Unified smoothing functions for absolute value equation associated with second-order cone

Transcription

1 Applied Numerical Mathematics, vol. 135, January, pp. 06-7, 019 Unified smoothing functions for absolute value equation associated with second-order cone Chieu Thanh Nguyen 1 Department of Mathematics National Taiwan Normal University Taipei 11677, Taiwan. B. Saheya College of Mathematical Science Inner Mongolia Normal University Hohhot 0100, Inner Mongolia, P. R. China Yu-Lin Chang 3 Department of Mathematics National Taiwan Normal University Taipei 11677, Taiwan. Jein-Shan Chen 4 Department of Mathematics National Taiwan Normal University Taipei 11677, Taiwan. February 19, 018 (revised on July 1, 018) Abstract In this paper, we eplore a unified way to construct smoothing functions for solving the absolute value equation associated with second-order cone (SOCAVE). Numerical comparisons are presented, which illustrate what kinds of smoothing functions 1 thanhchieu90@gmail.com. saheya@imnu.edu.cn. The author s work is supported by Natural Science Foundation of Inner Mongolia (Award Number: 017MS015). 3 ylchang@math.ntnu.edu.tw. 4 Corresponding author. jschen@math.ntnu.edu.tw. The author s work is supported by Ministry of Science and Technology, Taiwan. 1

2 work well along with the smoothing Newton algorithm. In particular, the numerical eperiments show that the well known loss function widely used in engineering community is the worst one among the constructed smoothing functions, which indicates that the other proposed smoothing functions can be employed for solving engineering problems. Keywords. Second-order cone, absolute value equations, smoothing Newton algorithm. 1 Introduction Recently, the paper 36 investigates a family of smoothing functions along with a smoothingtype algorithm to tackle the absolute value equation associated with second-order cone (SOCAVE) and shows the efficiency of such approach. Motivated by this article, we continue to ask two natural questions. (i) Whether there are other suitable smoothing functions that can be employed for solving the SOCAVE? (ii) Is there a unified way to construct smoothing functions for solving the SOCAVE? In this paper, we provide affirmative answers for these two queries. In order to smoothly convey the story of how we figure out the answers, we begin with recalling where the SOCAVE comes from. The standard absolute value equation (AVE) is in the form of A + B = b, (1) where A IR n n, B IR n n, B 0, and b IR n. Here means the componentwise absolute value of vector IR n. When B = I, where I is the identity matri, the AVE (1) reduces to the special form: A = b. It is known that the AVE (1) was first introduced by Rohn in 41, but was termed by Mangasarian 34. During the past decade, there has been many researchers paying attention to this equation, for eample, Caccetta, Qu and Zhou, Hu and Huang 1, Jiang and Zhang 0, Ketabchi and Moosaei 1, Mangasarian 6, 7, 8, 9, 30, 31, 3, 33, Mangasarian and Meyer 34, Prokopyev 37, and Rohn 43. We elaborate more about the developments of the AVE. Mangasarian and Meyer 34 show that the AVE (1) is equivalent to the bilinear program, the generalized LCP (linear complementarity problem), and to the standard LCP provided 1 is not an eigenvalue of A. With these equivalent reformulations, they also show that the AVE (1) is NP-hard in its general form and provide eistence results. Prokopyev 37 further improves the above equivalence which indicates that the AVE (1) can be equivalently recast as LCP without any assumption on A and B, and also provides a relationship with mied integer programming. In general, if solvable, the AVE (1) can have either unique solution or

3 multiple (e.g., eponentially many) solutions. Indeed, various sufficiency conditions on solvability and non-solvability of the AVE (1) with unique and multiple solutions are discussed in 34, 37, 4. Some variants of the AVE, like the absolute value equation associated with second-order cone and the absolute value programs, are investigated in 14 and 46, respectively. Recently, another type of absolute value equation, a natural etension of the standard AVE (1), is considered 14, 35, 36. More specifically the following absolute value equation associated with second-order cones, abbreviated as SOCAVE, is studied: A + B = b, () where A, B IR n n and b IR n are the same as those in (1); denotes the absolute value of coming from the square root of the Jordan product of and. What is the difference between the standard AVE (1) and the SOCAVE ()? Their mathematical formats look the same. In fact, the main difference is that in the standard AVE (1) means the componentwise i of each i IR, i.e., = ( 1,,, n ) T IR n ; however, in the SOCAVE () denotes the vector satisfying := associated with second-order cone under Jordan product. To understand its meaning, we need to introduce the definition of second-order cone (SOC). The second-order cone in IR n (n 1), also called the Lorentz cone, is defined as K n := { ( 1, ) IR IR n 1 1 }, where denotes the Euclidean norm. If n = 1, then K n is the set of nonnegative reals IR +. In general, a general second-order cone K could be the Cartesian product of SOCs, i.e., K := K n 1 K nr. For simplicity, we focus on the single SOC K n because all the analysis can be carried over to the setting of Cartesian product. The SOC is a special case of symmetric cones and can be analyzed under Jordan product, see 8. In particular, for any two vectors = ( 1, ) IR IR n 1 and y = (y 1, y ) IR IR n 1, the Jordan product of and y associated with K n is defined as y := T y. y y The Jordan product, unlike scalar or matri multiplication, is not associative, which is a main source of complication in the analysis of optimization problems involved SOC, see 4, 6, 9 and references therein for more details. The identity element under this Jordan product is e = (1, 0,..., 0) T IR n. With these definitions, means the Jordan product of with itself, i.e., := ; and with K n denotes the unique vector such that =. In other words, the vector in the SOCAVE () is computed by :=. 3

