A Two-Parametric Class of Merit Functions for the Second-Order Cone Complementarity Problem

Abstract

We propose a two-parametric class of merit functions for the second-order cone complementarity problem (SOCCP) based on the one-parametric class of complementarity functions. By the new class of merit functions, the SOCCP can be reformulated as an unconstrained minimization problem. The new class of merit functions is shown to possess some favorable properties. In particular, it provides a global error bound if 𝐹 and 𝐺 have the joint uniform Cartesian 𝑃-property. And it has bounded level sets under a weaker condition than the most available conditions. Some preliminary numerical results for solving the SOCCPs show the effectiveness of the merit function method via the new class of merit functions.

1. Introduction

We consider the following second-order cone complementarity problem (SOCCP) of finding (𝑥,𝑦,𝜁)∈𝑅^𝑛×𝑅^𝑛×𝑅^𝑛 such that (𝜁)(𝜁),𝑥,𝑦=0,𝑥∈𝐾,𝑦∈𝐾,𝑥=𝐹,𝑦=𝐺 (1) where ⟨⋅,⋅⟩ is the Euclidean inner product and 𝐹∶𝑅^𝑛→𝑅^𝑛 and 𝐺∶𝑅^𝑛→𝑅^𝑛 are continuously differentiable mappings. Here 𝐾⊂𝑅^𝑛 is the Cartesian product of second-order cones (SOC); that is, 𝐾=𝐾^𝑛1×𝐾^𝑛2×⋯×𝐾^𝑛𝑚 with 𝑚,𝑛₁,…,𝑛_𝑚≥1, 𝑛=𝑛₁+𝑛₂+⋯+𝑛_𝑚, and the 𝑛_𝑖-dimensional SOC defined by 𝐾^𝑛𝑖∶=𝑥_𝑖=𝑥_𝑖1;𝑥_𝑖2∈𝑅×𝑅^𝑛𝑖−1∶𝑥_𝑖1−𝑥_𝑖2≥0, (2) with ‖⋅‖ denoting the Euclidean norm.

Recently great attention has been paid to the SOCCP, since it has a variety of engineering and management applications, such as filter design, antenna array weight design, truss design, and grasping force optimization in robotics [1, 2]. Furthermore, the SOCCP contains a wide class of problems, such as nonlinear complementarity problems (NCP) and second-order cone programming (SOCP) [3, 4]. For example, the SOCCP with 𝑛₁=𝑛₂=⋯=𝑛_𝑚=1 and 𝐺(𝜁)=𝜁 for any 𝜁∈𝑅^𝑛 is the NCP, and the KKT conditions for the SOCP reduce to the SOCCP.

There have been various methods for solving SOCCPs [5], such as interior point methods [6–8], (noninterior continuation) smoothing Newton methods [4, 9–12], and smoothing-regularization methods [13]. Recently, there is an alternative approach [14, 15] based on reformulating the SOCCP as an unconstrained smooth minimization problem. In that approach, it aims to find a smooth function 𝜓∶𝑅^𝑛×𝑅^𝑛→𝑅₊ such that 𝜓𝑥,𝑦=0⟺𝑥,𝑦=0,𝑥∈𝐾,𝑦∈𝐾. (3) Such a 𝜓 is called a merit function for the SOCCP. Thus the SOCCP is equivalent to the following unconstrained smooth (global) minimization problem: min_{𝜁∈𝑅𝑛}𝜓(𝐹(𝜁),𝐺(𝜁)). (4)

A popular choice of 𝜓 is the Fischer-Burmeister (FB) merit function 𝜓_FB1𝑥,𝑦∶=2𝜙_FB𝑥,𝑦², (5) where 𝜙_FB∶𝑅^𝑛×𝑅^𝑛→𝑅^𝑛 is the vector-valued FB function defined by 𝜙_FB𝑥,𝑦∶=𝑥²+𝑦²−𝑥−𝑦, (6) with 𝑥²=𝑥∘𝑥 denoting the Jordan product between 𝑥 and itself and √𝑥 being a vector such that (√𝑥)²=𝑥. The function 𝜓_FB is shown to be a merit function for the SOCCP in [14].

In this paper, we consider the two-parametric class of merit functions defined by 𝜓_𝜏1,𝜏2𝑥,𝑦∶=𝜏₁𝜓₀𝑥,𝑦+𝜓_𝜏2𝑥,𝑦, (7) where 𝜓₀∶𝑅^𝑛×𝑅^𝑛→𝑅₊ and 𝜓_𝜏∶𝑅^𝑛×𝑅^𝑛→𝑅₊ are given, respectively, by 𝜓₀1𝑥,𝑦∶=2𝑥∘𝑦₊²,𝜓_𝜏1𝑥,𝑦∶=2𝜙_𝜏𝑥,𝑦₊², (8)(9) with (⋅)₊ denoting the metric projection on the second-order cone 𝐾, 𝜏₁>0 and 𝜏₂∈(0,4). Here 𝜙_𝜏∶𝑅^𝑛→𝑅^𝑛 is the one-parametric class of SOC complementarity functions [16] defined by 𝜙_𝜏𝑥,𝑦∶=𝑥²+𝑦²+(𝜏−2)𝑥∘𝑦−𝑥−𝑦, (10) where 𝜏∈(0,4) is an arbitrary but fixed parameter. When 𝜏=2, 𝜙_𝜏 reduces to the vector-valued FB function given by (6), and as 𝜏→0, it becomes a multiple of the vector-valued residual function 𝜙_NR𝑥,𝑦∶=𝑥−𝑥−𝑦₊. (11) Thus, the one-parametric class of vector-valued functions (10) covers two popular second-order cone complementarity functions. Hence the two-parametric class of merit functions defined as (7)–(10) includes a broad class of merit functions.

We will show that the SOCCP can be reformulated as the following unconstrained smooth (global) minimization problem: min_{𝜁∈𝑅𝑛}𝑓(𝜁)∶=𝜓_𝜏1,𝜏2(𝐹(𝜁),𝐺(𝜁)). (12)

If 𝜏₁=1 and 𝜏₂=2, the function 𝑓 in (12) induced by the new class of merit functions 𝜓_𝜏1,𝜏2 reduces to [17] 𝑓_LT(𝜁)∶=𝜓₀(𝐹(𝜁)(𝜁+1,𝐺))2𝜙_FB(𝐹(𝜁),𝐺(𝜁))₊², (13) with 𝜓₀ given as (8). It has been shown that 𝑓_LT provides a global error bound if 𝐹 and 𝐺 are jointly strongly monotone, and it has bounded level sets if 𝐹 and 𝐺 are jointly monotone and a strictly feasible solution exists [17]. In contrast, the merit function 𝜓_FB lacks these properties.

