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A descent method for a reformulation of the second-order cone complementarity problem
Jein-Shan Chen
a,∗,1, Shaohua Pan
baDepartment of Mathematics, National Taiwan Normal University, Taipei 11677, Taiwan bSchool of Mathematical Sciences, South China University of Technology, Guangzhou 510640, China
Received 27 July 2006; received in revised form 31 January 2007
Abstract
Analogous to the nonlinear complementarity problem and the semi-definite complementarity problem, a popular approach to solving the second-order cone complementarity problem (SOCCP) is to reformulate it as an unconstrained minimization of a certain merit function overRn. In this paper, we present a descent method for solving the unconstrained minimization reformulation of the SOCCP which is based on the Fischer–Burmeister merit function (FBMF) associated with second-order cone [J.-S. Chen, P. Tseng, An unconstrained smooth minimization reformulation of the second-order cone complementarity problem, Math. Programming 104 (2005) 293–327], and prove its global convergence. Particularly, we compare the numerical performance of the method for the symmetric affine SOCCP generated randomly with the FBMF approach [J.-S. Chen, P. Tseng, An unconstrained smooth minimization reformulation of the second-order cone complementarity problem, Math. Programming 104 (2005) 293–327]. The comparison results indicate that, if a scaling strategy is imposed on the test problem, the descent method proposed is comparable with the merit function approach in the CPU time for solving test problems although the former may require more function evaluations.
© 2007 Elsevier B.V. All rights reserved.
MSC: 26B05; 90C33
Keywords: Second-order cone; Complementarity; Merit function; Descent method
1. Introduction
In this paper, we consider the following second-order cone complementarity problem (SOCCP) of finding ∈ Rn satisfying
F (), = 0, F () ∈ K, ∈ K, (1)
where·, · is the Euclidean inner product, F : Rn→ Rnis a smooth (i.e., continuously differentiable) mapping, and K is the Cartesian product of second-order cones (SOC), also called Lorentz cones[9]. In other words,
K = Kn1× · · · × Knm, (2)
∗Corresponding author. Tel.: +886 2 29325417; fax: +886 2 29332342.
E-mail addresses:jschen@math.ntnu.edu.tw(J.-S. Chen),shhpan@scut.edu.cn(S. Pan).
1Member of Mathematics Division, National Center for Theoretical Sciences, Taipei Office. The author’s work is partially supported by National Science Council of Taiwan.
0377-0427/$ - see front matter © 2007 Elsevier B.V. All rights reserved.
doi:10.1016/j.cam.2007.01.029
where m, n1, . . . , nm1, n1+ · · · + nm= n, and
Kni := {(x1, x2) ∈ R × Rni−1| x2x1}, (3)
with · denoting the Euclidean norm and K1denoting the set of nonnegative realsR+. A special case of (2) is K = Rn+, the nonnegative orthant inRn, which corresponds to m = n and n1= · · · = nm= 1. If K = Rn+, then (1) reduces to the nonlinear complementarity problem (NCP). The NCP plays a fundamental role in optimization theory and has many applications in engineering and economics; see, e.g.,[7,10–12]. Unless otherwise stated, in the first three sections of this paper, we assumeK = Knfor simplicity, i.e.,K is a single second-order cone (all the analysis can be carried over to the case whereK is a product of second-order cones without difficulty).
Various methods have been proposed for solving the SOCCP. They include interior-point methods[1,24,26,28,30], re- formulating SOC constraints as smooth convex constraints[31], and (noninterior) smoothing Newton methods[5,16,20].
These methods require solving a nontrivial system of linear equations at each iteration. In the recent paper[4], an alter- native approach based on reformulating the SOCCP as an unconstrained smooth minimization problem was studied.
In particular, they were finding a smooth function : Rn× Rn → R+such that
(x, y) = 0 ⇐⇒ x, y = 0, x ∈ Kn, y ∈ Kn. (4)
We call such a a merit function. Then SOCCP can be expressed as an unconstrained smooth (global) minimization problem:
∈Rminn(F (), ). (5)
Various gradient methods such as conjugate gradient methods and quasi-Newton methods[2,15]can be applied to (5).