4 As remarked in the literature, the significance of the AVE (1) arises from the fact that the AVE is capable of formulating many optimization problems such as linear programs, quadratic programs, bimatri games, and so on. Likewise, the SOCAVE () plays a similar role in various optimization problems involving second-order cones. There has been many numerical methods proposed for solving the standard AVE (1) and the SOCAVE (). Please refer to 36 for a quick review. Basically, we follow the smoothing Newton algorithm employed in 36 to deal with the SOCAVE (). This kind of algorithm has been a powerful tool for solving many other optimization problems, including symmetric cone complementarity problems 11, 3, 4, the system of inequalities under the order induced by symmetric cone 18, 5, 47, and so on. It is also employed for the standard AVE (1) in 19, 44. The new upshot of this paper lies on discovering more suitable smoothing functions and eploring a unified way to construct smoothing functions. Of course, the numerical performance among different smoothing functions are compared. These are totally new to the literature and are the main contribution of this paper. To close this section, we recall some basic concepts and background materials regarding the second-order cone, which will be used in the subsequent analysis. More details can be found in 4, 6, 8, 9, 14. First, we recall the epression of the spectral decomposition of with respect to SOC. For = ( 1, ) IR IR n 1, the spectral decomposition of with respect to SOC is given by = λ 1 ()u (1) + λ ()u (), (3) where λ i () = 1 + ( 1) i for i = 1, and u (i) = ( 1 1 1, ( 1) i T T ) if 0, ( 1, ( 1) i ω ) T T if = 0, (4) with ω IR n 1 being any vector satisfying ω = 1. The two scalars λ 1 () and λ () are called spectral values of ; while the two vectors u (1) and u () are called the spectral vectors of. Moreover, it is obvious that the spectral decomposition of IR n is unique if 0. It is known that the spectral values and spectral vectors posses the following properties: (i) u (1) u () = 0 and u (i) u (i) = u (i) for i = 1, ; (ii) u (1) = u () = 1 and = 1 (λ 1() + λ ()). Net is the concept about the projection onto second-order cone. Let + denote the projection of onto K n, and be the projection of onto the dual cone (K n ) of K n, where the dual cone (K n ) is defined by (K n ) := {y IR n, y 0, K n }. In fact, the dual cone of K n is itself, i.e., (K n ) = K n. Due to the special structure of 4

5 K n, the eplicit formula of projection of = ( 1, ) IR IR n 1 onto K n is obtained in 4, 6, 8, 9, 10 as below: if K n, + = 0 if K n, u otherwise, where u = 1 + ( ) 1 +. Similarly, the epression of can be written out as 0 if K n, = if K n, where w = w otherwise, It is easy to verify that = + + and 1 ( ) 1. + = (λ 1 ()) + u (1) + (λ ()) + u () = ( λ 1 ()) + u (1) + ( λ ()) + u (), where (α) + = ma{0, α} for α IR. As for the epression of associated with SOC. There is an alternative way via the so-called SOC-function to obtain the epression of, which can be found in 3, 5. In any case, it comes out that = (λ 1 ()) + + ( λ 1 ()) + u (1) + (λ ()) + + ( λ ()) + u () = λ1 () u (1) + λ () u (). Unified smoothing functions for SOCAVE As mentioned in Section 1, we employ the smoothing Newton method for solving the SOCAVE (), which needs a smoothing function to work with. Indeed, a family of smoothing functions was already considered in 36. In this section, we look into what kinds of smoothing functions can be employed to work with the smoothing Newton algorithm for solving the SOCAVE (). Definition.1. A function φ : IR ++ IR IR is called a smoothing function of t if it satisfies the following: (i) φ is continuously differentiable at (, t) IR ++ IR; (ii) lim 0 φ(, t) = t for any t IR. Given a smoothing function φ, we further define a vector-valued function Φ : IR ++ IR n IR n as Φ(, ) = φ (, λ 1 ()) u (1) + φ (, λ ()) u () (5) 5

6 where IR ++ is a parameter, λ 1 (), λ () are the spectral values of, and u (1), u () are the spectral vectors of. Consequently, Φ is also smooth on IR ++ IR n. Moreover, it is easy to verify that lim Φ(, ) = λ 1() u (1) λ () u () = which means each function Φ(, ) serves as a smoothing function of associated with SOC. With this observation, for the SOCAVE (), we further define the function H(, ) : IR ++ IR n IR IR n by H(, ) = A + BΦ(, ) b, IR ++ and IR n. (6) Proposition.1. Suppose that = ( 1, ) IR IR n 1 has the spectral decomposition as in (3)-(4). Let H : IR ++ IR n IR n be defined as in (6). Then, (a) H(, ) = 0 if and only if solves the SOCAVE (); (b) H is continuously differentiable at (, ) IR ++ IR n with the Jacobian matri given by H 1 0 (, ) = B Φ(,) A + B Φ(,) (7) where Φ(, ) Φ(, ) = φ(, λ 1()) φ(, 1 ) = u (1) + φ(, λ ()) u (), I if = 0, b c T c ai + (b a) T if 0, with a = φ(, λ ()) φ(, λ 1 ()), λ () λ 1 () b = 1 ( φ(, λ ()) c = 1 ( φ(, λ ()) + φ(, λ 1()) φ(, λ 1()) ), (8) ). Proof. (a) First, we observe that H(, ) = 0 = 0 and A + BΦ(, ) b = 0 A + B b = 0 and = 0. 6

