on an open interval must be SOC-monotone if it is -matrix monotone. In addition, from the following proposition, we also have that the compound of two simple SOC-monotone functions is SOC-monotone. Proposition 3. If f : J J and g : J IR with J, J IR are SOC-monotone on J and J, respectively, then the function g f : J IR is SOC-monotone on J. Proof It is easy to verify that for all x, y IR n, x K n y if and only if λ i x λ i y with i =,. In addition, g is monotone on J since it is SOC-monotone. From the two facts, we immediately obtain the result. 3

12 70 J Glob Optim : Characterizations of SOC-convex functions In this section, we exploit Peirce decomposition to derive some characterizations for SOCconvex functions. Let f C J with J being an open interval in IR and dom f soc IR n. For any x dom f soc and h IR n,ifx = 0, then from Proposition. we have that h T A xh h T A xh [. = f h x T h h h h T A n xh Since h T h, h h K n, from Proposition. it follows that f is SOC-convex if and only if f x 0. By the arbitrariness of x, f is SOC-convex if and only if f is convex on J. In what follows, we assume that x = 0. Let x = λ xu x + λ xu x,whereu x = 0,υ i for i = 3,...,n, where and u x are given by 4 with x = x x.letui x υ 3,...,υn is any orthonormal set of vectors that span the subspace of IR n orthogonal to x. It is easy to verify that the vectors u x, u x, u 3 x,...,u n x are linearly independent. Hence, for any given h = h, h IR IR n, there exists µ i, i =,,...,n such that h = µ u x + µ u x + n i=3 ]. µ i u i x. From 9, we can verify that b + c and b c are the eigenvalues of A x with u x and u x being the corresponding eigenvectors, and a is the eigenvalue of multiplicity n with u i x = 0,υ i for i = 3,...,n being the corresponding eigenvectors. Therefore, where h T A xh = µ b c + µ b + c + a = f λ µ + f λ µ + a µ, 30 µ = n i=3 µ i. Similarly, we can verify that c + b a and c b + a are the eigenvalues of c b a x T x b a x x di + c d x x T x with u x and u x being the corresponding eigenvectors, and d is the eigenvalue of multiplicity n with u i x = 0,υ i for i = 3,...,n being the corresponding eigenvectors. Notice that Cx in can be decomposed the sum of the above matrix and d x x T. x 3 n i=3 µ i

13 J Glob Optim : Consequently, h T Cxh = µ c b + a + µ c + b a dµ µ + dµ. 3 In addition, by the definition of B i x, it is easy to compute that h T B i xh = h,i µ a d + µ a + d, 3 where h i = h,...,h,n.fromeqs.30 3 and the definition of A i x in Proposition., we thus have n h T A i xh =[h T Cxh] + h µ a d + µ a + d i= + µ µ h T Cxhµ a d + µ a + d =[h T Cxh] + µ µ +µ µ a d+µ a + d +µ µ h T Cxhµ a d + µ a + d =[h T Cxh + µ µ µ a d + µ a + d] +µ µ a d + µ a + d =[ f λ µ + f λ µ + dµ ] +µ µ a d + µ a + d. 33 On the other hand, by Proposition., f is SOC-convex if and only if n A x O and h T A i xh h T A xh. 34 i= From 30and33 45, we have that f is SOC convex if and only if A x O and [ f λ µ + f λ µ + dµ] + µ µ a d + µ a + d [ f λ µ + f λ µ + a µ ]. 35 When n =, it is clear that µ = 0. Then, f is SOC-convex if and only if A x O and f λ f λ 0. From the previous discussions, we know that b c = f λ, b + c = f λ and a = f λ f λ λ λ are all eigenvalues of A x. Thus, f is SOC-convex if and only if f λ 0, f λ 0, f λ f λ, which by the arbitrariness of x is equivalent to saying that f is convex on J. When n 3, if µ = 0, then from the discussions above, we know that f is SOC-convex if and only if f is convex. If µ = 0, without loss of generality, we assume that µ =. Then the inequality 46 above is equivalent to 4 f λ f λ µ µ + a d + f λ µ a d + f λ µ a + d µ a d + µ a + d + µ µ a d 0 forany µ,µ. 36 3

