A Smoothing Method with Appropriate Parameter Control Based on Fischer-Burmeister Function for Second-Order Cone Complementarity Problems

Abstract

We deal with complementarity problems over second-order cones. The complementarity problem is an important class of problems in the real world and involves many optimization problems. The complementarity problem can be reformulated as a nonsmooth system of equations. Based on the smoothed Fischer-Burmeister function, we construct a smoothing Newton method for solving such a nonsmooth system. The proposed method controls a smoothing parameter appropriately. We show the global and quadratic convergence of the method. Finally, some numerical results are given.

1. Introduction

In this paper, we consider the second-order cone complementarity problem (SOCCP) of the following form: find𝑥,𝑦,𝑝∈𝐑^𝑛×𝐑^𝑛×𝐑^ℓsuchthat𝑥∈𝒦,𝑦∈𝒦,𝑥,𝑦=0,𝐹𝑥,𝑦,𝑝=0, (1) where ⟨⋅,⋅⟩ denotes the Euclidean inner product, ℓ≥0, 𝑛≥1, 𝐹∶𝐑^𝑛×𝐑^𝑛×𝐑^ℓ→𝐑^𝑛+ℓ is a continuously differentiable function, and 𝒦 denotes the Cartesian product of several second-order cones (SOCs), that is, 𝒦=𝒦^𝑛1×⋯×𝒦^𝑛𝑚⊂𝐑^𝑛 with 𝑚,𝑛₁,…,𝑛_𝑚≥1, 𝑛=𝑛₁+⋯+𝑛_𝑚, and 𝒦^𝑛𝑖𝑧∶=¹,𝑧₂∈𝐑×𝐑^𝑛𝑖−1∣𝑧₁≥𝑧₂₂⊂𝐑^𝑛𝑖𝑛_𝑖,𝐑≥2₊=𝑧₁∈𝐑∣𝑧₁≥0𝑛_𝑖=1. (2)

The SOCCP is a wide class of complementarity problems. For example, it involves the mixed complementarity problem (MCP) and the nonlinear complementarity problem (NCP) [1] as subclasses, since 𝒦=𝐑^𝑛₊ when 𝑛_𝑖=1 for each 𝑖=1,…,𝑚(=𝑛). Moreover, the second-order cone programming (SOCP) problem can be reformulated as an SOCCP by using the Karush-Kuhn-Tucker (KKT) conditions. Apart from them, some practical problems in the game theory [2, 3] and the architecture [4] can be reformulated as the SOCCP.

Much theoretical and algorithmic research has been made so far for solving the SOCCP. Fukushima et al. [5] showed that the natural residual function, also called the min function, and the Fischer-Burmeister function for the NCP can be extended to the SOCCP by using the Jordan algebra. They further constructed the smoothing functions for those SOC complementarity (C-) functions and analyzed the properties of their Jacobian matrices. Hayashi et al. [6] proposed a smoothing method based on the natural residual and showed its global and quadratic convergence. On the other hand, Chen et al. [7] proposed another smoothing method with the natural residual in which the smoothing parameter is treated as a variable in contrast to [6]. Moreover, they showed the global and quadratic convergence of their method. Similar to Chen et al., Narushima et al. [8] proposed a smoothing method treating a smoothing parameter as a variable. They used the Fischer-Burmeister function instead of the natural residual function and also provided the global and quadratic convergence of the method.

In the present paper, we propose a smoothing method with the Fischer-Burmeister function for solving SOCCP (1). The main difference from the existing methods is twofold. (i)We do not assume the special structure on the function 𝐹 in SOCCP (1). In [6, 7, 9, 10], the authors focused on the following type of SOCCP: 𝑥∈𝒦,𝑦∈𝒦,𝑥,𝑦=0,𝑦=𝑓(𝑥), (3) which is a special case of SOCCP (1) with ℓ=0,𝐹𝑥,𝑦,𝑝=𝑓(𝑥)−𝑦 (4) for some continuously differentiable function 𝑓∶𝐑^𝑛→𝐑^𝑛. In [11–13], the authors studied the following type of SOCCP: 𝑝𝑝,𝑥∈𝒦,𝑦∈𝒦,𝑥,𝑦=0,𝑥=𝑓,𝑦=𝑔 (5) which is obtained by letting =𝑓𝑝𝑔𝑝ℓ=𝑛,𝐹𝑥,𝑦,𝑝−𝑥−𝑦, (6) where 𝑓 and 𝑔∶𝐑^𝑛→𝐑^𝑛 are continuously differentiable functions. However, we assume neither (4) nor (6). Therefore, our method is applicable to a wider class of SOCCPs. (ii)In contrast to [8], we do not incorporate the smoothing parameter into the decision variable. We control the smoothing parameter appropriately in each iteration.

This paper is organized as follows. In Section 2, we give some preliminaries, which will be used in the subsequent analysis. In Section 3, we review the SOC C-function. In particular, we recall the property of the (smoothed) Fischer-Burmeister function. In Section 4, we propose an algorithm for solving the SOCCP and discuss its global and local convergence properties. In Section 5, we report some preliminary numerical results.

Throughout the paper, we use the following notations. Let 𝐑₊ and 𝐑₊₊ be the sets of nonnegative and positive reals. For a symmetric matrix 𝐴, we write 𝐴≽𝑂 (resp., 𝐴≻O) if 𝐴 is positive semidefinite (resp., positive definite). For any 𝑥, 𝑦∈𝐑^𝑛, we write 𝑥≽𝑦 (resp., 𝑥≻𝑦) if 𝑥−𝑦∈𝒦^𝑛 (resp., 𝑥−𝑦∈int𝒦^𝑛), and we denote by ⟨𝑥,𝑦⟩ the Euclidean inner product, that is, ⟨𝑥,𝑦⟩∶=𝑥^⊤𝑦. We use the symbol ‖⋅‖ to denote the usual ℓ₂-norm of a vector or the corresponding induced matrix norm. We often write 𝑥=(𝑥₁,𝑥₂)∈𝐑×𝐑^𝑛−1 (possibly 𝑥₂ vacuous), instead of 𝑥=(𝑥₁,𝑥^⊤₂)^⊤∈𝐑^𝑛. In addition, we often regard 𝐑^𝑝×𝐑^𝑞 as 𝐑^𝑝+𝑞. We sometimes divide a vector 𝑥∈𝐑^𝑛 according to the Cartesian structure of 𝒦, that is, 𝑥=(𝑥¹,…,𝑥^𝑚)∈𝐑^𝑛1×⋯×𝐑^𝑛𝑚=𝐑^𝑛 with 𝑥^𝑖∈𝐑^𝑛𝑖. For any Fréchet-differentiable function 𝐺∶𝐑^𝑛→𝐑^𝑚, we denote its transposed Jacobian matrix at 𝑥∈𝐑^𝑛 by ∇𝐺(𝑥)∈𝐑^𝑛×𝑚. For a given set 𝑆⊂𝐑^𝑛, int𝑆, bd 𝑆, and conv 𝑆 mean the interior, the boundary, and the convex hull of 𝑆 in 𝐑^𝑛, respectively.