Motivated by these works, we aim to study the two-parametric class of merit functions for the SOCCP defined as (7)–(10) and its favorable properties in this paper. We also prove that the class of merit functions provides a global error bound if 𝐹 and 𝐺 have the joint uniform Cartesian 𝑃-property, which will play an important role in analyzing the convergence rate of some iterative methods for solving the SOCCP. And it has bounded level sets under a rather weak condition, which ensures that the sequence generated by a descent method has at least one accumulation point.

The organization of this paper is as follows. In Section 2, we review some preliminaries including the Euclidean Jordan algebra associated with SOC and some results about the one-parametric class of SOC complementarity functions. In Section 3, based on the one-parametric class of SOC complementarity functions, we propose a two-parametric class of merit functions for the second-order cone complementarity problem (SOCCP), which is shown to possess some favorable properties. In Section 4, we show that the class of merit functions provides a global error bound if 𝐹 and 𝐺 have the joint uniform Cartesian 𝑃-property, and it has bounded level sets under a rather weak condition. Some preliminary numerical results are reported in Section 5. And we close this paper with some conclusions in Section 6.

In what follows, we denote the nonnegative orthant of 𝑅 by 𝑅₊. We use the symbol ‖⋅‖ to denote the Euclidean norm defined by ‖𝑥‖∶=𝑥^𝑇𝑥 for a vector 𝑥 or the corresponding induced matrix norm. For simplicity, we often use 𝑥=(𝑥₁;𝑥₂) for the column vector 𝑥=(𝑥₁,𝑥^𝑇₂)^𝑇. For the SOC 𝐾^𝑛, int𝐾^𝑛, and bd𝐾^𝑛 mean the topological interior and the boundary of 𝐾^𝑛, respectively.

2. Preliminaries

In this section, we recall some preliminaries, which include Euclidean Jordan algebra [3, 18] associated with the SOC 𝐾 and some results used in the subsequent analysis.

Without loss of generality, we may assume that 𝑚=1 and 𝐾=𝐾^𝑛 in Sections 2 and 3.

First, we recall the Euclidean Jordan algebra associated with the SOC and some useful definitions. The Euclidean Jordan algebra for the SOC 𝐾^𝑛 is the algebra defined by 𝑥∘𝑦=𝑥^𝑇𝑦;𝑥₁𝑦₂+𝑦₁𝑥₂,∀𝑥,𝑦∈𝑅^𝑛, (14) with 𝑒=(1,0,…,0)∈𝑅^𝑛 being its unit element. Given an element 𝑥=(𝑥₁;𝑥₂)∈𝑅×𝑅^𝑛−1, we define 𝐿(𝑥)=𝑥₁𝑥^𝑇₂𝑥₂𝑥₁𝐼, (15) where 𝐼 represents the (𝑛−1)×(𝑛−1) identity matrix. It is easy to verify that 𝑥∘𝑦=𝐿(𝑥)𝑦 for any 𝑦∈𝑅^𝑛. Moreover, 𝐿(𝑥) is symmetric positive definite (and hence invertible) if and only if 𝑥∈int𝐾^𝑛.

Now we give the spectral factorization of vectors in 𝑅^𝑛 associated with the SOC 𝐾^𝑛. Let 𝑥=(𝑥₁;𝑥₂)∈𝑅×𝑅^𝑛−1. Then 𝑥 can be decomposed as 𝑥=𝜆₁𝑢⁽¹⁾+𝜆₂𝑢⁽²⁾, (16) where 𝜆₁,𝜆₂ and 𝑢⁽¹⁾,𝑢⁽²⁾ are the spectral values and the associated spectral vectors of 𝑥 given by 𝜆_𝑖=𝑥₁+(−1)^𝑖𝑥₂,𝑢^(𝑖)=121;(−1)^𝑖𝑥₂𝑥₂,if𝑥₂1≠0,21;(−1)^𝑖𝜔,otherwise, (17) for 𝑖=1,2, with any 𝜔∈𝑅^𝑛−1 such that ‖𝜔‖=1. It is obvious that ‖𝑢⁽¹⁾‖=‖𝑢⁽²⁾‖=1/√2. By the spectral factorization, a scalar function can be extended to a function for the SOC. For any 𝑥∈𝑅^𝑛, we define 𝑥²=𝜆²₁𝑢⁽¹⁾+𝜆²₂𝑢⁽²⁾. (18) Since both eigenvalues of any 𝑥∈𝐾^𝑛 are nonnegative, we define √𝑥=𝜆₁𝑢⁽¹⁾+𝜆₂𝑢⁽²⁾. (19)

Lemma 1 (see [14]). Let 𝐶 be any closed convex cone in 𝑅^𝑛. For each 𝑥∈𝑅^𝑛, let 𝑥⁺_𝐶 and 𝑥⁻_𝐶 denote the nearest point (in the Euclidean norm) projection of 𝑥 onto 𝐶 and −𝐶^∗, respectively. The following results hold.(i)For any 𝑥∈𝑅^𝑛, we have 𝑥=𝑥⁺_𝐶+𝑥⁻_𝐶 and ‖𝑥‖²=‖𝑥⁺_𝐶‖²+‖𝑥⁻_𝐶‖².(ii)For any 𝑥∈𝑅^𝑛 and 𝑦∈𝐶, we have ⟨𝑥,𝑦⟩≤⟨𝑥⁺_𝐶,𝑦⟩.

Lemma 2 (see [19]). Let 𝑥=(𝑥₁;𝑥₂)∈𝑅×𝑅^𝑛−1 and 𝑦=(𝑦₁;𝑦₂)∈𝑅×𝑅^𝑛−1. Then we have 𝑥,𝑦≤√2𝑥∘𝑦₊. (20)

The following results, describing the special properties of the function 𝜙_𝜏 given as (10), will play an important role in the subsequent analysis.