14 7 J Glob Optim :59 79 NowweshowthatA x O and 36 holds if and only if f is convex on J and f λ a + d a d, 37 f λ a d a + d. 38 Indeed, if f is convex on J, then by the discussions above A x O clearly holds. If the inequalities 37and38 hold, then by the convexity of f we have a d.ifµ µ 0, then we readily have the inequality 36. If µ µ > 0, then using a d yields that f λ f λ µ µ a d. Combining with Eqs thus leads to the inequality 36. On the other hand, if A x O,then f must be convex on J by the discussions above, whereas if the inequality 36 holds for any µ,µ, then by letting µ = µ = 0 yields that a d. 39 Using the inequality 39 and letting µ = 0in36 then yields 37, whereas using 39and letting µ = 0in36 leads to 38. Thus, when n 3, f is SOC-convex if and only if f is convex on J and 37and38 hold. We notice that 37and38 are equivalent to f λ [ f λ f λ + f λ λ λ ] λ λ [ f λ f λ f λ λ λ ] λ λ 4 and f λ [ f λ f λ f λ λ λ ] λ λ [ f λ f λ + f λ λ λ ] λ λ 4. Therefore, f is SOC-convex if and only if f is convex on J, and f t 0 [ f t 0 f t f tt 0 t] t 0 t [ f t f t 0 f t 0 t t 0 ] t 0 t 4 t 0, t J. 40 Summing up the above analysis, we can characterize the SOC-convexity as follows. Theorem 4. Let f C J with J being an open interval in IR and dom f soc IR n.if n =, then f is SOC-convex if and only if f is convex; if n 3, then f is SOC-convex if and only if f is convex and the inequality 40 holds for any t 0, t J. By the formulas of divided differences, it is not hard to verify that f is convex on J and 40 holds for any t 0, t J if and only if [ f t 0, t 0, t 0 ] f t 0, t, t 0 f t, t 0, t 0 O. 4 f t, t, t 0 This, together with Theorem 4. and [, Theorem 6.6.5], leads to the following results. Corollary 4. Let f C J with J being an open interval in IR and dom f soc IR n. Then, f is SOC-convex of order n 3 if and only if it is -matrix convex, and f is SOC-convex of order n if it is -matrix convex. Corollary 4. implies that, if f is a twice continuously differentiable function defined on an open interval J and -matrix convex, then it must be SOC-convex. Similar to Corollary 3. a, when f is SOC-convex of order, it may not be -matrix convex. For example, 3

15 J Glob Optim : f t = t 3 is SOC-convex of order on 0, + by Example 3.3 c of [5],butitiseasyto verify that 4 does not hold for this function, and consequently, f is not -matrix convex. Using Corollary 4., we may prove that Conjecture 4. in [5] does not hold in general. Corollary 4. Let f :[0, + [0, + be continuous. If f is SOC-concave, then f is SOC-monotone. Conversely, if f is SOC-monotone, then it may not be SOC-concave. Proof The first part has been shown in [5]. We next consider the second part. Consider f t = cot π + t + π, t [0, +. Notice that cott is SOC-monotone on [π/,π, whereas π +t is SOC-monotone on [0, +. Hence, their compound function f t is SOC-monotone on [0, +.However, f t does not satisfy the inequality 40 forallt 0, +. For example, when t = 7.7 and t = 7.6, the left hand side of 40 equals , whereas the right hand side equals This shows that f t = cott is not SOC-concave of order n 3. Particularly, by Corollary 4. and [9, Theorem.3], we can establish the following characterizations for SOC-convex functions. Corollary 4.3 Let f C 4 J with J being an open interval in IR and dom f soc IR n.if f t >0 for every t J, then f is SOC-convex of order n with n 3 if and only if one of the following conditions holds. a For every t J, the matrix f t f 3 t 6 f 3 t 6 f 4 t 4 O. 4 b There is a positive concave function c on I such that f t = ct 3 for every t J. c There holds that [ f t 0 f t f tt 0 t] [ f t f t 0 f t 0 t t 0 ] t 0 t t 0 t 4 f t 0 f t. 43 Moreover, f is also SOC-convex of order under one of the above conditions. Proof For completeness, we here present their proofs. Notice that f is convex on J. By Corollary 4., it suffices to prove the following equivalence: 40 assertion a assertion b assertion c. 40 assertion a: From the previous discussions, we know that 40 is equivalent to 37and38. We expand 37 using Taylor s expansion at λ to the fourth order and get 3 4 f λ f 4 λ f 3 λ. We do the same for the inequality 38atλ and get the inequality 3 4 f λ f 4 λ f 3 λ. 3