For this approach to be effective, the choice of is crucial. In the case of NCP, a popular choice is
FB(a, b) = 1 2
n i=1
FB(ai, bi)2
for all a = (a1, . . . , an)T ∈ Rn and b = (b1, . . . , bn)T ∈ Rn, whereFB is the well-known Fischer–Burmeister NCP-function[13,14]defined by
FB(ai, bi) =
a2i + b2i − ai− bi.
It has been shown thatFBis smooth (even thoughFBis not differentiable) and is a merit function for NCP[8,21,22].
These two functions can be extended to the case of SOCCP via Jordan algebra shown as below. For any x =(x1, x2), y = (y1, y2) ∈ R × Rn−1, we define their Jordan product associated withKnas
x ◦ y := (x, y, y1x2+ x1y2).
The identity element under this product is e := (1, 0, . . . , 0)T ∈ Rn. We write x2to mean x ◦ x and write x + y to mean the usual componentwise addition of vectors. It is known that x2 ∈ Knfor all x ∈ Rn. Moreover, if x ∈ Kn, then there exists a unique vector inKn, denoted by x1/2, such that (x1/2)2= x1/2◦ x1/2= x. Then,
FB(x, y) := (x2+ y2)1/2− x − y (6)
is well defined for all (x, y) ∈ Rn× Rnand mapsRn× RntoRn. It was shown in[16]thatFB(x, y) = 0 if and only ifx, y = 0, x ∈ Kn, y ∈ Kn. Hence,FB: Rn× Rn→ R+given by
FB(x, y) := 12FB(x, y)2 (7)
is a merit function for SOCCP because FB satisfies (4) as well. Therefore, the SOCCP is equivalent to the global minimization problem:
∈RminnfFB() := FB(F (), ). (8)
It was also shown in the paper[4]that, like the NCP case,FBis smooth and, when∇F is positive semi-definite, every stationary point of (8) solves the SOCCP. For semi-definite complementarity problem (SDCP), which is a natural
extension of NCP whereRn+is replaced by the cone of positive semi-definite matricesSn+and the partial order is also changed bySn+(a partial order associated withSn+where ASn+B means B − A ∈ Sn+) accordingly, the aforementioned features hold for the following analog of the SDCP merit function studied in[32]:
YF(x, y) := 0(x, y) + FB(x, y), (9)
where0: R → [0, ∞) is any smooth function satisfying
0(t) = 0 ∀t 0 and 0(t) > 0 ∀t > 0. (10)
In[32],0(t) =14(max{0, t})4was considered. In fact, the functionYF, which was recently studied in[4], is also a SOCCP version merit function that enjoys favorable properties as whatFBhas. Moreover,YFpossesses properties of bounded level sets and error bound.
In this paper, we focus on the following equivalent reformulation of SOCCP, which arises via the merit functionYF
defined as in (9) and (10):
∈Rminn fYF() := YF(F (), ). (11)
We are motivated by the work[32]showing a descent method for the SDCP. Thus, the main purpose of the paper is to explore the extension to SOCCP. In other words, we wish to adopt the algorithm therein to solve the equivalent reformulation (11) of the SOCCP and prove its global convergence (see Section 3). In particular, we also compare the numerical performance of the descent algorithm for the symmetric affine SOCCPs generated randomly with the Fischer–Burmeister merit function (FBMF) approach[4]. Here it is worth pointing out that the proposed algorithm does not work for the other reformulation (8). The reason is that fFB() lacks property of bounded level sets and does not provide error bound due to the absence of the term0.
Some words about our notation. Throughout this paper,Rndenotes the space of n-dimensional real column vectors.
For any differentiable function f : Rn → R, ∇f (x) denotes the gradient of f at x. For any differentiable mapping F : Rn→ Rm,∇F (x) is an n × m matrix which denotes the transposed Jacobian of F at x.
2. Preliminaries
As mentioned in the Introduction,YFsatisfies (4), so the SOCCP can be recast as an equivalent global minimization (11). It was shown in[4]that the function fYFis smooth, has bounded level sets, and provides error bound for the unconstrained minimization reformulation. Moreover, every stationary point of problem (11) is a solution of the SOCCP.