2. Some Preliminaries

In this section, we recall some background materials and preliminary results used in the subsequent sections.

First, we review the Jordan algebra associated with SOCs. For any 𝑥=(𝑥₁,𝑥₂)∈𝐑×𝐑^𝑛−1 and 𝑦=(𝑦₁,𝑦₂)∈𝐑×𝐑^𝑛−1(𝑛≥1), the Jordan product associated with 𝒦^𝑛 is defined as 𝑥∘𝑦∶=𝑥^⊤𝑦,𝑦₁𝑥₂+𝑥₁𝑦₂. (7) When 𝑛=1, that is, the second components 𝑥₂ and 𝑦₂ are vacuous, we interpret that the second component in (7) is also vacuous. We will write 𝑥² to mean 𝑥∘𝑥 and write 𝑥+𝑦 to mean the usual componentwise addition of vectors 𝑥 and 𝑦. For the Jordan product, the identity element 𝑒∈𝐑^𝑛 is defined by 𝑒∶=(1,0,…,0)^⊤. It is easily seen that 𝑥∘𝑒=𝑒∘𝑥=𝑥 for any 𝑥∈𝐑^𝑛. For any 𝑥∈𝒦^𝑛, we define 𝑥^1/2 as 𝑥^1/2𝑥∶=𝜍,²2𝜍,𝜍=12𝑥₁+𝑥²₁−𝑥₂²if𝑥≠0,0if𝑥=0. (8) Note that (𝑥^1/2)²=𝑥^1/2∘𝑥^1/2=𝑥 and that for 𝑛=1, 𝑥∘𝑦=𝑥₁𝑦₁, 𝑒=1, and 𝑥^1/2=𝜍=√𝑥₁. Although the Jordan product is not associative, associativity holds under the inner product in the sense that 𝑥,𝑦∘𝑧=𝑦,𝑧∘𝑥=𝑧,𝑥∘𝑦forany𝑥,𝑦,𝑧∈𝐑^𝑛. (9) In addition, it follows readily from the definition of 𝒦^𝑛 that ⟨𝑥,𝑦⟩≥0 (resp., ⟨𝑥,𝑦⟩>0) for any 𝑥, 𝑦≽0 (resp., 𝑥,𝑦≻0).

For each 𝑥=(𝑥₁,𝑥₂)∈𝐑×𝐑^𝑛−1(𝑛≥1), we define the symmetric matrix 𝐿_𝑥 by 𝐿_𝑥𝑥∶=¹𝑥^⊤₂𝑥₂𝑥₁𝐼, (10) which can be viewed as a linear mapping having the following properties.

Property 1. There holds that (a)𝐿_𝑥𝑦=𝑥∘𝑦=𝑦∘𝑥=𝐿_𝑦𝑥 and 𝐿_𝑥+𝑦=𝐿_𝑥+𝐿_𝑦 for any 𝑥,𝑦∈𝐑^𝑛; (b)𝑥≽0⇔𝐿_𝑥≽𝑂, and 𝑥≻0⇔𝐿_𝑥≻𝑂; (c)𝐿_𝑥 is invertible whenever 𝑥∈int𝒦^𝑛 with 𝐿_𝑥⁻¹=1(𝑥)𝑥det¹−𝑥^⊤₂−𝑥₂det(𝑥)𝑥₁𝑥𝐼+²𝑥^⊤₂𝑥₁, (11)where det(𝑥)∶=𝑥²₁−‖𝑥₂‖² denotes the determinant of 𝑥.

An important character of the Jordan algebra is its spectral factorization. By the spectral factorization associated with SOC, any 𝑥=(𝑥₁,𝑥₂)∈𝐑×𝐑^𝑛−1(𝑛≥1) can be decomposed as 𝑥=𝜆₁(𝑥)𝑠^{1}+𝜆₂(𝑥)𝑠^{2}, (12) where 𝜆₁(𝑥), 𝜆₂(𝑥) and 𝑠^{1}, 𝑠^{2} are the spectral values and the associated spectral vectors of 𝑥 given by 𝜆_𝑗(𝑥)∶=𝑥₁+(−1)^𝑗𝑥₂,𝑠^{𝑗}1∶=21,(−1)^𝑗𝑥₂𝑥₂if𝑥₂1≠0,21,(−1)^𝑗𝑠₂if𝑥₂=0, (13) for 𝑗=1,2, with 𝑠₂ being any vector in 𝐑^𝑛−1 satisfying ‖𝑠₂‖=1. If 𝑥₂≠0, the decomposition (12) is unique. We note again that when 𝑛=1 (viz., 𝑥=𝑥₁), we have 𝜆₁(𝑥)=𝜆₂(𝑥)=𝑥₁, 𝑠^{1}=𝑠^{2}=1/2. The spectral factorization associated with SOC leads to a number of interesting properties, some of which are as follows.

Property 2. For any 𝑥=(𝑥₁,𝑥₂)∈𝐑×𝐑^𝑛−1(𝑛≥1), let 𝜆₁(𝑥), 𝜆₂(𝑥) and 𝑠^{1}, 𝑠^{2} be the spectral values and the associated spectral vectors at 𝑥. Then the following statements hold.(a)𝑥∈𝒦^𝑛⇔𝜆₁(𝑥)≥0, 𝑥∈int𝒦^𝑛⇔𝜆₁(𝑥)>0, 𝑥∈bd𝒦^𝑛⇔𝜆₁(𝑥)=0.(b)𝑥²=𝜆₁(𝑥)²𝑠^{1}+𝜆₂(𝑥)²𝑠^{2}∈𝒦^𝑛.(c) Let 𝑥∈𝒦^𝑛. Then 𝑥^1/2=𝜆₁(𝑥)𝑠^{1}+𝜆₂(𝑥)𝑠^{2}∈𝒦^𝑛. Moreover, 𝑥∈int𝒦^𝑛⇔𝑥^1/2∈int𝒦^𝑛, 𝑥∈bd𝒦^𝑛⇔𝑥^1/2∈bd𝒦^𝑛.