7 This indicates that is a solution to the SOCAVE () if and only if (, ) is a solution to H(, ) = 0. (b) Since Φ(, ) is continuously differentiable on IR ++ IR n, it is clear that H(, ) is continuously differentiable on IR ++ IR n. Thus, it remains to compute the Jacobian matri of H(, ). Note that Φ(, ) = φ(, λ 1 ())u (1) + φ(, λ ())u (), 1 φ(, λ 1 ()) + φ(, λ ()) φ(, λ = 1 ()) T + φ(, λ if ()) T 0, 1 φ(, λ 1 ()) + φ(, λ ()) φ(, λ 1 ())ω T + φ(, λ ())ω T if = 0. φ(, λ 1 ()) + φ(, λ () ( φ(, λ 1 ()) + φ(, λ ())). if 0, = 1 ( φ(, λ 1 ()) + φ(, λ ())) n φ(, λ 1 ()) + φ(, λ ()) if = where := (,, n ) IR n 1, ω = (ω,, ω n ) IR n 1. From chain rule, it is trivial that Φ(, ) = φ(, λ 1()) u (1) + φ(, λ ()) u () In order to compute Φ(,), for simplicity, we denote τ 1 (, ) Φ(, ) := 1 τ (, ). τ n (, ) To proceed, we discuss two cases. (i) For 0, we compute τ 1 (, ) = φ(, λ 1()) = φ(, λ 1()) λ 1 () = φ(, λ 1()) λ 1 (). + φ(, λ ()) λ 1 () + φ(, λ ()) λ () + φ(, λ ()) λ () 7 := b λ ()

8 and τ 1 (, ) = φ(, λ 1()) i i = φ(, λ 1()) λ 1 () λ 1 () i = φ(, λ 1()) i λ 1 () Moreover, = = τ i (, ) = ( φ(, λ ()) λ () ( φ(, λ ()) ( φ(, λ ()) + φ(, λ ()) i + φ(, λ ()) λ () + φ(, λ ()) λ () φ(, λ 1()) λ 1 () φ(, λ 1()) λ () i i ) i ) i := c i, i =,, n. φ(, λ ) 1()) i = c i, i =,, n. Similarly, we have ( ) ( τ (, ) φ(, λ ()) = φ(, λ ) 1()) + (φ(, λ ()) φ(, λ 1 ())) = b ( 1 + (φ(, λ ()) φ(, λ 1 ())) ) 3 = a + (b a), where a means a := φ(, λ ()) φ(, λ 1 ()). In general, mimicking the same derivation λ () λ 1 () yields { τ i (, ) a + (b a) i i = if i = j, j (b a) i j if i j. To sum up, we obtain which is the desired result. Φ(, ) (ii) For = 0, it is clear to see = b c T c ai + (b a) T τ 1 (, ) = φ(, 1) and τ 1(, ) i = 0 for i =,, n. 8

9 Since τ i (, ) = 0 for i =,, n, it gives τ i(,) = 0. Moreover, τ (, ) τ (, 1,, 0,, 0) τ (, 1, 0,, 0) = lim 0 φ(, 1 + ) φ(, 1 ) = lim 0 Thus, we obtain = lim 0 = lim 0 = lim 0 = φ(, 1). which is equivalent to saying φ(, 1 + ) φ(, 1 ) φ(, 1 + ) φ(, 1 ) ( ) ( ) φ(, 1 + ) ( 1 + ) { τ i (, ) = j + φ(, 1 ) ( 1 ) φ(, 1) if i = j, 0 if i j. (as L Hopital s rule) Φ(, ) = φ(, 1) I. From all the above, we conclude that φ(, 1 ) Φ(, ) I if = 0, = b c T c ai + (b a) if T 0, Thus, the proof is complete. Now, we are ready to answer the question about what kind of smoothing functions can be adopted in the smoothing type algorithm. Two technical lemmas are needed towards the answer. Lemma.1. Suppose that M, N IR n n. Let σ min (M) denote the minimum singular value of M, and σ ma (N) denote the maimum singular value of N. Then, the following hold. (a) σ min (M) > σ ma (N) if and only if σ min (M T M) > σ ma (N T N). (b) If σ min (M T M) > σ ma (N T N), then M T M N T N is positive definite. 9

10 Proof. The proof is straightforward or can be found in usual tetbook of matri analysis, so we omit it here. Lemma.. Let A, S IR n n and A be symmetric. Suppose that the eigenvalues of A and SS T are arranged in non-increasing order. Then, for each k = 1,,, n, there eists a nonnegative real number θ k such that λ min (SS T ) θ k λ ma (SS T ) and λ k (SAS T ) = θ k λ k (A). Proof. Please see 15, Corollary for a proof. We point out that the crucial key, which guarantees a smoothing function can be employed in the smoothing type algorithm, is the nonsingularity of the Jacobian matri H (, )) given in (7). As below, we provide under what condition the Jacobian matri H (, )) is nonsingular. Theorem.1. Consider a SOCAVE () with σ min (A) > σ ma (B). Let H be defined as in (6). Suppose that φ : IR ++ IR IR is a smoothing function of t. If 1 d φ(, t) 1 dt is satisfied, then the Jacobian matri H (, ) is nonsingular for any > 0. Proof. From the epression of H (, ) given as in (7), we know that H (, ) is nonsingular if and only if the matri A + B Φ(,) is nonsingular. Thus, it suffices to show that the matri A + B Φ(,) is nonsingular under the conditions. Suppose not, that is, there eists a vector 0 v IR n such that Φ(, ) A + B v = 0 which implies that T Φ(, ) v T A T Av = v T B T Φ(, ) B v. (9) For convenience, we denote C := Φ(,). Then, it follows that v T A T Av = v T C T B T BCv. Applying Lemma., there eists a constant ˆθ such that λ min (C T C) ˆθ λ ma (C T C) and λ ma (C T B T BC) = ˆθλ ma (B T B). Note that if we can prove that 0 λ min (C T C) λ ma (C T C) 1, we will have λ ma (C T B T BC) λ ma (B T B). Then, by the assumption that the minimum singular value of A strictly eceeds the maimum singular value of B (i.e., σ min (A) > 10