In this section, we review some basic concepts and properties that will be used for proving the convergence results of the descent algorithm later. Since the work of[4]already includes as special cases the following lemmas, we here omit the proofs.
Lemma 2.1 (Chen and Tseng[4, Proposition 3.2]). Let FB, FBbe given by (6) and (7), respectively, andYFbe given by (9) and (10). ThenFBandYFare both smooth onRn× Rn.
Lemma 2.2 (Chen and Tseng[4, Proposition 4.2]). Let YF be given by (9) and (10) and fYF() be defined as (11). Then, for every ∈ Rn such that ∇F () is positive semi-definite, either fYF() = 0 or ∇fYF() = 0 with
d(), ∇fYF() < 0, where
d() := −(0(F (), ) + ∇xFB(F (), )). (12)
In what follows, we say that F is monotone if
F () − F (), − 0 ∀, ∈ Rn
and F is strongly monotone if there exists > 0 such that
F () − F (), − − 2 ∀, ∈ Rn.
It is well known that, when F is differentiable, F is monotone if and only if ∇F () is positive semi-definite for all
∈ Rnwhile F is strongly monotone if and only if ∇F () is positive definite for all ∈ Rn.
Lemma 2.3 (Chen and Tseng[4, Proposition 5.2]). Suppose that F is a differentiable and monotone mapping from RntoRn. Suppose also that the SOCCP (1) is strictly feasible, i.e., there exists ∈ Rnsuch that F (), ∈ int(Kn).
Then the level set
L() := { ∈ Rn | fYF()}
is nonempty and bounded for all0, where fYFis given by (11).
Remark 2.1. It is known that Lemma 2.3 is also true if the conditions of monotonicity and strict feasibility are replaced by strong monotonicity.
We next recall some basic results about the spectral factorization associated withKn. Any x = (x1, x2) ∈ R × Rn−1, admits a spectral factorization of the form
x = 1(x) · u(1)x + 2(x) · u(2)x , (13)
wherei(x) and u(i)x for i = 1, 2 are the spectral values and the associated spectral vectors of x given by
i(x) = x1+ (−1)ix2,
u(i)x =
⎧⎪
⎨
⎪⎩ 1 2
1, (−1)i x2
x2
if x2= 0, 1
2(1, (−1)iw2) if x2= 0
(14)
with w2being any vector inRn−1satisfyingw2 = 1. If x2 = 0, the factorization is unique. The set {u(1)x , u(2)x } is called a Jordan frame and has the following properties.
Property 2.1. For any x = (x1, x2) ∈ R × Rn−1with the spectral values1(x), 2(x) and spectral vectors u(1)x , u(2)x given as in (14), we have
(a) u(1)x and u(2)x are orthogonal under Jordan product and have length 1/√ 2, i.e., u(1)x ◦ u(2)x = 0, u(1)x = u(2)x =√12.
(b) u(1)x and u(2)x are idempotent under Jordan product, i.e., u(i)x ◦ u(i)x = u(i)x for i = 1, 2.
The spectral factorization (13)–(14) of x as well as x2and x1/2 has various interesting properties; see[16]. For instance, for any x = (x1, x2) ∈ R × Rn−1, with spectral values1(x), 2(x) and spectral vectors u(1)x , u(2)x , the following results hold:
(1) x2= 21(x)u(1)x + 22(x)u(2)x ∈ Kn.
(2) If x ∈ Kn, then 01(x)2(x) and x1/2=√
1(x)u(1)x +√
2(x)u(2)x .
To close this section, we present a property ofFBassociated with the spectral value.
Lemma 2.4 (Cheng and Tseng[4, Lemma 9(a)]). For any {(xk, yk)}∞k=1 ⊆ Rn×Rn, let1(xk)2(xk) and 1(yk)
2(yk) denote the spectral values of xk and yk, respectively. Then, if1(xk) → −∞ or 1(yk) → −∞, we have
FB(xk, yk) → ∞.
3. Main results
In this section, we propose a descent method for solving the unconstrained minimization reformulation (11) of the SOCCP and prove its global convergence. The proposed method uses d() defined as (12) as its direction. Now let us describe the algorithm.