In what follows, we recall some definitions for functions and matrices. The semismoothness is a generalized concept of the smoothness, which was originally introduced by Mifflin [14] for functionals, and extended to vector-valued functions by Qi and Sun [15]. For vector-valued functions associated with SOC, see also the work of Chen et al. [16]. Now we give the definition of the Clarke subdifferential [17].

Definition 1. Let 𝐻∶𝐑^𝑛→𝐑^𝑚 be a locally Lipschitzian function. The Clarke subdifferential of 𝐻 at 𝑥 is defined by (𝑥)𝑥𝜕𝐻∶=convlim_→𝑥∇𝐻(𝑥)∣𝑥∈𝒟_𝐻, (14) where 𝒟_𝐻 is the set of points at which 𝐻 is differentiable.

Note that if 𝐻 is continuously differentiable at 𝑥, then 𝜕𝐻(𝑥)={∇𝐻(𝑥)}. We next give the definitions of the semismoothness and the strong semismoothness.

Definition 2. A directionally differentiable and locally Lipschitzian function 𝐻∶𝐑^𝑛→𝐑^𝑚 is said to be semismooth at 𝑥 if 𝐻^′(𝑥;𝑑)−𝑉^⊤𝑑=𝑜(‖𝑑‖) (15) for any sufficiently small 𝑑∈𝐑^𝑛⧵{0} and 𝑉∈𝜕𝐻(𝑥+𝑑), where 𝐻^′(𝑥;𝑑)∶=lim_𝜏→+0𝐻(𝑥+𝜏𝑑)−𝐻(𝑥)𝜏 (16) is the directional derivative of 𝐻 at 𝑥 along the direction 𝑑. In particular, if 𝑜(‖𝑑‖) can be replaced by 𝑂(‖𝑑‖²), then function 𝐻 is said to be strongly semismooth.

It is known that if 𝐻 is (strongly) semismooth, then 𝐻^′(𝑥;𝑑)−(𝐻(𝑥+𝑑)−𝐻(𝑥))=𝑜(‖𝑑‖)𝑂‖𝑑‖² (17) holds (see [18], e.g.).

The definitions below for a function can be found in [10, 13, 19].

Definition 3 (see [10, 13]). A function 𝐹=(𝐹¹,…,𝐹^𝑚) with 𝐹^𝑖∶𝐑^𝑛→𝐑^𝑛𝑖 is said to have (a)the Cartesian 𝑃₀-property, if for any 𝑥=(𝑥¹,…,𝑥^𝑚), 𝑦=(𝑦¹,…,𝑦^𝑚)∈𝐑^𝑛1×⋯×𝐑^𝑛𝑚 with 𝑥≠𝑦, there exists an index 𝜈∈{1,…,𝑚} such that 𝑥^𝜈≠𝑦^𝜈 and ⟨𝑥^𝜈−𝑦^𝜈,𝐹^𝜈(𝑥)−𝐹^𝜈(𝑦)⟩≥0; (b)the uniform Cartesian 𝑃-property, if there exists a constant 𝜌>0 such that, for any 𝑥=(𝑥¹,…,𝑥^𝑚), 𝑦=(𝑦¹,…,𝑦^𝑚)∈𝐑^𝑛1×⋯×𝐑^𝑛𝑚, there exists an index 𝜈∈{1,…,𝑚} such that ⟨𝑥^𝜈−𝑦^𝜈,𝐹^𝜈(𝑥)−𝐹^𝜈(𝑦)⟩≥𝜌‖𝑥−𝑦‖².
By the definitions, it is clear that the Cartesian 𝑃-property implies the Cartesian 𝑃₀-property. Definition 3 is associated with SOCCP (3), while the following definitions are associated with SOCCP (5).

Definition 4 (see [19]). Let 𝐹∶𝐑^𝑛→𝐑^𝑛 and 𝐺∶𝐑^𝑛→𝐑^𝑛 be functions such that 𝐹=(𝐹¹,…,𝐹^𝑚), 𝐺=(𝐺¹,…,𝐺^𝑚) with 𝐹^𝑖∶𝐑^𝑛→𝐑^𝑛𝑖 and 𝐺^𝑖∶𝐑^𝑛→𝐑^𝑛𝑖. Then, 𝐹 and 𝐺 are said to have (a)the joint uniform Jordan 𝑃-property, if there exists a constant 𝜌>0 such that 𝜆₂𝐹(𝑥)−𝐹𝑦∘𝐺(𝑥)−𝐺𝑦≥𝜌𝑥−𝑦²forany𝑥,𝑦∈𝐑^𝑛; (18)(b)the joint Cartesian weak coerciveness, if there exists a vector 𝑥∈𝐑^𝑛 such that lim_{‖𝑥‖→∞}max_{1≤𝑖≤𝑚}𝐹^𝑖(𝑥),𝐺^𝑖(𝑥)−𝐺^𝑖(𝑥)‖𝑥−𝑥‖=+∞. (19)

Next we recall the concept of linear growth of a function, which is weaker than the global Lipschitz continuity.

Definition 5 (see [19]). A function 𝐹∶𝐑^𝑛→𝐑^𝑛 is said to have linear growth, if there exists a constant 𝐶>0 such that ‖𝐹(𝑥)‖≤‖𝐹(0)‖+𝐶‖𝑥‖ for any 𝑥∈𝐑^𝑛.

The following definitions for a matrix are originally given in [8], which is a generalization of the mixed 𝑃₀-property [1].

Definition 6 (see [8]). Let 𝑀∈𝐑^{(𝑛+ℓ)×(2𝑛+ℓ)} be a matrix partitioned as follows: 𝑀=𝑀₁𝑀₂𝑀₃, (20) where 𝑀₁, 𝑀₂∈𝐑^{(𝑛+ℓ)×𝑛} and 𝑀₃∈𝐑^{(𝑛+ℓ)×ℓ}. Then, 𝑀 is said to have (a) the Cartesian mixed 𝑃₀-property, if the following statements hold: (1)𝑀₃ has full column rank;(2) for any 𝑥=(𝑥¹,…,𝑥^𝑚), 𝑦=(𝑦¹,…,𝑦^𝑚)∈𝐑^𝑛1×⋯×𝐑^𝑛𝑚 with (𝑥,𝑦)≠0 and 𝑝∈𝐑^ℓ such that 𝑀₁𝑥+𝑀₂𝑦+𝑀₃𝑝=0, there exists an index 𝜈∈{1,…,𝑚} such that (𝑥^𝜈,𝑦^𝜈)≠0 and ⟨𝑥^𝜈,𝑦^𝜈⟩≥0; (b) the Cartesian mixed 𝑃-property, if (1) of (a) and the following statement hold:(2^′) for any 𝑥=(𝑥¹,…,𝑥^𝑚), 𝑦=(𝑦¹,…,𝑦^𝑚)∈𝐑^𝑛1×⋯×𝐑^𝑛𝑚 with (𝑥,𝑦)≠0 and 𝑝∈𝐑^ℓ such that 𝑀₁𝑥+𝑀₂𝑦+𝑀₃𝑝=0, there exists an index 𝜈∈{1,…,𝑚} such that ⟨𝑥^𝜈,𝑦^𝜈⟩>0.
In the case ℓ=0 (i.e., 𝑀₃ is vacuous) and 𝑀=[𝑀₁−𝐼], 𝑀 has the Cartesian mixed 𝑃₀ (𝑃)-property if and only if 𝑀₁ has the Cartesian 𝑃₀ (𝑃)-property (see [10, 13], e.g.). By the definitions, it is clear that the Cartesian mixed 𝑃-property implies the Cartesian mixed 𝑃₀-property. Moreover, when 𝑛₁=⋯=𝑛_𝑚=1, the Cartesian mixed 𝑃₀-property reduces to the mixed 𝑃₀-property (see [1, page 1013]).
We now introduce the Cartesian mixed Jordan 𝑃₀ (𝑃)-property.