11 σ ma (B)) and applying Lemma.1, we obtain v T A T Av > v T C T B T BCv. This contradicts the identity (9), which shows the Jacobian matri H (, ) is nonsingular for > 0. Thus, in light of the above discussion, it suffices to claim 0 λ min (C T C) λ ma (C T C) 1. To this end, we discuss two cases. Case 1: For = 0, we compute that C = φ(, 1) I. Since 1 φ(, 1) that 0 λ(c T C) 1 for > 0. Then, the claim is done. 1, it is clear Case : For 0, using the fact that the matri M T M is always positive semidefinite for any matri M IR m n, we see that the inequality λ min (C T C) 0 always holds. In order to prove λ ma (C T C) 1, we need to further argue that the matri I C T C is positive semidefinite. First, we write out 1 b c bc T I C T C = bc (1 a )I + (a b c ). T If 1 < φ(,λ i()) < 1, then we obtain ( φ(, b + c = 1 ) ( ) λ1 ()) φ(, λ ()) + < 1. This indicates that 1 b c > 0. By considering 1 b c as an 1 1 matri, this says 1 b c is positive definite. Hence, its Schur complement can be computed as below: (1 a )I + (a b c ) T 4b c T 1 b c ( = (1 a ) I ) ) T + (1 b c 4b c T 1 b c. (10) On the other hand, by the Mean Value Theorem, we have φ(, λ ()) φ(, λ 1 ()) = φ(, ξ) (λ () λ 1 ()), ξ where ξ (λ 1 (), λ ()). To proceed, we need to further discuss two subcases. (1) When 1 < φ(,ξ) < 1, we know φ(, λ ξ ()) φ(, λ 1 ()) < λ () λ 1 (). This together with (8) implies that 1 a > 0 for any > 0. In addition, for any > 0, we observe that (1 b c ) 4b c = (1 (b c) )(1 (b + c) ) ( ) ( ) φ(, λ1 ()) φ(, λ ()) = 1 1 > 0. 11

12 With all of these, we verify that the Schur complement ( of 1 b ) c given as in (10) is a linear positive combination of the matrices I T and T, which yields that the Schur complement (10) of 1 b c is positive semidefinite. Hence, the matri I C T C is also positive semidefinite, which is equivalent to saying 0 λ min (C T C) λ ma (C T C) 1. () When φ(,ξ) ξ = ±1, we have 1 a = 0, and (1 b c ) 4b c > 0. Since the matri T is positive semidefinite, the matri I C T C is positive semidefinite. Hence, 0 λ min (C T C) λ ma (C T C) 1. { { φ(,λ1 ()) If either 1 = ±1 φ(,λ1 ()) φ(,λ ()) or 1 = ±1 φ(,λ = ±1 ()), then we have b = ±1, c = 0 or = 1 b = 0, c = 1, which yields b + c = 1. Again, two subcases are needed. (1) When 1 < φ(,ξ) < 1, we have φ(, λ ξ ()) φ(, λ 1 ()) < λ () λ 1 (). This implies that 1 a > 0 for any > 0. Therefore I C T C = (1 a ) ( ) I T Since the matri I T is positive semidefinite, the matri I C T C is positive semidefinite. Hence, 0 λ min (C T C) λ ma (C T C) 1. () When φ(,ξ) ξ = ±1, we have I C T C = 0, which leads to λ(c T C) = 1. From all the above, the proof is complete., We point out that the condition σ min (A) > σ ma (B) in Theorem.1 guarantees the unique solution according to 35, Theorem 4.1. From Theorem.1, we realize that for a SOCAVE () with σ min (A) > σ ma (B), any smoothing function of t with 1 d φ(, t) 1 will be good for serving in the smoothing Newton algorithm when dt solving the above SOCAVE. With this, it is easy to find or construct smoothing functions of t satisfying the above condition. One popular approach is a smoothing approimation via convolution for the absolute value function 1,, 38, 45, which is described as below. First, we construct a smoothing approimation for the plus function (t) + = ma{0, t}. Then, we consider the piecewise continuous function d(t) with finite number of pieces, which is a density (kernel) function. In other words, it satisfies d(t) 0 and 1 + d(t)dt = 1.

13 With this d(t), we further define ŝ(t, ) := 1 d ( + t ), where is a positive parameter. If t d(t)dt < +, then a smoothing approimation for (t) + is formed. In particular, ˆp(t, ) = + (t s) + ŝ(s, )ds = t (t s)ŝ(s, )ds (t) +. The following are four well-known smoothing functions for the plus function 1, 38: ( ) ˆφ 1 (, t) = t + ln 1 + e t. (11) ˆφ (, t) = ˆφ 3 (, t) = ˆφ 4 (, t) = where the corresponding kernel functions are t if t ( ), 1 t + if < t <, (1) 0 if t. 4 + t + t. (13) d 1 (t) = d (t) = d 3 (t) = d 4 (t) = t if t >, t if 0 t, 0 if t < 0. e (1 + e ). { 1 if 1 1, 0 otherwise.. ( + 4) 3 { 1 if 0 1, 0 otherwise. Net, in light of t = (t) + + ( t), the smoothing function of t via convolution can be written as ˆp( t, ) = ˆp(t, ) + ˆp( t, ) = + t s ŝ(s, )ds. Analogous to (11)-(14), we achieve the following smoothing functions for t : ( ) ( φ 1 (, t) = ln 1 + e t + ln 1 + e t φ (, t) = (14) ). (15) t if t, t + 4 if < t <, t if t. (16) φ 3 (, t) = 4 + t. (17) { t if t, φ 4 (, t) = t (18) if t >. 13