Algorithm 3.1. (Step 0) Choose0∈ Rn,0, ∈ (0, 1/2), ∈ (0, 1) and set k := 0.
(Step 1) If fYF(k), then stop.
(Step 2) Compute d(k) := −(0(F (k), k)k+ ∇xFB(F (k), k)).
(Step 3) Find a step-size tk := mk, where mkis the smallest nonnegative integer m satisfying Armijo’s rule:
fYF(k+ md(k))(1 − 2m) fYF(k). (15)
(Step 4) Setk+1:= k+ tkd(k), k := k + 1 and go to Step 1.
Note that the above algorithm is∇F -free, i.e., there is no need to compute the Jacobian matrix of F , and moreover, the computation work in each iteration is very small, i.e., only several vector multiplications. In fact, this type of algorithm was also studied for the NCP (see[17]) and the SDCP (see[32]) and the most remarkable feature of this type of algorithm is that not only the step-size but also the search direction itself is adjusted via Armijo’s rule. In practical experience, is usually chosen close to zero, and is usually chosen in (101,12) depending on the confidence we have on the quality of the initial step-size (see[2]).
Next, we prove the global convergence of Algorithm 3.1. Without any loss of generality, we suppose = 0 so that the algorithm generates an infinite sequence{k}.
Proposition 3.1. Suppose that F is monotone and the SOCCP (1) is strictly feasible. Then the sequence {k} generated by Algorithm 3.1 has at least one accumulation point, and any accumulation point is a solution of the SOCCP (1).
Proof. The proof is standard and can be found in[2]. For completeness, we here present its proof by the following three steps.
(i) First, we show that, wheneverk is not a solution, there exists a nonnegative integer mkin Step 3 of Algorithm 3.1. Suppose not, then for any positive integer m, we have
fYF(k+ md(k)) − fYF(k) > − 2mfYF(k).
Dividing by mon both sides and letting m → ∞ yields
∇fYF(k), d(k)0. (16)
Since F is monotone which is equivalent to ∇F () is positive semi-definite, the inequality (16) contradicts Lemma 2.2. Hence, we can find an integer mkin Step 3.
(ii) Secondly, we show that the sequence{k} generated by the algorithm has at least one accumulation point. By the descent property of Algorithm 3.1, the sequence{fYF(k)}k∈N is decreasing. Hence by Lemma 2.3, we have that {k} is bounded, and consequently has at least one accumulation point.
(iii) Finally, we prove that any accumulation point of {k} is a solution of the SOCCP (1). Let ∗ be an arbitrary accumulation point of{k}k∈N. In other words, there is a subsequence{k}k∈K converging to∗, where K is a subset of N. We know d(·) is continuous (since 0andFBare smooth) which implies{d(k)}k∈Kconverges to d(∗). Next, we need to discuss two cases. First, we consider the case where there exists a constant ¯ such that mk ¯ > 0 for all k ∈ K. Then, from (15), we have
fYF(k+1)(1 − ¯ 2)fYF(k)
for all k ∈ K and the entire sequence {fYF(k)}k∈Kis decreasing. Thus, we obtain fYF(∗) = 0 (by taking the limit) which says∗is a solution of the SOCCP (1). Now, we consider the other case where there exists a further
subsequence such that mk → 0. Note that by Armijo’s rule (15) in Step 3, we have fYF(k+ mk−1d(k)) − fYF(k) > − 2(mk−1)fYF(k).
Dividing by mk−1both sides and passing the limit on the further subsequence, we obtain
∇fYF(∗), d(∗)0,
which yields that∗is a solution of the SOCCP (1) by Lemma 2.2.
Proposition 3.2. Let F be a continuously differentiable and strongly monotone mapping. Then the sequence {k} generated by Algorithm 3.1 converges to the unique solution of the SOCCP (1).
Proof. The proof is routine (see[7]); however, we present it for completeness. We know that the property of bounded level sets is also held when F is strongly monotone, so following the same arguments as in the proof of Proposition 3.1, we again obtain that{k} has at least one accumulation point and any accumulation point is a solution of the SOCCP (1).