Definition 7. Let 𝑀∈𝐑^{(𝑛+ℓ)×(2𝑛+ℓ)} be a matrix partitioned as follows: 𝑀=𝑀₁𝑀₂𝑀₃, (21) where 𝑀₁, 𝑀₂∈𝐑^{(𝑛+ℓ)×𝑛} and 𝑀₃∈𝐑^{(𝑛+ℓ)×ℓ}. Then, 𝑀 is said to have (a) the Cartesian mixed Jordan 𝑃₀-property, if the following statements hold: (1)𝑀₃ has full column rank;(2) for any 𝑥=(𝑥¹,…,𝑥^𝑚), 𝑦=(𝑦¹,…,𝑦^𝑚)∈𝐑^𝑛1×⋯×𝐑^𝑛𝑚 with (𝑥,𝑦)≠0 and 𝑝∈𝐑^ℓ such that 𝑀₁𝑥+𝑀₂𝑦+𝑀₃𝑝=0, there exists an index 𝜈∈{1,…,𝑚} such that (𝑥^𝜈,𝑦^𝜈)≠0 and 𝑥^𝜈∘𝑦^𝜈≽0; (b) the Cartesian mixed Jordan 𝑃-property, if (1) of (a) and the following statement hold: (2^′) for any 𝑥=(𝑥¹,…,𝑥^𝑚), 𝑦=(𝑦¹,…,𝑦^𝑚)∈𝐑^𝑛1×⋯×𝐑^𝑛𝑚 with (𝑥,𝑦)≠0 and 𝑝∈𝐑^ℓ such that 𝑀₁𝑥+𝑀₂𝑦+𝑀₃𝑝=0, there exists an index 𝜈∈{1,…,𝑚} such that 𝑥^𝜈∘𝑦^𝜈≻0.
Note that the relation 𝑥^𝜈∘𝑦^𝜈≽0 (resp., 𝑥^𝜈∘𝑦^𝜈≻0) can be rewritten as 𝜆₁(𝑥^𝜈∘𝑦^𝜈)≥0 (resp., 𝜆₁(𝑥^𝜈∘𝑦^𝜈)>0). By the definitions, it is clear that the Cartesian mixed Jordan 𝑃-property implies the Cartesian mixed Jordan 𝑃₀-property, and that the Cartesian mixed Jordan 𝑃₀ (𝑃)-property implies the Cartesian mixed 𝑃₀ (𝑃)-property. Similar to the Cartesian mixed 𝑃₀-property, in the case 𝑛₁=⋯=𝑛_𝑚=1, the Cartesian mixed Jordan 𝑃₀-property also reduces to the mixed 𝑃₀-property.

3. SOC C-Function and Its Smoothing Function

In this section, we introduce the SOC C-function and its smoothing function. In Section 3.1, we give the concept of the SOC C-function to transform the SOCCP into a system of equations. We focus on the Fischer-Burmeister function as an SOC C-function and review some properties of the smoothed Fischer-Burmeister function in Section 3.2.

3.1. SOC C-Function

First, we recall the concept of the SOC C-function.

Definition 8. A function 𝜙∶𝐑^𝑟×𝐑^𝑟→𝐑^𝑟 is said to be an SOC complementarity (C-) function, if the following holds: 𝜙𝑥,𝑦=0⟺𝑥∈𝒦^𝑟,𝑦∈𝒦^𝑟,𝑥,𝑦=0. (22)
Let 𝜙∶𝐑^𝑛×𝐑^𝑛→𝐑^𝑛 be defined as 𝜙𝜙𝑥,𝑦∶=¹𝑥¹,𝑦¹⋮𝜙^𝑚𝑥^𝑚,𝑦^𝑚, (23) where 𝑥 and 𝑦∈𝐑^𝑛 are divided as 𝑥=(𝑥¹,…,𝑥^𝑚) and 𝑦=(𝑦¹,…,𝑦^𝑚) with 𝑥^𝑖, 𝑦^𝑖∈𝐑^𝑛𝑖, 𝑖=1,…,𝑚, and 𝜙^𝑖∶𝐑^𝑛𝑖×𝐑^𝑛𝑖→𝐑^𝑛𝑖 are SOC C-functions. Then it follows from (22) that 𝜙𝑥,𝑦=0⟺𝑥∈𝒦,𝑦∈𝒦,𝑥,𝑦=0. (24) Accordingly, SOCCP (1) is reformulated as a system of equations 𝐻(𝑥,𝑦,𝑝)=0, where 𝐻∶𝐑^𝑛×𝐑^𝑛×𝐑^ℓ→𝐑^2𝑛+ℓ is defined by 𝐻𝜙𝐹𝑥,𝑦,𝑝∶=𝑥,𝑦𝑥,𝑦,𝑝. (25) Moreover, we also give a merit function Ψ∶𝐑^𝑛×𝐑^𝑛×𝐑^ℓ→𝐑 defined by Ψ1𝑥,𝑦,𝑝∶=2𝐻𝑥,𝑦²=12𝜙𝑥,𝑦²+12𝐹𝑥,𝑦,𝑝². (26) Note that Ψ(𝑥,𝑦,𝑝)≥0 for any (𝑥,𝑦,𝑝)∈𝐑^𝑛×𝐑^𝑛×𝐑^ℓ, and that Ψ(𝑥,𝑦,𝑝)=0 if and only if (𝑥,𝑦,𝑝) is a solution of SOCCP (1).
There are many kinds of SOC C-functions. The natural residual function 𝜙^𝑖_NR∶𝐑^𝑛𝑖×𝐑^𝑛𝑖→𝐑^𝑛𝑖 and the Fischer-Burmeister function 𝜙^𝑖_FB∶𝐑^𝑛𝑖×𝐑^𝑛𝑖→𝐑^𝑛𝑖 are respectively defined by 𝜙^𝑖_NR𝑥^𝑖,𝑦^𝑖∶=𝑥^𝑖−𝑥^𝑖−𝑦^𝑖₊,𝜙^𝑖_FB𝑥^𝑖,𝑦^𝑖∶=𝑥^𝑖+𝑦^𝑖−𝑥^𝑖2+𝑦^𝑖21/2, (27)(28) where [𝑧]₊ denotes the projection of 𝑧 onto the SOC 𝒦^𝑛𝑖. Fukushima et al. [5] showed that (22) holds for functions 𝜙^𝑖_NR and 𝜙^𝑖_FB. Chen et al. [7] and Hayashi et al. [6] proposed methods for solving SOCCP based on the natural residual function (27), whereas Narushima et al. [8] proposed methods for solving SOCCP based on the Fischer-Burmeister function (28).
In what follows, functions 𝜙_FB, 𝐻_FB, and Ψ_FB denote 𝜙, 𝐻, and Ψ with 𝜙^𝑖_FB, respectively. Also, functions 𝜙_NR, 𝐻_NR, and Ψ_NR denote 𝜙, 𝐻, and Ψ with 𝜙^𝑖_NR, respectively.
Recently, Bi et al. [20] showed the following inequality: 2−√2𝜙^𝑖_NR𝑥^𝑖,𝑦^𝑖≤𝜙^𝑖_FB𝑥^𝑖,𝑦^𝑖≤2+√2𝜙^𝑖_NR𝑥^𝑖,𝑦^𝑖forany𝑥^𝑖,𝑦^𝑖∈𝐑^𝑛𝑖. (29) We see from (29) that the level-boundedness of Ψ_FB is equivalent to that of Ψ_NR.