14 If we take a Epanechnikov kernel function { 3 K(t) = (1 4 t ) if t 1, 0 if otherwise, then we obtain the following smoothing function for t : t if t >, φ 5 (, t) = t4 + 3t + 3 if t, t if t <. (19) Moreover, taking a Gaussian kernel function K(t) = 1 π e t ŝ(t, ) := 1 ( ) t K 1 = t π e, for all t IR yields and it leads to the smoothing function 45 for t : ( ) t t φ 6 (, t) = terf + π e, (0) where the error function is defined by erf(t) = t e u du π 0 t IR. In summary, we have constructed si smoothing functions from the above discussions. Can all the above functions serve as smoothing functions for solving SOCAVE? The answer is affirmative because it is not hard to verify that each φ i possesses 1 d dt φ i(, t) 1. Thus, these si functions will be adopted for our numerical implementations. Accordingly, we need to define Φ i (, ) and H i (, ) based on each φ i. For subsequent needs, we only present the epression of each Jacobian matri H i(, ) without detailed derivations. Based on each φ i, let Φ i : IR IR n IR n for i = 1,,, 6 be similarly defined as in (5), i.e Φ i (, ) = φ i (, λ 1 ()) u (1) + φ i (, λ ()) u () (1) and H i : IR IR n IR n for i = 1,,, 6 be similarly defined as in (6), i.e H i (, ) =, IR A + BΦ i (, ) b ++ and IR n. () Then, each H i is continuously differentiable on IR ++ IR n with the Jacobian matri given by H i(, 1 0 ) = B Φ i(,) A + B Φ i(,) (3) for all (, ) IR ++ IR n with = ( 1, ) IR IR n 1. Moreover, the differentation of each Φ i is epressed as below. 14

15 (1) The Jacobian of Φ 1 is characterized as below. Φ 1 (, ) = φ 1(, λ 1 ()) = with u (1) φ 1 (, λ 1 ()) + λ 1() Φ 1 (, ) + φ 1(, λ ()) 1 e λ 1 () 1 + e λ 1 () = 1 e 1 1 e +1 u () u (1) + φ 1 (, λ ()) + λ () I if = 0, b 1 c T 1 c 1 a 1 I + (b 1 a 1 ) if T 0, a 1 = φ 1(, λ ()) φ 1 (, λ 1 ()), λ () λ 1 () ) (e b 1 = 1 λ 1 () 1 + e λ() 1, e λ 1 () + 1 e λ () + 1 ) (1 c 1 = 1 e λ 1 () + e λ() 1. e λ 1 () + 1 e λ () + 1 () The Jacobian of Φ is characterized as below. 1 e λ () 1 + e λ () u (). Φ (, ) = φ (, λ 1 ()) u (1) + φ (, λ ()) u () with φ (, λ i ()) Φ (, ) = ( λi () 0 if λ i () ), + 1 if < λ 4 i() <, 0 if λ i (). di if = 0, = b c T c a I + (b a ) if T 0, 15

16 with a = φ (, λ ()) φ (, λ 1 ()), λ () λ 1 () 0 if λ () > λ 1(), 1 if λ () > λ 1 (), λ 1 () + 1 if λ () b = > λ 1() >, λ 1 ()+λ () if > λ () > λ 1 () >, λ () 1 if > λ () > λ 1(), 1 if λ 1 () < λ (), 1 if λ () > λ 1(), 0 if λ () > λ 1 (), 1 c = λ 1() if λ () > λ 1() >, λ () λ 1 () if > λ () > λ 1 () >, λ () + 1 if > λ () > λ 1(), 0 if λ 1 () < λ (), d = 1 if 1, 1 if < 1 <, 1 if 1. (3) The Jacobian of Φ 3 is characterized as below. with Φ 3 (, ) = = Φ 3 (, ) b 3 = 1 ( λ 1() u(1) λ () u() I if = 0, b 3 c T 3 c 3 a 1 I + (b 1 a 1 ) T if 0, a 3 = φ 3(, λ ()) φ 3 (, λ 1 ()), λ () λ 1 () ( λ 1 () 4 + λ 1() + c 3 = 1 λ 1 () 4 + λ 1() + λ () 4 + λ () ) ) λ (). 4 + λ (), (4) The Jacobian of Φ 4 is characterized as below. Φ 4 (, ) = φ 4(, λ 1 ()) u (1) + φ 4(, λ ()) u () 16

17 with with φ 4 (, λ i ()) Φ 4 (, ) 1 if λ i () >, ( ) = 1 λi () if λ i (), 1 if λ i () <. ei if = 0, = b 4 c T 4 c 4 a 4 I + (b 4 a 4 ) if T 0, a 4 = φ 4(, λ ()) φ 4 (, λ 1 ()), λ () λ 1 () b 4 = c 4 = e = 0 if λ () > > > λ 1 (), 1 if λ () > λ 1 () >, λ 1 () + 1 if λ () > λ 1 (), λ 1 ()+λ () if λ () > λ 1 (), λ () 1 if λ () > λ 1 (), 1 if λ 1 () < λ () <. 1 if λ () > > > λ 1 (), 0 if λ () > λ 1 () >, 1 λ 1() if λ () > λ 1 (), λ () λ 1 () if λ () > λ 1 (), λ () + 1 if λ () > λ 1 (), 0 if λ 1 () < λ () <, 1 if 1 >, 1 if 1, 1 if 1 <. (5) The Jacobian of Φ 5 is characterized as below. with Φ 5 (, ) φ 5 (, λ i ()) Φ 5 (, ) = φ 5(, λ 1 ()) u (1) = 3 8 ( ( λi () + φ 5(, λ ()) u () 0 if λ i () >, ) ) 1 if λ i (), 0 if λ i () <. ei if = 0, = b 5 c T 5 c 5 a 5 I + (b 5 a 5 ) if T 0, 17