16 74 J Glob Optim :59 79 The above two inequalities are precisely 3 4 f t f 4 t f 3 t t J, 44 which is clearly equivalent to saying that the matrix in a is positive semidefinite. Assertion a assertion b: Take ct =[f t] /3 for t J. Thenc is a positive function and f t = ct 3. By twice differentiation, we obtain f 4 t = ct 5 [c tt] 3ct 4 c t. Substituting the last equality into the matrix in a then yields that 6 ct 7 c t 0, which, together with ct >0foreveryt J, implies that c is concave. Assertion b assertion c: We first prove the following fact: if f t is strictly positive for every t J and the function ct = [ f t ] /3 is concave on J,then [ f t 0 f t f tt 0 t] t 0 t f t 0 /3 f t /3 t 0, t J. 45 Indeed, using the concavity of the function c, it follows that [ f t 0 f t f tt 0 t] t 0 t = = u 0 0 u 0 0 u 0 0 f [t + u t 0 t] du du c u t + u t 0 3 du du u ct + u ct 0 3 du du. Notice that gt = /t t > 0 has the second-order derivative g t = /t 3. Hence, [ f t 0 f t f tt 0 t] t 0 t = = u 0 0 g u ct + u ct 0 du du gct 0 gct ct 0 ct g ct ct 0 ct ct 0 ctct = f t 0 /3 f t /3, which implies the inequality 45. Now exchanging t 0 with t in 45, we obtain [ f t f t 0 f t 0 t t 0 ] t 0 t f t /3 f t 0 /3 t, t 0 J. 46 Since f is convex on J by the given assumption, the left hand sides of the inequalities 45 and 46 are nonnegative, and their product satisfies the inequality of 43. Assertion c 40: We introducing a function F : J IR defined by Ft = f t 0 [ f t 0 f t f tt 0 t] [ f t f t 0 f t 0 t t 0 ] 3 t 0 t

17 J Glob Optim : if t = t 0, and otherwise Ft 0 = 0. We next prove that F is nonnegative on J. It is easy to verify that such Ft is differentiable on J, and moreover, F t = f t 0 f tt t 0 t t 0 [ f t f t 0 f t 0 t t 0 ] f t f t 0 + t t 0 3 [ f t f t 0 f t 0 t t 0 ] = f t 0 f tt t 0 t t 0 3 [ f t f t 0 f t 0 t t 0 ][ f t 0 f t f tt 0 t] [ = t t 0 4 f t 0 f t t t 0 4 f t f t 0 f t 0 t t 0 ] f t 0 f t f tt 0 t. Using the inequality in part c, we can verify that Ft has a minimum value 0 at t = t 0, and therefore, Ft is nonnegative on J. This implies the inequality Examples and applications The SOC-monotonicity and the SOC-convexity are often involved in the solution methods of convex SOCPs; for example, the proximal-like methods proposed in [5]. In this section, we focus on their applications in the augmented Lagrangian methods for the convex SOCP: min f ζ s.t. Aζ + b K n 0, 47 where f : IR n IR is a convex function, A IR n m has the full column rank, and b IR n. Let ϕ : b, + IR be a function with the following favorable properties: i ϕ is strictly convex, strictly monotone increasing and twice continuously differentiable on its domain domϕ := b, + with < b < 0; ii ϕ0 = 0, ϕ 0 = and lim t b ϕ t =, lim t + ϕ t = 0; iii lim ε 0 εϕt/ε = 0foranyt > 0; iv ϕ is SOC-monotone on b, + and ϕ is SOC-concave on b, +. For any ε>0, we use such ϕ to define the function ϕ ε : bε, + IR by ϕ ε t = εϕt/ε. 48 Then, by the properties i iii and the SOC-monotonicity of ϕ, we can rewrite 47as min f ζ s.t. ϕ ε socaζ + b K n Moreover, by the SOC-convexity of ϕ, it is not difficult to verify that 49 isalsoaconvex programming problem. Consequently, its Lagrangian, given by L ε x, y := f x y,ϕ ε soc Ax + b 50 is convex for any x domϕ ε soc. Similar to the augmented Lagrangian method for nonlinear convex programming in nonnegative orthant cone, we can design the augmented Lagrangian 3