3.2. Smoothed FB Function and Its Properties

In this section, we consider the smoothing function associated with the Fischer-Burmeister function and give its properties and Jacobian matrix.

Since 𝐻_FB is not differentiable in general, we cannot apply conventional methods such as Newton’s method or Newton-based methods. We therefore consider the smoothed Fischer-Burmeister function 𝜙_𝑡∶𝐑^2𝑛→𝐑^𝑛, which was originally proposed by Kanzow [21] for solving NCP and generalized by Fukushima et al. [5] to SOCCP. Let 𝜙^𝑖_𝑡∶𝐑^2𝑛𝑖→𝐑^𝑛𝑖 be defined by 𝜙^𝑖_𝑡𝑥,𝑦∶=𝑥^𝑖+𝑦^𝑖−2𝑡²𝑒^𝑖+𝑥^𝑖2+𝑦^𝑖21/2 (30) for each 𝑖=1,…,𝑚, where 𝑡 is a smoothing parameter and 𝑒^𝑖∶=(1,0,…,0)^⊤∈𝐑^𝑛𝑖. Then, the smoothed Fischer-Burmeister function 𝜙_𝑡∶𝐑^2𝑛→𝐑^𝑛 is defined as 𝜙_𝑡𝜙𝑥,𝑦∶=^1𝑡𝑥¹,𝑦¹⋮𝜙^𝑚_𝑡𝑥^𝑚,𝑦^𝑚. (31) Also, the smoothing function 𝐻_𝑡∶𝐑^2𝑛+ℓ→𝐑^2𝑛+ℓ and the merit function Ψ_𝑡∶𝐑^2𝑛+ℓ→𝐑 are defined as 𝐻_𝑡𝜙𝑥,𝑦,𝑝∶=^𝑡𝐹Ψ𝑥,𝑦𝑥,𝑦,𝑝_𝑡,1𝑥,𝑦,𝑝∶=2𝐻_𝑡𝑥,𝑦,𝑝²=12𝜙_𝑡𝑥,𝑦²+12𝐹𝑥,𝑦,𝑝², (32)(33) respectively. Clearly, 𝜙₀(𝑥,𝑦)≡𝜙_FB(𝑥,𝑦), and so 𝐻₀(𝑥,𝑦,𝑝)≡𝐻_FB(𝑥,𝑦,𝑝) and Ψ₀(𝑥,𝑦,𝑝)≡Ψ_FB(𝑥,𝑦,𝑝). We note that 𝜙^𝑖_𝑡𝑥^𝑖,𝑦^𝑖−𝜙^𝑖_FB𝑥,𝑦≤√2|𝑡| (34) holds for any 𝑡∈𝐑 and (𝑥^𝑖,𝑦^𝑖)∈𝐑^2𝑛𝑖 (see [5] or [22]). From definition (32) of 𝐻_𝑡 and (34), it follows that 𝐻_𝑡𝑥,𝑦,𝑝−𝐻_FB=𝜙𝑥,𝑦,𝑝_𝑡𝑥,𝑦−𝜙_FB=𝑥,𝑦_𝑚𝑖=1𝜙^𝑖_𝑡𝑥^𝑖,𝑦^𝑖−𝜙^𝑖_FB𝑥^𝑖,𝑦^𝑖21/2≤√2𝑚|𝑡| (35) for any 𝑡∈𝐑 and (𝑥,𝑦,𝑝)∈𝐑^2𝑛+ℓ.