Lemma 3 (see [16]). For any 𝑥=(𝑥₁;𝑥₂), 𝑦=(𝑦₁;𝑦₂)∈𝑅×𝑅^𝑛−1, if 𝑧=(𝑧₁;𝑧₂)=𝑥²+𝑦²+(𝜏−2)(𝑥∘𝑦)∉int𝐾^𝑛, then 𝑥²₁=𝑥₂²,𝑦²₁=𝑦₂²,𝑥₁𝑦₁=𝑥^𝑇₂𝑦₂,𝑥₁𝑦₂=𝑦₁𝑥₂,𝑥²₁+𝑦²₁+(𝜏−2)𝑥₁𝑦₁=𝑥₁𝑥₂+𝑦₁𝑦₂+(𝜏−2)𝑥₁𝑦₂=𝑥₂²+𝑦₂²+(𝜏−2)𝑥^𝑇₂𝑦₂. (21) If, in addition, (𝑥,𝑦)≠(0,0), then 𝑧₂≠0, and furthermore, 𝑥^𝑇₂𝑧₂𝑧₂=𝑥₁,𝑥₁𝑧₂𝑧₂=𝑥₂,𝑦^𝑇₂𝑧₂𝑧₂=𝑦₁,𝑦₁𝑧₂𝑧₂=𝑦₂. (22)

Lemma 4 (see [20]). For any 𝑥,𝑦∈𝑅^𝑛, let 𝜙_𝜏 be defined as in (10). Then, 2𝜙_𝜏𝑥,𝑦²≥2𝜙_𝜏𝑥,𝑦₊²≥4−𝜏2(−𝑥)₊²+−𝑦₊². (23)

3. A Two-Parametric Class of Merit Functions

In this section, we study the two-parametric class of merit functions 𝜓_𝜏1,𝜏2 given by (7)–(10). As we will see, 𝜓_𝜏1,𝜏2 has some favorable properties. The most important property is that the SOCCP can be reformulated as the global minimization of the function 𝑓(𝜁) given as (12). Moreover, the function 𝑓 provides a global error bound and bounded level sets under weak conditions, which will be shown in the next section.

Proposition 5. Let 𝜓_𝜏 be given by (9). Then, 𝜓_𝜏𝑥,𝑦=0,𝑥,𝑦≤0⟺𝑥,𝑦=0,𝑥∈𝐾^𝑛,𝑦∈𝐾^𝑛. (24)

Proof. Suppose 𝑥∈𝐾^𝑛, 𝑦∈𝐾^𝑛, and ⟨𝑥,𝑦⟩=0. Thus by Proposition 3.1 [16], we have 𝜙_𝜏(𝑥,𝑦)=0 and therefore 𝜓_𝜏(𝑥,𝑦)=(1/2)‖𝜙_𝜏(𝑥,𝑦)₊‖²=0, ⟨𝑥,𝑦⟩≤0. Conversely, we assume 𝜓_𝜏(𝑥,𝑦)=0 and ⟨𝑥,𝑦⟩≤0. Then 𝜙_𝜏(𝑥,𝑦)₊=0 implies 𝜙_𝜏∶=𝜙_𝜏(𝑥,𝑦)∈−𝐾^𝑛. From (10), we obtain 𝑥+𝑦=𝑥²+𝑦²+(𝜏−2)𝑥∘𝑦−𝜙_𝜏. (25) Squaring both sides yields (4−𝜏)𝑥∘𝑦=−2𝑥²+𝑦²+(𝜏−2)𝑥∘𝑦∘𝜙_𝜏+𝜙_𝜏². (26) Taking the trace of both sides and using the fact tr(𝑥∘𝑦)=2⟨𝑥,𝑦⟩, we have 2(4−𝜏)𝑥,𝑦=−4𝑥²+𝑦²+(𝜏−2)𝑥∘𝑦,𝜙_𝜏+2𝜙_𝜏². (27) Since 𝑥²+𝑦²+(𝜏−2)(𝑥∘𝑦)∈𝐾^𝑛 and 𝜙_𝜏∈−𝐾^𝑛, we obtain −4𝑥²+𝑦²+(𝜏−2)𝑥∘𝑦,𝜙_𝜏≥0, (28) and thus the right hand side of (27) is nonnegative. Then by the assumption ⟨𝑥,𝑦⟩≤0, we have ⟨𝑥,𝑦⟩=0. This together with (27) implies 𝜙_𝜏(𝑥,𝑦)=0. Therefore, it follows from Proposition 3.1 [16] that 𝑥∈𝐾^𝑛 and 𝑦∈𝐾^𝑛.

Proposition 6. The function 𝜓_𝜏 given by (9) is differentiable at any (𝑥,𝑦)∈𝑅^𝑛×𝑅^𝑛. Moreover, ∇_𝑥𝜓_𝜏(0,0)=∇_𝑦𝜓_𝜏(0,0)=0; if 𝑥²+𝑦²+(𝜏−2)(𝑥∘𝑦)∈int𝐾^𝑛, then ∇_𝑥𝜓_𝜏𝑥,𝑦=𝐿_{𝑥+((𝜏−2)/2)𝑦}𝐿_𝑧⁻¹−𝐼𝜙_𝜏𝑥,𝑦₊,∇_𝑦𝜓_𝜏𝑥,𝑦=𝐿_{𝑦+((𝜏−2)/2)𝑥}𝐿_𝑧⁻¹−𝐼𝜙_𝜏𝑥,𝑦₊; (29) if (𝑥,𝑦)≠(0,0) and 𝑥²+𝑦²+(𝜏−2)(𝑥∘𝑦)∉int𝐾^𝑛, then 𝑥²₁+𝑦²₁+(𝜏−2)𝑥₁𝑦₁≠0, and ∇_𝑥𝜓_𝜏=𝑥𝑥,𝑦¹+((𝜏−2)/2)𝑦₁𝑥²₁+𝑦²₁+(𝜏−2)𝑥₁𝑦₁−1𝜙_𝜏𝑥,𝑦₊,∇_𝑦𝜓_𝜏=𝑦𝑥,𝑦¹+((𝜏−2)/2)𝑥₁𝑥²₁+𝑦²₁+(𝜏−2)𝑥₁𝑦₁−1𝜙_𝜏𝑥,𝑦₊. (30)