On the other hand, the strong monotonicity of F implies that the SOCCP (1) has at most one solution. To see this, say there are two solutions∗, ∗∈ Rnsuch that
F (∗), ∗ = 0,
F (∗) ∈ Kn, ∗∈ Kn and
F (∗), ∗ = 0,
F (∗) ∈ Kn, ∗∈ Kn. Since F is strongly monotone, we have F (∗) − F (∗), ∗− ∗ > 0. However,
F (∗) − F (∗), ∗− ∗ = F (∗), ∗ + F (∗), ∗ − F (∗), ∗ − F (∗), ∗
= − F (∗), ∗ − F (∗), ∗
0,
where the inequality is due to F (∗), ∗, F (∗), ∗are all inK. Hence, it is a contradiction and therefore there is at most one solution for the SOCCP (1).
From all the above, it says there is a unique solution∗, so the entire sequence{xk} must converge to ∗. Propositions 3.1 and 3.2 may not be so surprising since they seem as expected. Nonetheless, we do not take them for granted before we prove them even though we think they should be true. Now, the results of Propositions 3.1 and 3.2 do fill up the gap in the literature. We notice that Lemma 2.3 plays an important role in the proofs for them. In fact, the assumption of strict feasibility is necessary for Lemma 2.3 to be held. For example, when F () ≡ 0, every
∈ Knis a solution of SOCCP (1) and hence the solution set is unbounded. In the following, we continue a further study of considering another (weaker) condition to replace this kind of strict condition by F being a R01-function (will be defined in Definition 3.1) that is a new concept recently developed for linear and nonlinear transformations on Euclidean Algebra[18,25,29].
Definition 3.1. For a mapping F : Rn→ Rn, it is called a
(a) R01-function if for any sequence{k} such that
k → ∞, (−k)+
k → 0, (−F (k))+
k → 0, (17)
we have
lim inf
k→∞
k, F (k)
k2 > 0;
(b) R02-function if for any sequence{k} such that (17), we have
lim inf
k→∞
(k◦ F (k))
k2 > 0.
The above concepts are extensions of the ones defined for NCP and for SDCP. It is also known that every R01-function is R02-function[23, Lemma 4]; and if F has the uniform Jordan P -property (see[18,25,29]), then F is R02-function [23, Lemma 5]. However, it is not clear whether uniform P -property (see[18,25,29]) implies R02-function or not. With this new concept, Lemma 2.3 and Proposition 3.1 can be improved as Lemma 3.1 and Proposition 3.3, respectively.
These results are significant not only because they are brand-new but also because there is no need of the assumption of strict feasibility therein.
Lemma 3.1. Let fYFbe given as in (11). Suppose that F is a R01-function. Then the level set L() := { ∈ Rn | fYF()}
is bounded for all0.
Proof. We will prove this result by contradiction. Suppose there exists an unbounded sequence{k} ⊂ L() for some
0. It can be seen that the sequence of the smaller spectral values of {k} and {F (k)} are bounded below. In fact, if not, it follows from Lemma 2.4 that fYF(k) → ∞, which contradicts {k} ⊂ L(). Therefore, {(−k)+} and {(−F (k))+} are bounded above, which says the conditions in (17) are satisfied. Then, by the assumption of R01-function, we have
lim inf
k→∞
k, F (k)
k2 > 0.
This yieldsk, F (k) → ∞, and hence fYF(k) → ∞ by definition of fYFgiven as in (11). Thus, it is a contradiction to{k} ⊂ L().
Proposition 3.3. Let F be a continuously differentiable mapping. Suppose that F is R01-function. Then the sequence {k} generated by Algorithm 3.1 has at least one accumulation point, and any accumulation point is a solution of the SOCCP (1).
Proof. By applying Lemma 3.1 and following the same arguments as in Proposition 3.1, the desired results hold. We omit it.
From[23,29], the condition of R01-function is weaker than strong monotonicity, and it is also weaker than mono- tonicity plus strict feasibility in certain sense. However, it is not clear yet whether R01-function can be replaced by R02-function in our brand-new results.