In what follows, we write 𝑥^𝑖=((𝑥^𝑖)₁,(𝑥^𝑖)₂)∈𝐑×𝐑^𝑛𝑖−1 for any vector 𝑥^𝑖∈𝐑^𝑛𝑖. Moreover, for convenience, we use the following notation. For any 𝑥^𝑖, 𝑦^𝑖∈𝐑^𝑛𝑖 and any 𝑡∈𝐑, we write 𝑧^𝑖=(𝑥^𝑖,𝑦^𝑖)∈𝐑^2𝑛𝑖 and define the functions 𝑤^𝑖_𝑡, 𝑢^𝑖_𝑡∶𝐑^2𝑛𝑖→𝐑×𝐑^𝑛𝑖−1 by 𝑤^𝑖_𝑡=𝑤^𝑖_𝑡1,𝑤^𝑖_𝑡2∶=2𝑡²𝑒^𝑖+𝑥^𝑖2+𝑦^𝑖2,𝑢^𝑖_𝑡=𝑢^𝑖_𝑡1,𝑢^𝑖_𝑡2∶=2𝑡²𝑒^𝑖+𝑥^𝑖2+𝑦^𝑖21/2. (36) Furthermore, we drop the subscript for 𝑡=0 for simplicity, and thus, 𝑤^𝑖=𝑤^𝑖₁,𝑤^𝑖₂∶=𝑥^𝑖2+𝑦^𝑖2,𝑢^𝑖=𝑢^𝑖₁,𝑢^𝑖₂∶=𝑥^𝑖2+𝑦^𝑖21/2. (37) Direct calculation yields 𝑤^𝑖_𝑡1=2𝑡²+𝑥^𝑖2+𝑦^𝑖2=2𝑡²+𝑤^𝑖₁,𝑤^𝑖_𝑡2=2𝑥^𝑖₁𝑥^𝑖₂+𝑦^𝑖₁𝑦^𝑖₂=𝑤^𝑖₂. (38) Note that (𝑤^𝑖_𝑡)₂ is actually independent of 𝑡, so that hereafter we will write 𝑤^𝑖_𝑡=((𝑤^𝑖_𝑡)₁,(𝑤^𝑖)₂). We also easily get, for 𝑗=1,2, 𝜆_𝑗𝑤^𝑖_𝑡=2𝑡²+𝑥^𝑖2+𝑦^𝑖2+2(−1)^𝑗𝑥^𝑖₁𝑥^𝑖₂+𝑦^𝑖₁𝑦^𝑖₂𝑠^{𝑗}=121,(−1)^𝑗𝑤^𝑖₂, (39) where 𝜆₁(𝑤^𝑖_𝑡), 𝜆₂(𝑤^𝑖_𝑡) and 𝑠^{1}, 𝑠^{2} are the spectral values and the associated spectral vectors of 𝑤^𝑖_𝑡, respectively, with (𝑤^𝑖)₂∶=(𝑤^𝑖)₂/‖(𝑤^𝑖)₂‖ if (𝑤^𝑖)₂≠0, and otherwise, (𝑤^𝑖)₂ being any vector in 𝐑^𝑛𝑖−1 satisfying ‖(𝑤^𝑖)₂‖=1.

Now we review some propositions needed to establish convergence properties of the smoothing Newton method. The following proposition gives explicit expression of the transposed Jacobian matrix with 𝑡≠0.

Proposition 9 (see [5]). For any 𝑡≠0, 𝐻_𝑡 is continuously differentiable on 𝐑^2𝑛+ℓ, and its transposed Jacobian matrix is given by ∇𝐻_𝑡=∇𝑥,𝑦,𝑝diag_𝑥𝑖𝜙^𝑖_𝑡𝑥^𝑖,𝑦^𝑖𝑚_𝑖=1∇_𝑥𝐹∇𝑥,𝑦,𝑝diag_𝑦𝑖𝜙^𝑖_𝑡𝑥^𝑖,𝑦^𝑖𝑚_𝑖=1∇_𝑦𝐹𝑥,𝑦,𝑝𝑂∇_𝑝𝐹𝑥,𝑦,𝑝, (40) where diag{𝐷_𝑖}^𝑚_𝑖=1 denotes the block-diagonal matrix with block elements 𝐷_𝑖∈𝐑^{𝑛𝑖×𝑛𝑖}, and ∇_𝑥𝑖𝜙^𝑖_𝑡𝑥^𝑖,𝑦^𝑖=𝐼−𝐿_𝑥𝑖𝐿_𝑢⁻¹_𝑖𝑡,∇_𝑦𝑖𝜙^𝑖_𝑡𝑥^𝑖,𝑦^𝑖=𝐼−𝐿_𝑦𝑖𝐿_𝑢⁻¹_𝑖𝑡,𝐿_𝑢⁻¹_𝑖𝑡=𝑏^𝑖_𝑡−𝑐^𝑖_𝑡𝑤^𝑖⊤₂−𝑐^𝑖_𝑡𝑤^𝑖₂𝑎^𝑖_𝑡𝐼+𝑏^𝑖_𝑡−𝑎ⁱ_𝑡𝑤^𝑖₂𝑤^𝑖⊤₂ (41) with 𝑎^𝑖_𝑡2∶=𝜆₁𝑤^𝑖_𝑡+𝜆₂𝑤^𝑖_𝑡,𝑏^𝑖_𝑡1∶=21𝜆₁𝑤^𝑖_𝑡+1𝜆₂𝑤^𝑖_𝑡,𝑐^𝑖_𝑡1∶=21𝜆₁𝑤^𝑖_𝑡−1𝜆₂𝑤^𝑖_𝑡. (42)

In order to obtain the Newton step, the nonsingularity of ∇𝐻_𝑡 is important. The next proposition establishes the nonsingularity of ∇𝐻_𝑡.

Proposition 10 (see [8]). Let 𝑡 be an arbitrary nonzero number and let (𝑥,𝑦,𝑝)∈𝐑^2𝑛+ℓ be an arbitrary triple such that the Jacobian matrix ∇𝐹(𝑥,𝑦,𝑝)^⊤ has the Cartesian mixed 𝑃₀-property at (𝑥,𝑦,𝑝), that is, ∇_𝑝𝐹(𝑥,𝑦,𝑝) satisfies rank∇_𝑝𝐹𝑥,𝑦,𝑝=ℓ, (43) and ∇𝐹𝑥,𝑦,𝑝^⊤𝜉,𝜂,𝜑=0,𝜉,𝜂≠0,𝜑∈𝐑^ℓ𝜉=𝜉¹,…,𝜉^𝑚,𝜂=𝜂¹,…,𝜂^𝑚∈𝐑^𝑛1×⋯×𝐑^𝑛𝑚{}𝜉⟹𝑡ℎ𝑒𝑟𝑒𝑒𝑥𝑖𝑠𝑡𝑠𝑎𝑛𝑖𝑛𝑑𝑒𝑥𝜈∈1,…,𝑚𝑠𝑢𝑐ℎ𝑡ℎ𝑎𝑡^𝜈,𝜂^𝜈≠0,𝜉^𝜈,𝜂^𝜈≥0. (44) Then, the matrix ∇𝐻_𝑡(𝑥,𝑦,𝑝) given by (40) is nonsingular.

The local Lipschitz continuity and the (strong) semismoothness of 𝐻_FB play a significant role in establishing locally rapid convergence.

Proposition 11. The function 𝐻_FB is locally Lipschitzian on 𝐑^2𝑛+ℓ and, moreover, is semismooth on 𝐑^2𝑛+ℓ. In addition, if ∇𝐹 is locally Lipschitzian, then 𝐻_FB is strongly semismooth on 𝐑^2𝑛+ℓ.