18 with a 5 = φ 5(, λ ()) φ 5 (, λ 1 ()), λ () λ 1 () 0 if λ () > > > λ 1 (), 1 if λ () > λ 1 () >, ( ) 3 1 λ1 () λ 1 () + 1 if λ 4 () > λ 1 (), b 5 = 1 λ 3 1 ()+λ3 () + 3 λ 1 ()+λ () if λ () > λ 1 (), ( ) 3 1 λ () λ () 1 if λ 4 () > λ 1 (), 1 if λ 1 () < λ () <. 1 if λ () > > > λ 1 (), 0 if λ () > λ 1 () >, ( ) λ1 () 4 3 λ 1 () if λ 4 () > λ 1 (), c 5 = e = 1 λ 3 () λ3 1 () + 3 λ ()+λ 1 () ( 1 λ () 4 1 ( 1 if λ () > λ 1 (), ) λ () + 1 if λ 4 () > λ 1 (), 0 if λ 1 () < λ () <, 1 if 1 >, 1 ) if 1, 1 if 1 <. (6) The Jacobian of Φ 6 is characterized as below. with Φ 6 (, ) Φ 6 (, ) λ 1 = () π e u (1) λ + () π e u () ( ) erf 1 I if = 0, = b 6 c T 6 if 0, c 6 a 6 I + (b 6 a 6 ) T a 6 = φ 6(, λ ()) φ 6 (, λ 1 ()), λ () λ 1 () b 6 = 1 ( ( ) ( )) λ1 () λ () erf + erf, c 6 = 1 ( erf ( ) ( )) λ () λ1 () erf. 18

19 3 Smoothing Newton method In this section, we study the smoothing Newton algorithm based on the smoothing function Φ i (, ) for i {1,,..., 6} to solve the SOCAVE (), and show its convergence properties. First, we present the generic framework of the smoothing Newton algorithm. Algorithm 3.1. (A Smoothing Newton Algorithm) Step 0 Choose δ (0, 1), σ (0, 1), and 0 IR ++, 0 IR n. Set z 0 := ( 0, 0 ), e := (1, 0) IR IR n 1. Choose β > 1 satisfying min{1, H i (z 0 ) } β 0. Set k := 0. Step 1 If H i (z k ) = 0, stop. Otherwise, set τ k := min{1, H i (z k ) }. Step Compute z k = ( k, k ) IR IR n by H i (z k ) + H i(z k ) z k = 1 β τ k e, (4) where H i(z k ) denotes the Jacobian matri of H i (z k ) at ( k, k ) given by (7). Step 3 Let α k be the maimum of the values 1, δ, δ, such that H i (z k + α k z k ) 1 σ(1 1 β )α k H i (z k ). (5) Step 4 Set z k+1 := z k + α k z k and k := k + 1. Go to Step 1. Theorem.1 indicates the Newton equation (4) in Algorithm 3.1 is solvable. It paves a way to show that the linear search (5) in Algorithm 3.1 is well-defined which is presented in Theorem 3.1 as below. More specifically, from 17, Lemma 3.1, we know that there eists an ᾱ (0, 1 such that (5) holds for any α (0, ᾱ. This indicates that taking α = ma{1, δ, δ,... }, where δ (0, 1) in step 3, leads to (5) being well-defined. Indeed, the detailed arguments are very similar to those in 16, 18, 36, we only state it here and omit its proof. Theorem 3.1. Consider a SOCAVE () with σ min (A) > σ ma (B). IR IR n given by (4), the linear search (5) is well-defined. Then, for z Net, we discuss the convergence of Algorithm 3.1. To this end, we need the following results whose arguments are also similar to the ones in 18, Remark.1. In particular, for Theorem 3.(d), we provide a proof in light of the structure of each φ i so that the readers can look into the analytic difference among them. 19

20 Theorem 3.. Consider a SOCAVE () with σ min (A) > σ ma (B). Let H i be defined as in (). Suppose that the sequence {z k } is generated by Algorithm 3.1. Then, the following results hold. (a) The sequences { H i (z k ) } and {τ k } are monotonically non-increasing. (b) β k τ k for all k. (c) The sequence { k } is monotonically non-increasing and k > 0 for all k. (d) The sequence {z k } is bounded. Proof. (a) From definition of the line search in (5) and τ k := min{1, H i (z k ) }, it is clear that { H i (z k ) } and {τ k } are monotonically non-increasing. (b) We prove this conclusion by induction. First, by Algorithm 3.1, it is clear that τ0 β 0 with τ 0, β and 0 chosen in Algorithm 3.1. Secondly, we suppose that τk β k for some k. Then, for k + 1, we have k+1 τ k+1 β = k + α k k τ k+1 β τk = (1 α k ) k + α k β τ k+1 β (1 α k ) τ k β + α τk k β τ k+1 β 0, where the second equality holds due to the Newton equation (4), and the second inequality holds due to part (a). Hence, it follows that β k τk for all k. (c) From the iterative scheme z k+1 = z k + α k z k, we know k+1 = k + α k k. By the Newton equations (4) and the line search as in (5) again, it follows that τk k+1 = (1 α k ) k + α k β (1 α k) τ k β + α τk k β > 0 for all k. On the other hand, we have k+1 = (1 α k ) k + α k τ k β (1 α k) k + α k k k, where the first inequality holds due to part (b). Hence, the sequence { k } is monotonically non-increasing and k > 0 for all k. 0