4. Numerical results
In this section, we report our computational experience with solving the symmetric affine SOCCPs generated ran- domly by the proposed algorithm, and compare the numerical performance with the FBMF approach[4]. Unless otherwise stated, the function fYFin Algorithm 3.1 is always defined as in (11), whereYF is defined by (9) and (10) with0(t) =12(max{0, t})2.
The symmetric affine SOCCP is stated as follows: finding ∈ Rnsuch that
F (), = 0, ∈ K, F () = M + q ∈ K, (18)
where M ∈ Rn×n and q ∈ Rnare a given symmetric positive semidefinite matrix and a vector, respectively. In our experiments, the matrix M and the vector q are generated by the following procedure. Elements of q were chosen randomly from the interval[−1, 1] and the matrix M was obtained by setting M = NNT, where N is a square matrix
whose nonzero elements are chosen randomly from the interval [−1, 1]. In this procedure, the number of nonzero elements of N is determined so that the nonzero density of M can be approximately estimated.
All experiments were done at a PC with 2.8 GHz CPU and 512 MB memory. The computer codes were all written in Matlab 6.1. To improve the numerical behavior of Algorithm 3.1, we replaced the standard Armijo rule by the nonmonotone line search as described in[19], i.e., we computed the smallest nonnegative integer m such that
fYF(k+ md(k))Wk− 2mfYF(k), (19)
whereWkis given by
Wk= max
j =k−mk,...,k fYF(j)
and where, for given nonnegative integers ˆm and s, we set mk=
0 if k s,
min{mk−1+ 1, ˆm} otherwise. (20)
Throughout the experiments, unless otherwise stated, we used the following parameters:
ˆm = 5, s = 5, = 0.3 and = 1.0e − 4. (21)
For the FBMF approach[4], we chose a limited-memory BFGS algorithm with five limited-memory vector-updates [3]to solve the unconstrained minimization reformulation (8) for the SOCCP (1). For the scaling matrix H0= I in the BFGS algorithm, we adopted the choice of = pTq/pTq recommended by[27, p. 226], where p := − oldand q := ∇fFB() − ∇fFB(old). To ensure convergence, we revert to the steepest descent direction −∇fFB() whenever the current direction fails to satisfy the sufficient descent condition
∇fFB()T − 10−4∇fFB().
In addition, we also employed the same nonmonotone line search as above to seek a suitable step-length, except that the parameter is chosen as 0.2.
During the experiments, we started Algorithm 3.1 and the FBMF approach with the starting point0=0.001(1, 1, . . . , 1)Tand terminated the iterate once one of the following conditions is satisfied:
(1) max{(), |F ()T|}10−4, where represents fYFor fFB. (2) The number of iteration is over 50 000.
(3) The step-length is lower than 10−16.
We have done the following three groups of experiments.
Experiment A: Testing the influence of the scaling strategy on Algorithm 3.1 and the FBMF method. Note that, when the matrix M and the vector q in (18) are replaced by
M =¯ M
w and ¯q = q
w, (22)
where w 1 is a constant, the optimal solution of problem (18) does not change. Hence, in this experiment, we generated 10 test problems with sparsity 0.5% and 10% and m = 10, n1= n2= · · · = nm= 100, and then solved each problem and their different scaled formulations with Algorithm 3.1 and the FBMF approach. Numerical results are summarized inTables 1 and 2, where NO. represents the number of problem, Den denotes the approximate sparsity of M, Nf and Time, respectively, denote the total number of function evaluations and the CPU time for solving each problem.
FromTables 1 and 2, we see that, when w > 1, i.e., imposing the scaling strategy on the original problems, Algorithm 3.1 and the FBMF approach require much less function evaluations. Therefore, the scaling strategy in (22) can greatly improve the numerical performance of Algorithm 3.1 and the merit function approach. In particular, for those problems to which Algorithm 3.1 fails due to too small step-length, using the scaling strategy can yield satisfying solutions. This implies that Algorithm 3.1 has more dependence on the scaling strategy than the MF approach.