Proof
Case 1. If (𝑥,𝑦)=(0,0), then for any ℎ=(ℎ₁;ℎ₂), 𝑘=(𝑘₁;𝑘₂)∈𝑅×𝑅^𝑛−1, let 𝜇₁≤𝜇₂ be the spectral values of ℎ²+𝑘²+(𝜏−2)(ℎ∘𝑘) and let 𝑣⁽¹⁾,𝑣⁽²⁾ be the corresponding spectral vectors. Then, ℎ²+𝑘²+()=√𝜏−2)(ℎ∘𝑘−ℎ−𝑘𝜇₁𝑣⁽¹⁾+√𝜇₂𝑣⁽²⁾≤√−ℎ−𝑘𝜇₁𝑣⁽¹⁾+√𝜇₂𝑣⁽²⁾+‖ℎ‖+‖𝑘‖≤2𝜇₂+‖ℎ‖+‖𝑘‖. (31) It follows from the definition of spectral value that 𝜇₂=‖ℎ‖²+‖𝑘‖²+(𝜏−2)ℎ^𝑇𝑘+2ℎ₁ℎ₂+𝑘₁𝑘₂+(𝜏−2)ℎ₁𝑘₂+𝑘₁ℎ₂≤2‖ℎ‖²+2‖𝑘‖²|ℎ𝑘‖‖ℎ‖+3𝜏−2|‖‖‖≤5²+‖𝑘‖². (32) Combining (31) and (32) together with Lemma 4 yields that 2𝜓_𝜏(ℎ,𝑘)−𝜓_𝜏(=𝜙0,0)_𝜏(ℎ,𝑘)₊²≤𝜙_𝜏(ℎ,𝑘)²=ℎ²+𝑘²+(𝜏−2)(ℎ∘𝑘)−ℎ−𝑘²≤10‖ℎ‖²+‖𝑘‖²+‖ℎ‖+‖𝑘‖²=𝑂‖ℎ‖²+‖𝑘‖². (33) This shows that 𝜓_𝜏(𝑥,𝑦) is differentiable at (0,0) with ∇_𝑥𝜓_𝜏(0,0)=∇_𝑦𝜓_𝜏(0,0)=0.
Case 2. If 𝑥²+𝑦²+(𝜏−2)(𝑥∘𝑦)∈int𝐾^𝑛, let 𝑔∶𝑅^𝑛→𝑅₊ be defined by 𝑔(𝑧)=(1/2)‖𝑧₊‖² for any 𝑧∈𝑅^𝑛. By the proof of Proposition 3.2 [17], 𝑔(𝑧) is continuously differentiable and ∇𝑔(𝑧)=𝑧₊. Since 𝜓_𝜏(𝑥,𝑦)=𝑔(𝜙_𝜏(𝑥,𝑦)) and 𝜙_𝜏(𝑥,𝑦) is differentiable at any (𝑥,𝑦) satisfying 𝑥²+𝑦²+(𝜏−2)(𝑥∘𝑦)∈int𝐾^𝑛 by Proposition 3.2 [16], 𝜓_𝜏 is differentiable in this case and ∇_𝑥𝜓_𝜏𝑥,𝑦=∇_𝑥𝜙_𝜏𝑥,𝑦∇𝑔𝜙_𝜏=𝐿𝑥,𝑦_{𝑥+((𝜏−2)/2)𝑦}𝐿_𝑧⁻¹−𝐼𝜙_𝜏𝑥,𝑦₊,∇_𝑦𝜓_𝜏𝑥,𝑦=∇_𝑦𝜙_𝜏𝑥,𝑦∇𝑔𝜙_𝜏=𝐿𝑥,𝑦_{𝑦+((𝜏−2)/2)𝑥}𝐿_𝑧⁻¹−𝐼𝜙_𝜏𝑥,𝑦₊. (34)
Case 3. If (𝑥,𝑦)≠(0,0) and 𝑥²+𝑦²+(𝜏−2)(𝑥∘𝑦)∉int𝐾^𝑛, it follows from Lemma 3 that 𝑥²₁+𝑦²₁+(𝜏−2)𝑥₁𝑦₁=𝑥₁𝑥₂+𝑦₁𝑦₂+(𝜏−2)𝑥₁𝑦₂≠0. (35) In this case, direct calculations together with Lemma 3 yield 𝜙_𝜏=𝑥,𝑦𝑥²₁+𝑦²₁+(𝜏−2)𝑥₁𝑦₁−𝑥₁−𝑦₁𝑥₁𝑥₂+𝑦₁𝑦₂+((𝜏−2)/2)𝑥₁𝑦₂+((𝜏−2)/2)𝑦₁𝑥₂𝑥²₁+𝑦²₁+(𝜏−2)𝑥₁𝑦₁−𝑥₂−𝑦₂. (36) Thus, the bigger spectral value 𝜆₂ of 𝜙_𝜏(𝑥,𝑦) and its corresponding spectral vector 𝑢⁽²⁾ are given as 𝜆₂=𝑥²₁+𝑦²₁+(𝜏−2)𝑥₁𝑦₁−𝑥₁−𝑦₁+𝑤₂,𝑢⁽²⁾=12𝑤1;²𝑤₂, (37) where 𝑤₂=𝑥₁𝑥₂+𝑦₁𝑦₂+((𝜏−2)/2)𝑥₁𝑦₂+((𝜏−2)/2)𝑦₁𝑥₂𝑥²₁+𝑦²₁+(𝜏−2)𝑥₁𝑦₁−𝑥₂−𝑦₂. (38) By the spectral factorization, we have 𝜙_𝜏𝑥,𝑦₊=𝜙_𝜏𝑥,𝑦⟺𝜙_𝜏𝑥,𝑦∈𝐾^𝑛,𝜙_𝜏𝑥,𝑦₊=0⟺𝜙_𝜏𝑥,𝑦∈−𝐾^𝑛,𝜙_𝜏𝑥,𝑦₊=𝜆₂𝑢⁽²⁾⟺𝜙_𝜏𝑥,𝑦∉𝐾^𝑛∪−𝐾^𝑛. (39) Therefore, we prove the differentiability of 𝜓_𝜏 in this case by considering the following three subcases.