21 (d) From part (a), we know the sequence { H i (z k ) } is bounded, which means there is a constant C such that H i (z k ) C. Thus, C H i (z k ) A k + BΦ i ( k, k ) b A k BΦi ( k, k ) b = ( k ) T A T A k Φ i ( k, k ) T B T BΦ i ( k, k ) b λ min (A T A) k λ ma (B T B) Φ i ( k, k ) b = λ min (A T A) k λ ma (B T B) φ i ( k, λ 1 ( k ))u (1) + φ i ( k, λ ( k ))u () b = λ min (A T A) k λ ma (B T B) φ i ( k, λ 1 ( k )) u (1) + φ i ( k, λ ( k )) u () b = λ min (A T A) k λ ma (B T B) 1 φ i ( k, λ 1 ( k )) + φ i ( k, λ ( k )) b. On the other hand, for i = 1,, 3, 4, 5, we see that φ i ( k, λ 1 ( k )) + φ i ( k, λ ( k )) = f i (, λ j ( k )) + λ 1( k ) + λ ( k ). j=1 (i) For i = 1, we have f 1 (, λ j ( k )) = 4 λ k ln(e j ( k ) k + 1) ln(e λ j (k ) k + 1). It is known that the function g(t) = 4 ln(e t + 1) ln(e t + 1) is bounded for all t IR. It follows that there eists N 1 such that j=1 f 1(, λ j ( k )) k N 1. (ii) For i =, 4, 5, it is easy to verify that there eist N i such that j=1 f i(, λ j ( k )) k N i. For i = 3, we have j=1 f i(, λ j ( k )) = 8 k := k N 3, which yields φ i ( k, λ 1 ( k )) + φ i ( k, λ ( k )) k N i + λ 1( k ) + λ ( k ) ( ) = N i k + k ( ) Ni k + k. This together with H i (z k ) C implies that k C + λ ma (B T N B) i k + b λmin (A T A) λ ma (B T B) 1

22 holds for all k. Thus, the sequence { k } is bounded. (iv) For i = 6, we know that erf(t) 1 and 0 < e t 1. Thus, it leads to φ 6( k, λ 1 ( k )) + φ 6( k, λ ( k )) ( ) ( λ 1 ( k ) + π k + λ ( k ) + ( k π + k ) ) π k where the last inequality is due to λ 1 ( k ) + λ ( k ) k. Then, it follow that k C + λ ma (B T B) k π + b λmin (A T A) λ ma (B T B) holds for all k. Thus, the sequence { k } is bounded. From all the above, the proof is complete. Now, we shall show that any sequence {z k } is generated by Algorithm 3.1 convergent to a solution to the SOCAVE (). In the net theorem we demonstrate that under our assumptions. The proof is essentially similar to a result 13, 36, Theorem 4.1. Hence, we omit the detailed proof and only present the convergence result. Theorem 3.3. Consider a SOCAVE () with σ min (A) > σ ma (B). Suppose that {z k } is generated by Algorithm 3.1. Then, any accumulation point of {z k } is a solution to the SOCAVE (). Algorithm 3.1 possesses the local quadratic convergence rate. In fact, we can achieve it by similar arguments as those in 36, 40, Theorem 8. Theorem 3.4. Consider a SOCAVE () with σ min (A) > σ ma (B). Let H i be defined as in (6) and z be the unique solution to SOCAVE (). Suppose that all V H i (z ) are nonsingular. Then, the whole sequence {z k } converges to z, and z k+1 z = O( z k z ). 4 Numerical Results In this section, we report some numerical results via five numerical eamples to evaluate the efficiency of Algorithm 3.1. First, In our eperiments, we set parameters as 0 = 0.1, 0 = rand(n, 1), δ = 0.5, σ = 10 5 and β = ma(1, 1.01 τ 0 /).

23 We stop the iterations when H(z k ) 10 6 or the number of iterations eceeds 100. All the eperiments are done on a PC with Intel(R) CPU of.40ghz and RAM of 4.00GHz, and all the programming codes are written in Matlab and run in Matlab environment. For each problem, we implement the smoothing Newton Algorithm 3.1 with si different smoothing functions φ 1 (, t), φ (, t), φ 3 (, t), φ 4 (, t), φ 5 (, t), φ 6 (, t), respectively. Each problem is randomly generated 50 times and the average results are listed in Tables, where n denotes the size of problem, itn denotes the average number of iterations, time denotes the average value of the CPU time in seconds and fails means the number of failures. Secondly, in order to compare the performance of smoothing function φ i (, t), for i = 1,, 3, 4, 5, 6 in the smoothing Newton Algorithm 3.1, we adopt the performance profile which is introduced in 7 as a means. In other words, we regard Algorithm 3.1 corresponding to a smoothing function φ i (, t), for i = 1,, 3, 4, 5, 6 as a solver, and assume that there are n s solvers and n p test problems from the test set P which is generated randomly. We are interested in using the iteration number and computing time as performance measure for Algorithm 3.1 with different φ i (, t). For each problem p and solver s, let f p,s = iteration number(or computing time) required to solve problem p by solver s. We employ the performance ratio r p,s := f p,s min{f p,s : s S}, where S is the four solvers set. We assume that a parameter r p,s r M for all p, s are chosen, and r p,s = r M if and only if solver s does not solve problem p. In order to obtain an overall assessment for each solver, we define ρ s (τ) := 1 n p size{p P : r p,s τ}, which is called the performance profile of the number of iteration for solver s. Then, ρ s (τ) is the probability for solver s S that a performance ratio f p,s is within a factor τ IR of the best possible ratio. The performance profiles of each problem are depicted in Figures Problem 4.1. Consider the SOCAVE () which is generated in the following way: first choose two random matrices B, C IR n n from a uniformly distribution on 10, 10 for every element. We compute the maimal singular value σ 1 of B and the minimal singular value σ of C, and let σ := min{1, σ /σ 1 }. Net, we divide C by σ multiplied by a random number in the interval 0, 1, and the resulting matri is denoted as A. 3