18 76 J Glob Optim :59 79 method for 47 by solving a sequence of convex subproblems min L soc εx, y k with y k K n 0. x domϕ ε In what follows, we employ the characterizations established in the last two sections to present some examples of SOC-monotone functions and SOC-convex functions, from which we can choose a suitable one to design the augmented Lagrangian method for 47. Proposition 5. a For any fixed σ IR, the function f t = is SOC-monotone on σ t σ, +. b For any fixed σ IR, the function f t = t σ is SOC-monotone on [σ, +. c For any fixed σ IR, the function f t = lnt σ is SOC-monotone on σ, +. d For any fixed σ 0, the function f t = t is SOC-monotone on σ, +. t + σ Proof a For any t, t σ, +, it is clear that σ t σ t σ t O. σ t σ t σ t From Theorem 3., we then obtain the desired result. b If x K n σ e,thenx σ e / K n 0. Thus, by Theorem 3., it suffices to show t σ t σ t t σ t t σ t σ t t O for any t, t > 0, t σ which is equivalent to proving that 4 t σ t σ t σ + t σ 0. This inequality holds by 4 t σ t σ t σ + t σ for any t, t σ, +. c By Theorem 3., it suffices to prove that for any t, t σ, +, t σ t t ln t σ t σ O, t t ln t σ t σ which is equivalent to showing that t σt σ [ t σ ] t t ln t σ 0. t σ Notice that ln t t t > 0, and hence it is easy to verify that [ ] t t ln t σ t σ t σt σ. Consequently, the desired result follows. 3

19 J Glob Optim : d Since for any fixed σ 0andanyt, t σ, +, there holds that σ σ + t σ σ + t σ + t O, σ σ + t σ + t σ σ + t we immediately obtain the desired result from Theorem 3.. The proof is complete. Notice that Proposition 5. a, b and d were established in [5] by the definition of SOC-monotonicity, and Proposition 5. c was shown in [5] by the definition of SOCmonotonicity and a transformation technique. However, we here use the characterization of Theorem 3. to convert their proofs into proving some simple inequalities on IR. Proposition 5. a For any fixed σ IR, the function f t = t σ r with r 0 is SOC-convex on σ, + if and only if 0 r. b For any fixed σ IR, the function f t = t σ r with r 0 is SOC-convex on [σ, + if and only if r, and f is SOC-concave on [σ, + if and only if 0 r. c For any fixed σ IR, the function f t = lnt σ is SOC-concave on σ, +. d For any fixed σ 0, the function f t = t is SOC-concave on σ, +. t + σ Proof a For any fixed σ IR, by a simple computation, we have that f t f 3 t rr +t σ r rr + r t σ r 3 6 f 3 t f 4 t = 6 rr + r t σ r 3 rr +r +r + 3t σ r The sufficient and necessary condition for the above matrix being positive semidefinite is r r + r + r + 3t σ r 6 r r + r + t σ r 6 0, which is equivalent to requiring 0 r. By Corollary 4.3, it then follows that f is SOC-convex on σ, + if and only if 0 r. b For any fixed σ IR, by a simple computation, we have that f t f 3 t rr t σ r rr r t σ r 3 6 f 3 t f 4 t = 6 rr r t σ r 3 rr r r 3t σ r The sufficient and necessary condition for the above matrix being positive semidefinite is r and r r r r 3t σ r 6 4 r r r t σ r 6 0, 5 8 whereas the sufficient and necessary condition for it being negative semidefinite is 0 r and r r r r 3t σt r 6 4 r r r t σ r