(i) If 𝜙_𝜏(𝑥,𝑦)∉𝐾^𝑛∪−𝐾^𝑛, then 𝜙_𝜏(𝑥,𝑦)₊=𝜆₂𝑢⁽²⁾, where 𝜆₂,𝑢⁽²⁾ are given by (37). Then we have 𝜓_𝜏=1𝑥,𝑦2𝜙_𝜏(𝑥,𝑦)₊²=14𝜆²₂=14𝑥²₁+𝑦²₁+(𝜏−2)𝑥₁𝑦₁−𝑥₁−𝑦₁²+2𝑥²₁+𝑦²₁+(𝜏−2)𝑥₁𝑦₁−𝑥₁−𝑦₁𝑤₂+𝑤₂². (40) It is obvious that 𝜓_𝜏 is differentiable in this case. Moreover, by Lemma 3, we have ∇_𝑥1𝑤₂=𝑥₂+𝜏−22𝑦₂𝑥²₁+𝑦²₁+(𝜏−2)𝑥₁𝑦₁−𝑥₁𝑥₂+𝑦₁𝑦₂+𝜏−22𝑥₁𝑦₂+𝜏−22𝑦₁𝑥₂×𝑥₁+((𝜏−2)/2)𝑦₁𝑥²₁+𝑦²₁+(𝜏−2)𝑥₁𝑦₁×𝑥²₁+𝑦²₁+(𝜏−2)𝑥₁𝑦₁⁻¹=𝑥₂+𝜏−22𝑦₂𝑥²₁+𝑦²₁+(𝜏−2)𝑥₁𝑦₁−𝑥₁𝑥₂+𝑦₁𝑦₂+𝜏−22𝑥₁𝑦₂+𝜏−22𝑦₁𝑥₂×𝑥₁+𝜏−22𝑦₁×𝑥²₁+𝑦²₁+(𝜏−2)𝑥₁𝑦₁⁻³∇=0,_𝑥2𝑤₂=𝑥₁+((𝜏−2)/2)𝑦₁𝑥²₁+𝑦²₁+(𝜏−2)𝑥₁𝑦₁−1. (41) Therefore, the derivative of 𝜓_𝜏 with respect to 𝑥₁ is 𝜕𝜕𝑥₁𝜓_𝜏=1𝑥,𝑦42𝑥²₁+𝑦²₁+(𝜏−2)𝑥₁𝑦₁−𝑥₁−𝑦₁×𝑥₁+((𝜏−2)/2)𝑦₁𝑥²₁+𝑦²₁+(𝜏−2)𝑥₁𝑦₁𝑥−1+2¹+((𝜏−2)/2)𝑦₁𝑥²₁+𝑦²₁+(𝜏−2)𝑥₁𝑦₁−1𝑤₂+2𝑥²₁+𝑦²₁+(𝜏−2)𝑥₁𝑦₁−𝑥₁−𝑦₁×𝑤^𝑇₂∇_𝑥1𝑤₂𝑤₂+2𝑤^𝑇₂∇_𝑥1𝑤₂=12𝑥₁+((𝜏−2)/2)𝑦₁𝑥²₁+𝑦²₁+(𝜏−2)𝑥₁𝑦₁×−1𝑥²₁+𝑦²₁+(𝜏−2)𝑥₁𝑦₁−𝑥₁−𝑦₁+𝑤₂, (42) and the gradient of 𝜓_𝜏 with respect to 𝑥₂ is ∇_𝑥2𝜓_𝜏=1𝑥,𝑦42𝑥²₁+𝑦²₁+(𝜏−2)𝑥₁𝑦₁−𝑥₁−𝑦₁×∇_𝑥2𝑤₂⋅𝑤₂𝑤₂+2∇_𝑥2𝑤₂⋅𝑤₂=12𝑥²₁+𝑦²₁+(𝜏−2)𝑥₁𝑦₁−𝑥₁−𝑦₁×𝑥₁+((𝜏−2)/2)𝑦₁𝑥²₁+𝑦²₁+(𝜏−2)𝑥₁𝑦₁𝑤−1²𝑤₂+𝑥₁+((𝜏−2)/2)𝑦₁𝑥²₁+𝑦²₁+(𝜏−2)𝑥₁𝑦₁−1𝑤₂=12𝑥₁+((𝜏−2)/2)𝑦₁𝑥²₁+𝑦²₁+(𝜏−2)𝑥₁𝑦₁×−1𝑥²₁+𝑦²₁+(𝜏−2)𝑥₁𝑦₁−𝑥₁−𝑦₁+𝑤₂𝑤₂𝑤₂. (43) Then it follows from (37), (42), and (43) that ∇_𝑥𝜓_𝜏 can be rewritten as ∇_𝑥𝜓_𝜏=𝜕𝑥,𝑦𝜕𝑥₁𝜓_𝜏∇𝑥,𝑦_𝑥2𝜓_𝜏=𝑥𝑥,𝑦₁+((𝜏−2)/2)𝑦₁𝑥²₁+𝑦²₁+(𝜏−2)𝑥₁𝑦₁−1𝜆₂𝑢⁽²⁾=𝑥₁+((𝜏−2)/2)𝑦₁𝑥²₁+𝑦²₁+(𝜏−2)𝑥₁𝑦₁−1𝜙_𝜏𝑥,𝑦₊. (44) Similarly, we can show that ∇_𝑦𝜓_𝜏=𝑦𝑥,𝑦¹+((𝜏−2)/2)𝑥₁𝑥²₁+𝑦²₁+(𝜏−2)𝑥₁𝑦₁−1𝜙_𝜏𝑥,𝑦₊. (45)
(ii) If 𝜙_𝜏(𝑥,𝑦)∈𝐾^𝑛, we have 𝜙_𝜏(𝑥,𝑦)₊=𝜙_𝜏(𝑥,𝑦) and thus 𝜓_𝜏(𝑥,𝑦)=(1/2)‖𝜙_𝜏(𝑥,𝑦)‖². Then by Proposition 3.2 [16], the gradient of 𝜓_𝜏 is ∇_𝑥𝜓_𝜏=𝑥𝑥,𝑦¹+((𝜏−2)/2)𝑦₁𝑥²₁+𝑦²₁+(𝜏−2)𝑥₁𝑦₁−1𝜙_𝜏=𝑥𝑥,𝑦₁+((𝜏−2)/2)𝑦₁𝑥²₁+𝑦²₁+(𝜏−2)𝑥₁𝑦₁−1𝜙_𝜏𝑥,𝑦₊,∇_𝑦𝜓_𝜏=𝑦𝑥,𝑦¹+((𝜏−2)/2)𝑥₁𝑥²₁+𝑦²₁+(𝜏−2)𝑥₁𝑦₁−1𝜙_𝜏=𝑦𝑥,𝑦₁+((𝜏−2)/2)𝑥₁𝑥²₁+𝑦²₁+(𝜏−2)𝑥₁𝑦₁−1𝜙_𝜏𝑥,𝑦₊. (46) If there exists (𝑥^′,𝑦^′) such that 𝜙_𝜏(𝑥^′,𝑦^′)∉𝐾^𝑛∪−𝐾^𝑛 and 𝜙_𝜏(𝑥^′,𝑦^′)→𝜙_𝜏(𝑥,𝑦), it follows from (44)–(46) that ∇_𝑥𝜓_𝜏𝑥^′,𝑦^′⟶∇_𝑥𝜓_𝜏,∇𝑥,𝑦_𝑦𝜓_𝜏𝑥^′,𝑦^′⟶∇_𝑦𝜓_𝜏𝑥,𝑦. (47) Therefore, 𝜓_𝜏 is differentiable in this subcase.
(iii) If 𝜙_𝜏(𝑥,𝑦)∈−𝐾^𝑛, we have 𝜙_𝜏(𝑥,𝑦)₊=0 and thus 𝜓_𝜏(𝑥,𝑦)=(1/2)‖𝜙_𝜏(𝑥,𝑦)₊‖²=0. Then it is obvious that the gradient of 𝜓_𝜏 is ∇_𝑥𝜓_𝜏𝑥𝑥,𝑦=0=¹+((𝜏−2)/2)𝑦₁𝑥²₁+𝑦²₁+(𝜏−2)𝑥₁𝑦₁−1𝜙_𝜏𝑥,𝑦₊,∇_𝑦𝜓_𝜏𝑦𝑥,𝑦=0=¹+((𝜏−2)/2)𝑥₁𝑥²₁+𝑦²₁+(𝜏−2)𝑥₁𝑦₁−1𝜙_𝜏𝑥,𝑦₊. (48) If there exists (𝑥^′,𝑦^′) such that 𝜙_𝜏(𝑥^′,𝑦^′)∉𝐾^𝑛∪−𝐾^𝑛 and 𝜙_𝜏(𝑥^′,𝑦^′)→𝜙_𝜏(𝑥,𝑦), it follows from (44), (45), and (48) that ∇_𝑥𝜓_𝜏𝑥^′,𝑦^′⟶0=∇_𝑥𝜓_𝜏,∇𝑥,𝑦_𝑦𝜓_𝜏𝑥^′,𝑦^′⟶0=∇_𝑦𝜓_𝜏𝑥,𝑦. (49) Therefore, 𝜓_𝜏 is differentiable in this subcase.