Proof. It follows from [23, Corollary 3.3] that 𝜙^𝑖_FB is globally Lipschitzian and strongly semismooth. Since 𝐹 is a continuously differentiable function, 𝐻_FB is locally Lipschitzian on 𝐑^2𝑛+ℓ. Also, the (strong) semismoothness of 𝐻_FB can be easily shown from the strong semismoothness of 𝜙^𝑖_FB and the (local Lipschitz) continuity of ∇𝐹.

We define function Θ_𝑖∶𝐑^2𝑛𝑖→𝐑₊ by Θ_𝑖(𝑥^𝑖,𝑦^𝑖)∶=‖𝐿_𝑥𝑖𝐿_𝑦𝑖‖. It is easily seen that Θ_𝑖(𝑥^𝑖,𝑦^𝑖)=0 if and only if (𝑥^𝑖,𝑦^𝑖)=(0,0). Now we partition 𝐑^2𝑛𝑖 as 𝐑^2𝑛𝑖=𝒵^𝑖₁∪𝒵^𝑖₂∪{(0,0)}, where 𝒵^𝑖₁∶=𝑧^𝑖=𝑥^𝑖,𝑦^𝑖∈𝐑^2𝑛𝑖∣𝑤^𝑖=𝑥^𝑖2+𝑦^𝑖2∈int𝒦^𝑛𝑖,𝒵^𝑖₂∶=𝑧^𝑖=𝑥^𝑖,𝑦^𝑖∈𝐑^2𝑛𝑖∣𝑤^𝑖=𝑥^𝑖2+𝑦^𝑖2∈bd𝒦^𝑛𝑖,𝑤^𝑖≠0. (45) In order to achieve locally rapid convergence of the method, we need to control the parameter 𝑡 so that the distance between ∇𝐻_𝑡 and 𝜕𝐻_FB is sufficiently small. The following proposition is helpful to control the parameter 𝑡 appropriately.

Proposition 12 (see [22]). Let (𝑥,𝑦,𝑝) be any point in 𝐑^2𝑛+ℓ. Let 𝜃_𝑖(𝑥^𝑖,𝑦^𝑖) be any function such that Θ_𝑖(𝑥^𝑖,𝑦^𝑖)≤𝜃_𝑖(𝑥^𝑖,𝑦^𝑖). Let 𝛿>0 be given. Let 𝑡∶𝐑^2𝑛×𝐑₊→𝐑₊∪{+∞} be defined by 𝑡𝑥,𝑦,𝛿∶=min𝑡_𝑖𝑥^𝑖,𝑦^𝑖,𝛿∣𝑖=1,…,𝑚, (46) where 𝑡_𝑖𝑥^𝑖,𝑦^𝑖𝜆,𝛿∶=¹𝑤^𝑖𝛿2𝜃_𝑖𝑧^𝑖2−𝜆₁𝑤^𝑖𝛿²𝑖𝑓zⁱ=𝑥^𝑖,𝑦^𝑖∈𝒵^𝑖₁𝜃,𝛿<ⁱ𝑧^𝑖𝜆₁𝑤^𝑖,𝑤^𝑖₁𝛿2𝜃_𝑖𝑧^𝑖2𝜃_𝑖𝑧^𝑖−𝛿2𝑤^𝑖₁𝑖𝑓𝑧^𝑖=𝑥^𝑖,𝑦^𝑖∈𝒵^𝑖₂,𝛿<2𝜃_i𝑧^𝑖2𝑤^𝑖₁,+∞𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒. (47) Then, for any 𝑡∈𝐑 such that 0<|𝑡|≤𝑡(𝑥,𝑦,𝛿), dist∇𝐻_𝑡𝑥,𝑦,𝑝,𝜕𝐻_FB𝑥,𝑦,𝑝<𝛿, (48) where dist(𝑋,𝑆) denotes min{‖𝑋−𝑌‖∣𝑌∈𝑆}.

4. Smoothing Newton Method and Its Convergence Properties

In this section, we first propose an algorithm of the smoothing Newton method based on the Fischer-Burmeister function and its smoothing function. We then prove its global and 𝑄-superlinear (𝑄-quadratic) convergence.

4.1. Algorithm

We provide the smoothing Newton algorithm based on the Fischer-Burmeister function. In what follows, we write 𝑣^(𝑘)=(𝑥^(𝑘),𝑦^(𝑘),𝑝^(𝑘)) and 𝑧^(𝑘)=(𝑥^(𝑘),𝑦^(𝑘)) for simplicity.

Algorithm 13.   
   Step 0. Choose 𝜂, 𝜌∈(0,1), 𝜂∈(0,𝜂], 𝜎∈(0,1/2), 𝑟>1, 𝜅>0, and 𝜅>0.
Choose 𝑣⁽⁰⁾=(𝑥⁽⁰⁾,𝑦⁽⁰⁾,𝑝⁽⁰⁾)∈𝐑^2𝑛+ℓ and 𝛽₀∈(0,∞). Let 𝑡₀∶=‖𝐻_FB(𝑣⁽⁰⁾)‖. Set 𝑘∶=0.
Step 1. If a stopping criterion, such as ‖𝐻_FB(𝑣^(𝑘))‖=0, is satisfied, then stop.
Step 2.
Step 2.0. Set 𝑣̂⁽⁰⁾∶=𝑣^(𝑘) and 𝑗∶=0.
Step 2.1. Compute 𝑑^(𝑗)∈𝐑^2𝑛+ℓ by solving 𝐻_𝑡𝑘𝑣̂^(𝑗)+∇𝐻_𝑡𝑘𝑣̂^(𝑗)⊤𝑑^(𝑗)=0. (49)
Step 2.2. If ‖𝐻_𝑡𝑘(𝑣̂^(𝑗)+𝑑^(𝑗))‖≤𝛽_𝑘, then let 𝑣^(𝑘+1)∶=𝑣̂^(𝑗)+𝑑^(𝑗) and go to Step 3. Otherwise, go to Step 2.3.
Step 2.3. Let 𝑙_𝑗 be the smallest nonnegative integer 𝑙 satisfying Ψ_𝑡𝑘𝑣̂^(𝑗)+𝜌^𝑙𝑑^(𝑗)≤1−2𝜎𝜌^𝑙Ψ_𝑡𝑘𝑣̂^(𝑗). (50)
Let 𝜏_𝑗∶=𝜌^𝑙𝑗 and 𝑣̂^(𝑗+1)∶=𝑣̂^(𝑗)+𝜏_𝑗𝑑^(𝑗).
Step 2.4. If 𝐻_𝑡𝑘𝑣̂^(𝑗+1)≤𝛽_𝑘, (51) then let 𝑣^(𝑘+1)∶=𝑣̂^(𝑗+1) and go to Step 3. Otherwise, set 𝑗∶=𝑗+1 and go back to Step 2.1.
Step 3. Update the parameters as follows: 𝛿_𝑘+1∶=𝐻_FB𝑣^(𝑘+1),𝑡_𝑘+1∶=min𝜅𝛿^𝑟_𝑘+1,𝑡₀𝜂^𝑘+1,𝑡𝑧^(𝑘+1),𝜅𝛿_𝑘+1,𝛽_𝑘+1∶=𝛽₀𝜂^𝑘+1. (52)
Step 4. Set 𝑘∶=𝑘+1. Go back to Step 1.
Note that the proposed algorithm consists of the outer iteration steps and the inner iteration steps. Step 2 is the inner iteration steps with the variable 𝑣̂ and the counter 𝑗, while the outer iteration steps have the variable 𝑣 and the counter 𝑘.
From Step 3 of Algorithm 13 and (48), the following inequality holds: min∇𝐻_𝑡𝑘𝑣^(𝑘)−𝑌∣𝑌∈𝜕𝐻_FB𝑣^(𝑘)≤𝜅𝐻_FB𝑣^(𝑘). (53) Letting 𝑣^∗ be a solution of SOCCP (1), we have 𝐻_FB(𝑣^∗)=0. Therefore, from (53) and the local Lipschitz continuity of 𝐻_FB, the following holds: min∇𝐻_𝑡𝑘𝑣^(𝑘)−𝑌∣𝑌∈𝜕𝐻_FB𝑣^(𝑘)≤𝜅𝐻_FB𝑣^(𝑘)−𝐻_FB𝑣^∗=𝑂𝑣^(𝑘)−𝑣^∗. (54)
In the rest of this section, we consider convergence properties of Algorithm 13. In Section 4.2, we prove the global convergence of the algorithm, and in Section 4.3, we investigate local behavior of the algorithm. For this purpose, we make the following assumptions.