20 78 J Glob Optim :59 79 It is easily shown that 5 holds if and only if r, and 53 holds if and only if 0 r. By Corollary 4.3, this shows that f is SOC-convex on σ, + if and only if r, and f is SOC-concave on σ, + if and only if 0 r. This together with the definition of SOC-convexity then yields the desired result. c Notice that for any t >σ, there always holds that f t f 3 t 6 f 3 t f 4 t = t σ 3t σ 3 O t σ 3 4t σ 4 Consequently, from Corollary 4.3 a we obtain that f is SOC-concave on σ, +. d For any t > σ, it is easy to compute that f t f 3 t 6 f 3 t f 4 t = t + σ 3 t + σ 4 O. 6 4 t + σ 4 t + σ 5 By Corollary 4.3, we then have that the function f is SOC-concave on σ, +. From Propositions 5. and 5., we see that the functions ln + t, t + and are SOC-monotone as well as SOC-concave. Furthermore, it is easy to verify t + that they also satisfy the properties i iii. Thus, with the three functions, we can design the augmented Lagrangian method for solving the convex SOCP 47, which in fact corresponds to the modified barrier functions method proposed by R. Ployak [7] for nonlinear programming over nonnegative orthant cones. We will leave the details of this method for a future research topic. In addition, we should point out that the SOC-monotonicity and the SOC-convexity often give a great help to establish some inequalities involved the partial order K n. For example, from Proposition 5. b, we readily recover the Eq. 3.9 of [8, Proposition 3.4]. Acknowledgements The authors would like to thank two anonymous referees for their helpful comments and suggestions on the revision of this paper. References. Aujla, J.S., Vasudeva, H.L.: Convex and monotone operator functions. Ann. Pol. Math. 6, 995. Bhatia, R.: Matrix Analysis. Springer, New York Bhatia, R., Parthasarathy, K.P.: Positive definite functions and operator inequalities. Bull. Lond. Math. Soc. 3, Brinkhuis, J., Luo, Z.-Q., Zhang, S.: Matrix convex functions with applications to weighted centers for semi-definite programming, submitted manuscript Chen, J.-S.: The convex and monotone functions associated with second-order cone. Optimization 55, Chen, J.-S., Chen, X., Tseng, P.: Analysis of nonsmooth vector-valued functions associated with secondorder cones. Math. Program. 0, Faraut, J., Korányi, A.: Analysis on Symmetric Cones, Oxford Mathematical Monographs. Oxford University Press, New York Fukushima, M., Luo, Z.-Q., Tseng, P.: Smoothing functions for second-order cone complementarity problems. SIAM J. Optim., Hansen, F., Tomiyama, J.: Differential analysis of matrix convex functions. Linear Algebra Appl. 40,

21 J Glob Optim : Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge 985. Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis. Cambridge University Press, Cambridge 99. Korányi, A.: Monotone functions on formally real Jordan algebras. Math. Ann. 69, Kwong, M.-K.: Some results on matrix monotone functions. Linear Algebra Appl. 8, Löwner, K.: Über monotone matrixfunktionen. Math. Z. 38, Pan, S.-H., Chen, J.-S.: A class of interior proximal-like algorithms for convex second-order cone programming, accepted by SIAM J. Optim. 9, Polik, I., Terlaky, T.: A Comprehensive Study of the S-Lemma. Advanced Optimization On-Line, Report No. 004/ Polyak, R.: Modified barrier functions: theory and methods. Math. Program. 54, Roos, K.: 9. Sun, D., Sun, J.: Semismooth matrix valued functions. Math. Oper. Res. 7, Tseng, P.: Merit function for semidefinite complementarity problems. Math. Program. 83,