Proposition 7. Let 𝜓_𝜏 be given by (9). For any 𝑥=(𝑥₁,𝑥₂), 𝑦=(𝑦₁,𝑦₂)∈𝑅×𝑅^𝑛−1, we have 𝑥,∇_𝑥𝜓_𝜏𝑥,𝑦+𝑦,∇_𝑦𝜓_𝜏𝑥,𝑦=𝜙_𝜏𝑥,𝑦₊²,∇_𝑥𝜓_𝜏𝑥,𝑦,∇_𝑦𝜓_𝜏𝑥,𝑦≥0, (50)(51) and the equality in (51) holds whenever 𝜓_𝜏(𝑥,𝑦)=0.

Proof. By following the proof of Lemma 4.1 [16] and using Proposition 6, we can show that the desired results hold.

Proposition 8. Let 𝑓∶𝑅^𝑛→𝑅₊ be given by (7)–(10) and (12). Then, the following results hold.(i)For all 𝜁∈𝑅^𝑛, we have 𝑓(𝜁)≥0 and 𝑓(𝜁)=0 if and only if 𝜁 solves the SOCCP.(ii)If 𝐹 and 𝐺 are differentiable, the function 𝑓 is differentiable with ∇𝑓(𝜁)=𝜏₁∇𝐹(𝜁)𝐿_𝐺(𝜁)+∇𝐺(𝜁)𝐿_𝐹(𝜁)(𝐹(𝜁)∘𝐺(𝜁))₊+∇𝐹(𝜁)∇_𝑥𝜓_𝜏2(𝐹(𝜁),𝐺(𝜁))+∇𝐺(𝜁)∇_𝑦𝜓_𝜏2(𝐹(𝜁),𝐺(𝜁)). (52)

Proof. (i) It is obvious that 𝑓(𝜁)≥0 for all 𝜁∈𝑅^𝑛. Now we prove 𝑓(𝜁)=0 if and only if 𝜁 solves the SOCCP. Suppose 𝑓(𝜁)=0. Then we have 𝜓_𝜏2(𝑥,𝑦)=0, and 𝜓₀(𝑥,𝑦)=0 which implies 𝑥∘𝑦∈−𝐾^𝑛 and therefore ⟨𝑥,𝑦⟩≤0. By Proposition 5, the last relation together with 𝜓_𝜏2(𝑥,𝑦)=0 yields ⟨𝑥,𝑦⟩=0, 𝑥∈𝐾^𝑛, 𝑦∈𝐾^𝑛. Therefore, 𝜁 solves the SOCCP. On the other hand, suppose that 𝜁 solves the SOCCP. Then ⟨𝑥,𝑦⟩=0,  𝑥∈𝐾^𝑛,  𝑦∈𝐾^𝑛, which are equivalent to 𝑥∘𝑦=0, 𝑥∈𝐾^𝑛,𝑦∈𝐾^𝑛 [4]. By Proposition 5, (7), (8), and (12), we have 𝜓_𝜏2(𝑥,𝑦)=0, 𝜓₀(𝑥,𝑦)=0, and therefore 𝑓(𝜁)=0.
(ii) From Lemma 3.1 [19], we have that the function 𝜓₀ is differentiable for all (𝑥,𝑦)∈𝑅^𝑛×𝑅^𝑛 with ∇_𝑥𝜓₀(𝑥,𝑦)=𝐿_𝑦⋅(𝑥∘𝑦)₊ and ∇_𝑦𝜓₀(𝑥,𝑦)=𝐿_𝑥⋅(𝑥∘𝑦)₊. Then, by the chain rule and direct calculations, the result follows.

4. Error Bound and Bounded Level Sets

By Proposition 8, we see that the SOCCP is equivalent to the global minimization of the function 𝑓(𝜁). In this section, we show that the function 𝑓 provides a global error bound for the solution of the SOCCP and has bounded level sets, under rather weak conditions.

In this section, we consider the general case that 𝐾⊂𝑅^𝑛 is the Cartesian product of SOCs; that is, 𝐾=𝐾^𝑛1×𝐾^𝑛2×⋯×𝐾^𝑛𝑚 with 𝑚,𝑛₁,…,𝑛_𝑚≥1, 𝑛=𝑛₁+𝑛₂+⋯+𝑛_𝑚. Thus, we obtain 𝜓_𝜏1,𝜏2𝑥,𝑦=^𝑚_𝑖=1𝜓_𝜏1,𝜏2𝑥_𝑖,𝑦_𝑖,𝑓(𝜁)=^𝑚_𝑖=1𝜓_𝜏1,𝜏2𝐹_𝑖(𝜁),𝐺_𝑖(𝜁), (53) and therefore the results in Sections 2 and 3 can be easily extended to the general case.

First, we discuss under what condition the function 𝑓 provides a global error bound for the solution of the SOCCP. To this end, we need the concepts of Cartesian 𝑃-properties introduced in [21] for a nonlinear transformation, which are natural extensions of the 𝑃-properties on Cartesian products in 𝑅^𝑛 established by Facchinei and Pang [22]. Recently, the Cartesian 𝑃-properties are extended to the context of general Euclidean Jordan algebra associated with symmetric cones [20].

Definition 9. The mappings 𝐹=(𝐹₁,…,𝐹_𝑚) and 𝐺=(𝐺₁,…,𝐺_𝑚) are said to have(i)the joint uniform Cartesian 𝑃-property if there exists a constant 𝜌>0 such that, for every 𝜁,𝜉∈𝑅^𝑛, there is an index 𝑖∈{1,2,…,𝑚} such that 𝐹_𝑖(𝜁)−𝐹_𝑖(𝜉),𝐺_𝑖(𝜁)−𝐺_𝑖(𝜉)≥𝜌𝜁−𝜉²; (54)(ii)the joint Cartesian 𝑃-property if, for every 𝜁,𝜉∈𝑅^𝑛 with 𝜁≠𝜉, there is an index 𝑖∈{1,2,…,𝑚} such that 𝜁_𝑖≠𝜉_𝑖,𝐹_𝑖(𝜁)−𝐹_𝑖(𝜉),𝐺_𝑖(𝜁)−𝐺_𝑖(𝜉)>0. (55)

Now we show that the function 𝑓 provides a global error bound for the solution of the SOCCP if 𝐹 and 𝐺 have the joint uniform Cartesian 𝑃-property.