Experiment B: Testing Algorithm 3.1 and the FBMF approach on the affine SOCCP (18) with various degree of sparsity. In this experiment, we generated 10 test problems with m = 1 and n = 1000 for each nonzero density 0.1%,
Table 1
Numerical results of Algorithm 3.1 for the scaled problems
NO. Den (%) w = 1 w = 10 w = 50 w = 100
Nf Time Nf Time Nf Time Nf Time
1 0.5 21 693 69.68 6716 21.71 4201 15.31 5608 21.89
2 0.5 55 916 175.9 26 234 85.81 17 836 68.15 24 073 98.15
3 0.5 11 897 44.57 989 3.82 803 3.40 1168 5.34
4 0.5 14 860 53.68 998 4.04 776 3.28 1047 5.07
5 0.5 13 260 48.53 553 2.01 553 2.32 733 3.20
6 10 – – 2238 89.67 237 10.09 99 4.46
7 10 – – 2518 95.54 264 10.64 114 5.21
8 10 – – 8592 344.4 228 10.26 162 7.23
9 10 – – – – 273 12.78 81 6.18
10 10 – – 1982 82.60 239 10.98 125 5.56
“–” means that the iteration was stopped since the step-length was less than 10−16.
Table 2
Numerical results of the FBMF method for the scaled problems
NO. Den (%) w = 1 w = 10 w = 50 w = 100
Nf Time Nf Time Nf Time Nf Time
1 0.5 8135 56.96 4346 30.01 3076 22.14 4649 30.56
2 0.5 9086 57.56 14 020 91.06 16 619 117.5 21 284 149.9
3 0.5 611 3.95 531 3.70 812 5.89 976 6.82
4 0.5 1030 7.65 677 5.17 493 4.09 970 6.96
5 0.5 769 5.56 403 2.98 583 4.09 591 4.32
6 10 6682 488.0 807 64.15 132 9.65 100 7.40
7 10 4668 337.7 737 56.85 247 19.21 185 16.37
8 10 5639 431.1 812 63.82 131 10.12 114 9.20
9 10 4616 347.4 723 57.21 112 9.20 81 6.18
10 10 5818 452.6 702 59.12 220 17.04 96 7.59
Table 3
Numerical results for the affine SOCCP with sparsity 0.1%
NO. Algorithm 3.1 MF method NO. Algorithm 3.1 MF method
Nf Time Nf Time Nf Time Nf Time
1 597 0.76 369 1.10 2 * * * *
3 539 0.85 325 0.98 4 * * * *
5 * * * * 6 * * * *
7 254 0.34 127 0.33 8 * * * *
9 * * * * 10 799 0.95 143 0.28
“∗” means that the iteration was stopped since the number of iteration was over 50 000.
0.5%, 1%, 10%, 50% and 80%, and then solved each problem with Algorithm 3.1 and the FBMF approach. Numerical results were summarized inTables 3 and 4, where Nf and Time are same as Experiment A, Gap denotes the value of
|F ()T| at the final iteration, and Scale inTable 4denotes the value of w in (22). In particular, the values of Gap, Nf and Time inTable 4are the averages of 10 trials for each sparsity.
FromTable 3, it appears that Algorithm 3.1 and the FBMF approach have similar numerical performance on those problems with sparsity 0.1%. However, fromTable 4, we see that, under the scaling strategy shown, Algorithm 3.1
Table 4
Numerical results for the affine SOCCP with different sparsity
Den (%) Scale Algorithm 3.1 MF approach
Gap Nf Time Gap Nf Time
0.5 1 9.16e − 5 1597.1 3.28 8.77e − 5 914.9 4.09
1 1 8.10e − 5 7401.4 28.76 6.51e − 5 5016.8 38.31
10 10 6.89e − 5 402.6 15.37 6.85e − 5 312.0 21.92
50 100 5.90e − 5 472.2 17.07 6.59e − 5 568.7 39.00
80 100 4.81e − 5 468.4 17.54 7.62e − 5 668.7 45.23
Table 5
Numerical results for the affine SOCCP with differentK
m Scale Algorithm 3.1 MF approach
Gap Nf Time Gap Nf Time
1 10 7.95e−5 201.3 7.64 8.759e−5 167.5 10.58
10 10 8.24e−5 497.6 17.68 8.74e−5 217.1 14.03
50 10 9.34e−5 1193 49.12 9.71e−5 266.5 19.42
100 100 5.75e−5 116.4 6.62 7.86e−5 138.7 11.44
200 100 4.54e−5 129.2 9.28 7.93e−5 149.2 15.69
500 100 7.09e−5 9719.4 1115.4 7.93e−5 149.2 15.69
Table 6
Numerical results of Algorithm 3.1 for different
NO. = 0.3 = 0.1 NO. = 0.3 = 0.1
Nf Time Nf Time Nf Time Nf Time
1 75 8.42 198 21.90 2 3081 355.5 142 15.23
3 19 001 2231.4 210 22.25 4 6644 769.0 179 19.09
5 69 7.93 178 19.00 6 75 8.37 804 88.67
7 62 169 7068.4 132 14.56 8 5939 689.6 295 32.23
9 77 8.54 208 23.03 10 64 7.12 144 16.03
always needed less CPU time than the FBMF approach although the former may require more function evaluations.