24 Accordingly, the minimum singular values of A eceeds the maimal singular value of B. We choose randomly b IR n on 0, 1 for every element. By Algorithm 3.1 in this paper, the resulting SOCAVE () is solvable. The initial point is chosen in the range 0, 1 entry-wisely. Note that a similar way to construct the problem was given in ρs(τ) ϕ 1 ϕ ϕ 3 ϕ 4 ϕ 5 ϕ τ Figure 1: Performance profile of iteration numbers of Problem ϕ ϕ ρs(τ) ϕ ϕ 4 ϕ ϕ τ Figure : Performance profile of computing time of Problem 4.1. Table 1 and Figures 1- show that function φ 1 (, t) performs worst, whereas the difference among other functions is very slight. Figure 1 demonstrates the performance profile of iteration numbers for Problem 4.1. The subplot in Figure 1 is the zoomed plot for upper-left part of the Figure 1. Figure shows the performance profile of computing time for Problem 4.1. The subplot in Figure is the zoomed plot for lower-left part of Figure. From this figure, we can see that the performance of of function φ 1 (, t) is also the worst one. 4

25 Table 1: Numerical results for Problem 4.1 φ1 φ φ3 φ4 φ5 φ6 n itn time fails itn time fails itn time fails itn time fails itn time fails itn time fails

26 Problem 4.. Consider the SOCAVE () which is generated in the following way: choose two random matrices C, D IR n n from a uniformly distribution on 10, 10 for every element, and compute their singular value decompositions C := U 1 S 1 V1 T and D := U S V T with diagonal matrices S 1 and S ; unitary matrices U 1, V 1, U and V. Then, we choose randomly b, c IR n on 0, 10 for every element. Net, we take a IR n by setting a i = c i + 10 for all i {1,..., n}, so that a b. Set A := U 1 Diag(a)V1 T and B := U Diag(b)V T, where Diag() denotes a diagonal matri with its i-th diagonal element being i. The gap between the minimal singular value of A and the maimal singular value of B is limited and can be very small. We choose randomly b IR n in 0, 10. The initial point is chosen in the range 0, 1 entry-wisely ρs(τ) ϕ 1 ϕ ϕ 3 ϕ 4 ϕ 5 ϕ τ Figure 3: Performance profile of iteration numbers of Problem ρs(τ) τ ϕ 1 ϕ ϕ 3 ϕ 4 ϕ 5 ϕ 6 Figure 4: Performance profile of computing time of Problem 4.. For Problem 4., as depicted in Figures 3-4 and Table, all the smoothing functions φ i (, t) for i = 1,, 3, 4, 5, 6 perform very well, no any discrepancy. 6

27 Table : Numerical results for Problem 4. φ1 φ φ3 φ4 φ5 φ6 n itn time fails itn time fails itn time fails itn time fails itn time fails itn time fails

28 Problem 4.3. Consider the SOCAVE () which is generated in the following way: choose two random matrices A, B IR n n from a uniformly distribution on 10, 10 for every element. In order to ensure that the SOCAVE () is solvable, we update the matri A by the following: let USV = svd(a). If min{s(i, i)} = 0 for i = 0, 1,, n, we make A = U(S E)V, and then A = λma(bt B)+0.01 λ min A. We choose randomly b IR n on (A T A) 0, 10 for every element. The initial point is chosen in the range 0, 1 entry-wisely ρs(τ) ϕ 1 ϕ ϕ 3 ϕ 4 ϕ 5 ϕ τ Figure 5: Performance profile of iteration numbers of Problem ρs(τ) ϕ 1 ϕ ϕ 3 ϕ 4 ϕ 5 ϕ τ Figure 6: Performance profile of computing time of Problem 4.3. For Problem 4.3, from the Table 3 and Figures 5-6, we see that φ 1 (, t) is obviously inferior to other functions. Moreover, in terms of the computing time, the function φ 4 (, t) is the best one, followed by φ 3 (, t). In summary, the function φ 1 (, t) is still the worst performer. Problem 4.4. We consider the SOCAVE () which is generated the same as Problem 4.1. But, here the SOC is given by K := K n 1 K nr, where n 1 = = n r = n r. 8

29 Table 3: Numerical results for Problem 4.3 φ1 φ φ3 φ4 φ5 φ6 n itn time fails itn time fails itn time fails itn time fails itn time fails itn time fails

30 ρs(τ) ϕ 1 ϕ ϕ 3 ϕ 4 ϕ 5 ϕ τ Figure 7: Performance profile of iteration numbers of Problem ϕ 1 ρs(τ) ϕ ϕ 3 ϕ 4 ϕ 5 ϕ τ Figure 8: Performance profile of computing time of Problem 4.4. Figures 7-8 show the performance profiles of Problem 4.4. Indeed, the performance profiles are similar to those for Problem 4.1 (only the cone structure is different). Again, the function φ 1 (, t) is still the worst performer and there is no significant difference among other five smoothing functions. Problem 4.5. We consider the SOCAVE () which is generated the same as Problem 4.3. But, here the SOC is given by K := K n 1 K nr, where n 1 = = n r = n r. Figures 9-10 show the performance profiles of Problem 4.5. performance of function φ 1 (, t) one more time. They verify the poor In summary, the function φ 1 (, t) is not a good choice to work with the smoothing Newton algorithm. Note that φ 1 (, t) is related to loss function and widely used in engineering like machine learning. However, for the SOCAVE, the numerical performance of function φ 1 (, t) is always the worst one. This is a very interesting phenomenon and 30

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