Proposition 10. Let 𝑓 be given by (7)–(10) and (12). Suppose that 𝐹 and 𝐺 have the joint uniform Cartesian 𝑃-property and the SOCCP has a solution 𝜁^∗. Then there exists a constant 𝜅>0 such that, for any 𝜁∈𝑅^𝑛, 𝜅𝜁−𝜁^∗2≤2𝜏₁+44−𝜏₂𝑓(𝜁)^1/2. (56)

Proof. Since 𝐹 and 𝐺 have the joint uniform Cartesian 𝑃-property, there exists a constant 𝜌>0 such that, for any 𝜁∈𝑅^𝑛, there is an index 𝑖∈{1,2,…,𝑚} such that 𝜌𝜁−𝜁^∗2≤𝐹_𝑖(𝜁)−𝐹_𝑖𝜁^∗,𝐺_𝑖(𝜁)−𝐺_𝑖𝜁^∗=𝐹_𝑖(𝜁),𝐺_𝑖(𝜁)+𝐹_𝑖𝜁^∗,−𝐺_𝑖(𝜁+)−𝐹_𝑖(𝜁),𝐺_𝑖𝜁^∗≤𝐹_𝑖(𝜁),𝐺_𝑖(𝜁)+𝐹_𝑖𝜁^∗,−𝐺_𝑖(𝜁)₊+−𝐹_𝑖(𝜁)₊,𝐺_𝑖𝜁^∗≤√2𝐹_𝑖(𝜁)∘𝐺_𝑖(𝜁)₊+𝐹_𝑖𝜁^∗−𝐺_𝑖(𝜁)₊+−𝐹_𝑖(𝜁)₊𝐺_𝑖𝜁^∗≤max√2,𝐹_𝑖𝜁^∗,𝐺_𝑖𝜁^∗×𝐹_𝑖(𝜁)∘𝐺_𝑖(𝜁)₊+−𝐹_𝑖(𝜁)₊+−𝐺_𝑖(𝜁)₊≤max√2,𝐹𝜁^∗,𝐺𝜁^∗×𝐹_𝑖(𝜁)∘𝐺_𝑖(𝜁)₊+−𝐹_𝑖(𝜁)₊+−𝐺_𝑖(𝜁)₊≤max√2,𝐹𝜁^∗,𝐺𝜁^∗×(𝐹(𝜁)∘𝐺(𝜁))₊+(−𝐹(𝜁))₊+(−𝐺(𝜁))₊, (57) where the second inequality is due to Lemma 1(ii) and the third inequality follows from Lemma 2. Setting 𝜅∶=𝜌/max{√2,‖𝐹(𝜁^∗)‖,‖𝐺(𝜁^∗)‖}, we obtain 𝜅𝜁−𝜁^∗2≤(𝐹(𝜁)∘𝐺(𝜁))₊+(−𝐺(𝜁))₊+(−𝐹(𝜁))₊. (58) By (7), (8), and (12), we have (𝐹(𝜁)∘𝐺(𝜁))₊=√2𝜓₀(𝐹(𝜁),𝐺(𝜁))^1/2≤2𝜏₁𝑓(𝜁)^1/2. (59) Moreover, we obtain from Lemma 4 that (−𝐹(𝜁))₊+(−𝐺(𝜁))₊≤√2(−𝐹(𝜁))₊²+(−𝐺(𝜁))₊^21/2≤44−𝜏₂𝜓_𝜏2(𝐹(𝜁),𝐺(𝜁))^1/2≤44−𝜏₂𝑓(𝜁)^1/2. (60) Combining (58), (59), and (60) yields the desired result.

To guarantee the boundedness of the level sets 𝐿_𝑓𝛾∶=𝜁∈𝑅^𝑛∣𝑓(𝜁)≤𝛾 (61) for any 𝛾≥0, we give the following condition.

Condition 11. For any sequence {𝜁^𝑘}⊆𝑅^𝑛 such that 𝜁^𝑘𝜁⟶+∞,−𝐹^𝑘₊<+∞,−𝐺𝜁^𝑘₊<+∞, (62) there holds that max_{1≤𝑖≤𝑚}𝜆_max𝐹_𝑖𝜁^𝑘∘𝐺_𝑖𝜁^𝑘₊⟶+∞. (63)

Proposition 12. If the mappings 𝐹 and 𝐺 satisfy Condition 11, then the level sets 𝐿_𝑓(𝛾) of 𝑓 for any 𝛾≥0 are bounded.

Proof. On the contrary, we assume that there exists an unbounded sequence {𝜁^𝑘}⊆𝐿_𝑓(𝛾) for some 𝛾≥0. Thus ‖𝜁^𝑘‖→+∞ and 𝜓_𝜏2(𝐹(𝜁^𝑘),𝐺(𝜁^𝑘))≤𝑓(𝜁^𝑘)≤𝛾 for all 𝑘. Then by Lemma 4, we have for all 𝑘 that −𝐹𝜁^𝑘₊²+−𝐺𝜁^𝑘₊²≤44−𝜏₂𝜙_𝜏2𝐹𝜁^𝑘,𝐺𝜁^𝑘2=84−𝜏₂𝜓_𝜏2𝐹𝜁^𝑘,𝐺𝜁^𝑘≤84−𝜏₂𝛾, (64) which implies ‖(−𝐹(𝜁^𝑘))₊‖<+∞ and ‖(−𝐺(𝜁^𝑘))₊‖+∞. Therefore, from Condition 11, there is 𝑗∈{1,2,…,𝑚} such that 𝜆_max[(𝐹_𝑗(𝜁^𝑘)∘𝐺_𝑗(𝜁^𝑘))₊]→+∞. It follows from (7), (8), and (12) that 𝜆_max𝐹_𝑗𝜁^𝑘∘𝐺_𝑗𝜁^𝑘₊≤√2𝐹_𝑗𝜁^𝑘∘𝐺_𝑗𝜁^𝑘₊=2𝜓₀𝐹_𝑗𝜁^𝑘,𝐺_𝑗𝜁^𝑘1/2≤2√𝜏₁𝑓𝜁^𝑘1/2, (65) and hence 𝑓(𝜁^𝑘)→+∞. This contradicts the fact that {𝜁^𝑘}⊆𝐿_𝑓(𝛾).

It should be noted that Condition 11 is rather weak to guarantee the boundedness of level sets of 𝑓. As far as we know, the weakest condition available to ensure the boundedness of level sets is the following condition given by [20].