In addition, we also observe that the number of function evaluations required by Algorithm 3.1 will become less when the sparsity of M becomes higher.
Experiment C: Testing Algorithm 3.1 and the FBMF approach on the affine SOCCP (18) with various Cartesian structures ofK. To construct SOCs of various types, we chose ni and m such that n1= n2= · · · = nmand n1+
· · · + nm= 2000. For each type of K, we solved 10 test problems with nonzero density 1% by Algorithm 3.1 and the FBMF approach, respectively. Numerical results were reported inTable 5,where Scale, Gap, Nf and Time are same as Experiment A, and particularly the values of Gap, Nf and Time are the averages of 10 trials for each type ofK.
FromTable 5, we see that, under the scaling strategy shown, Algorithm 3.1 is comparable with the FBMF method for the first five groups of test problems whether in the CPU time or in the number of function evaluations. For the last group of test problems, Algorithm 3.1 obviously required more CPU time and function evaluations than the FBMF approach. However, fromTable 6, we see that if Scale is still chosen as 100 but the parameter in the line search is chosen as 0.1 instead of 0.3, the numerical performance of Algorithm 3.1 will have a great improvement, and moreover, the CPU time and the number of function evaluations needed are comparable with those of the FBMF method.
To sum up, for the symmetric affine SOCCPs in (18), if a suitable scaling strategy and the parameter are used, Algorithm 3.1 will be comparable with, even superior to, the FBMF method in the CPU time for solving test prob-
lems although the former may require more function evaluations. Otherwise, the FBMF approach will be superior to Algorithm 3.1 whether in the CPU time or in the number of function evaluations.
5. Final Remarks
In this paper, we investigated a descent method for the equivalent reformulation (11) of the SOCCP which was also used for the NCP and the SDCP in literature, and proved its global convergence under some mild assumptions.
Numerical comparison with the FBMF approach[4]for symmetric affine SOCCPs generated randomly indicate that the descent method is comparable with, even to superior to, the FBMF approach in the CPU time if a suitable scaling strategy and the parameter in line search are adopted. We also expect that the method can be used to deal with large SOCCPs due to very small computational work per iteration. In addition, we notice that the proposed algorithm does not work for another reformulation (8) of the SOCCP since fFBlacks property of bounded level sets (Lemma 2.3) where0plays an important role therein.
Propositions 3.1 and 3.2 are more or less an afterthought of[4], nonetheless, it does parallel the extension to the SOCCP from the NCP and SDCP cases. On the other hand, this work does a further study based on replacing the conditions of monotonicity and strict feasibility by a new (and weaker under certain sense) so-called R01-function. More specifically, under the new so-called R01-function condition, the level sets of fYFare still bounded and the proposed descent algorithm still has global convergence. These results are significant not only because they are brand-new but also because there is no need of the assumption of strict feasibility therein.
One future topic is to analyze the convergence rate theoretically which is more intractable. Other direction like weakening conditions which guarantees the property of bounded level sets is also interesting and worthwhile. There may be the direction as one referee pointed out which is to apply this optimization method to real-life studies, for example[6]and references therein.
Acknowledgments
The authors thank the referees for their careful reading of the paper and helpful suggestions.
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