Condition 13 (see [20]). For any sequence {𝜁^𝑘}⊆𝑅^𝑛 such that 𝜁^𝑘𝜁→+∞,−𝐹^𝑘₊<+∞,−𝐺𝜁^𝑘₊<+∞, (66) there holds that max_{1≤𝑖≤𝑚}𝜆_max𝐹_𝑖𝜁^𝑘∘𝐺_𝑖𝜁^𝑘⟶+∞. (67)
It is obvious that 𝜆_max[𝐹_𝑖(𝜁^𝑘)∘𝐺_𝑖(𝜁^𝑘)]≤𝜆_max[(𝐹_𝑖(𝜁^𝑘)∘𝐺_𝑖(𝜁^𝑘))₊] for any 𝑖∈{1,2,…,𝑚}, and therefore Condition 13 implies Condition 11. It has been shown that the symmetric cone complementarity problem (SCCP) with the jointly monotone mappings and a strictly feasible point, or the SCCP with joint Cartesian 𝑅₀₂-property [23], all imply Condition 13 [20]. Hence they all implies Condition 11, since the SCCP includes the SOCCP. Therefore, Condition 11 is a weaker condition than the most available conditions to guarantee the boundedness of level sets.

5. Numerical Results

In this section, we employ the merit function method based on the unconstrained minimization reformulation (12) to solve the SOCCPs (1). All the experiments were performed on a desktop computer with Intel Pentium Dual T2390 CPU 1.86 GHz and 1.00 GB memory. The operating system was Windows XP and the implementations were done in MATLAB 7.0.1.

We adopt the L-BFGS method [24], a limited-memory quasi-Newton method, with 5 limited-memory vector-updates to solve the unconstrained minimization reformulation (12) where the two-parametric class of merit functions 𝜓_𝜏1,𝜏2 is given as (7)–(10). For the scaling matrix 𝐻⁰=𝛾𝐼 in the L-BFGS, we adopt 𝛾=𝑝^𝑇𝑞/𝑞^𝑇𝑞 as recommended by [25], where 𝑝∶=𝜁−𝜁^old,𝑞∶=∇𝜓_𝜏1,𝜏2(𝜁)−∇𝜓_𝜏1,𝜏2𝜁^old. (68) In the L-BFGS, we revert to the steepest descent direction −∇𝜓_𝜏1,𝜏2(𝜁) whenever 𝑝^𝑇𝑞≤10⁻⁵‖𝑝‖‖𝑞‖. In addition, we use the nonmonotone line search [26] to seek a suitable steplength. In detail, we compute the smallest nonnegative integer 𝑙_𝑘 such that 𝜓_𝜏1,𝜏2𝜁^𝑘+𝜌^𝑙𝑘𝑑^𝑘≤𝑊_𝑘+𝜎𝜌^𝑙𝑘∇𝜓_𝜏1,𝜏2𝜁^𝑘𝑇𝑑^𝑘, (69) where 𝑑^𝑘 denotes the direction in the 𝑘th iteration generated by the L-BFGS, 𝜌 and 𝜎 are parameters in (0,1), and 𝑊_𝑘 is given by 𝑊_𝑘=max_{𝑗=𝑘−𝑚𝑘,…,𝑘}𝜓_𝜏1,𝜏2𝜁^𝑗, (70) where, for a given nonnegative integer 𝑚 and 𝑠, we set 𝑚_𝑘=𝑚0,if𝑘≤𝑠,min_𝑘−1+1,𝑚otherwise. (71) Throughout the numerical experiments, we choose the following parameters: 𝜌=0.8,𝜎=0.01,𝑚=5,𝑠=5. (72) The algorithm is stopped whenever the number of function evaluations for 𝜓_𝜏1,𝜏2 is over 10000 or max{𝜓_𝜏1,𝜏2(𝜁),|⟨𝐹(𝜁),𝐺(𝜁)⟩|}≤10⁻⁶ as the stopping criterion.

The test problems are the randomly generated linear SOCCPs (1), where 𝐹(𝜁)=𝜁,𝐺(𝜁)=𝑀𝜁+𝑞, (73) with 𝑀∈𝑅^𝑛×𝑛 and 𝑞∈𝑅^𝑛. In detail, we generate a random matrix 𝑁=rand(𝑛,𝑛) and a random vector 𝑞=rand(𝑛,1), and then let 𝑀∶=𝑁^𝑇𝑁. Since the matrix 𝑀 is semidefinite positive, the generated problems (1) are the monotone linear SOCCPs. In the tables of test results, 𝑛 denotes the size of problems; NF denotes the (average) number of iterations; CPU(s) denotes the (average) CPU time in seconds; and Gap denotes the (average) value of |⟨𝐹(𝜁),𝐺(𝜁)⟩| when the algorithm terminates.

We solve the linear SOCCPs of different dimensions with size 𝑛 from 50 to 1000 and 𝑚=1. The random problems of each size are generated 10 times, and the test results with different parameters 𝜏₁>0 and 𝜏₂∈(0,4) are listed in Tables 1, 2, and 3. From the results of these tables, we give several observations.(i)All the random problems have been solved in very short CPU time.(ii)The problem size slightly affects the number of iterations.(iii)For the same dimension of linear SOCCPs, choices of different parameters 𝜏₁>0 and 𝜏₂∈(0,4) generally do not affect the number of iterations and the CPU time.

Table 1 

Numerical results for SOCCPs with (𝜏₁,𝜏₂)=(0.1,0.1).

Table 2 

Numerical results for SOCCPs with (𝜏₁,𝜏₂)=(1,2.0).

Table 3 

Numerical results for SOCCPs with (𝜏₁,𝜏₂)=(10,3.5).

6. Conclusions

In this paper, we have studied a two-parametric class of merit functions for the second-order cone complementarity problem based on the one-parametric class of complementarity functions. The new proposed class of merit functions includes a broad class of merit functions, since the one-parametric class of complementarity functions is closely related to the famous natural residual function and Fischer-Burmeister function. The new class of merit functions has been shown to possess some favorable properties. In particular, it provides a global error bound if 𝐹 and 𝐺 have the joint uniform Cartesian 𝑃-property. And it has bounded level sets under a weaker condition than the most available conditions [20, 23]. Some preliminary numerical results for solving the SOCCPs show the effectiveness of the merit function method via the new class of merit functions.

Acknowledgments

This research is supported by the National Natural Science Foundation of China (no. 71171150), China Postdoctoral Science Foundation (no. 2012M511651), and the Excellent Youth Project of Hubei Provincial Department of Education (no. Q20122709), China. The authors are grateful to the editor and the anonymous referees for their valuable comments on this paper. Particulary, the authors thank one of the referees for his helpful suggestions on numerical results, which have greatly improved this paper.