Assumption 1. (A1) The solution set 𝒮 of SOCCP (1) is nonempty and bounded. (A2) The function Ψ_FB is level-bounded, that is, for any 𝑣∈𝐑^2𝑛+ℓ, the level set {𝑣∈𝐑^2𝑛+ℓ∣Ψ_FB(𝑣)≤Ψ_FB(𝑣)} is bounded. (A3) For any 𝑡>0 and 𝑣∈𝐑^2𝑛+ℓ, ∇𝐻_𝑡(𝑣) is nonsingular.

From Proposition 10, Assumption (A3) holds if ∇𝐹(𝑣)^⊤ has the Cartesian mixed 𝑃₀-property for any 𝑣∈𝐑^2𝑛+ℓ. The following remarks correspond to SOCCPs (3) and (5).

Remark 14. The case of SOCCP (3). If ∇𝑓(𝑥)^⊤ has the Cartesian 𝑃₀-property, then ∇𝐹(𝑥,𝑦,𝑝)^⊤ with 𝐹 in (4) has the Cartesian mixed 𝑃₀-property and vice versa. Note that ∇𝑓(𝑥)^⊤ has the Cartesian 𝑃₀-property at any 𝑥∈𝐑^𝑛 if 𝑓 has the Cartesian 𝑃₀-property (see [10, 13] for the definition of the Cartesian 𝑃₀-property for a matrix).

Remark 15. The case of SOCCP (5). If ∇𝑓(𝑝) is nonsingular and (∇𝑓(𝑝)⁻¹∇𝑔(𝑝))^⊤ has the Cartesian 𝑃₀-property, then ∇𝐹(𝑥,𝑦,𝑝)^⊤ with 𝐹 in (6) has the Cartesian mixed 𝑃₀-property.

Note that Assumption (A2) is equivalent to the coerciveness in the sense that lim_{‖𝑣‖→∞}Ψ_FB(𝑣)=+∞. (55) We now give some sufficient conditions for Assumption (A2) in the case of SOCCP (3) or (5).

Lemma 16. Consider SOCCP (5). Let Ψ_FB∶𝐑^𝑛→𝐑 be a function such that Ψ_FB(𝑝)∶=(1/2)‖𝜙_FB(𝑓(𝑝),𝑔(𝑝))‖² for any 𝑝∈𝐑^𝑛. Then Ψ_FB is level-bounded if and only if Ψ_FB is level-bounded.

Proof. We use below condition (55) equivalent to the level-boundedness. We first assume that Ψ_FB is level-bounded and claim that Ψ_FB is level-bounded. Suppose the contrary. Then, we can find a sequence {𝑣^(𝑘)}={(𝑥^(𝑘),𝑦^(𝑘),𝑝^(𝑘))}⊂𝐑^3𝑛 such that the sequence {Ψ_FB(𝑣^(𝑘))} is bounded and ‖𝑣^(𝑘)‖→∞. If {‖𝑝^(𝑘)‖} is bounded, then we must have ‖(𝑥^(𝑘),𝑦^(𝑘))‖→∞. Thus, from the inequality 2Ψ_FB𝑣^(𝑘)≥𝑓𝑝^(𝑘)−𝑥^(𝑘),𝑔𝑝^(𝑘)−𝑦^(𝑘)≥𝑥^(𝑘),𝑦^(𝑘)−𝑓𝑝^(𝑘),𝑔𝑝^(𝑘) (56) and from the continuity of 𝑓 and 𝑔, we have Ψ_FB(𝑣^(𝑘))→∞. Since this is not possible, {‖𝑝^(𝑘)‖} is unbounded. By taking a subsequence if necessary, we may assume that ‖𝑝^(𝑘)‖→∞ as 𝑘→∞. Since 𝜙_FB is globally Lipschitzian by [23, Corollary 3.3], we have from (33) that, for any (𝑥,𝑦,𝑝)∈𝐑^3𝑛, 2Ψ_FB𝑝=𝜙_FB𝑓𝑝𝑝≤𝜙,𝑔_FB𝑓𝑝,𝑔𝑝−𝜙_FB𝑥,𝑦+𝜙_FB𝑓𝑝𝑝−+𝜙𝑥,𝑦≤𝐿,𝑔𝑥,𝑦_FB≤𝑥,𝑦𝐿²+1𝑓𝑝−𝑥,𝑔𝑝−𝑦²+𝜙_FB𝑥,𝑦²=𝐿²+12Ψ_FB𝑥,𝑦,𝑝, (57) where 𝐿>0 is a Lipschitz constant. Then, it follows from (57) and the level-boundedness of Ψ_FB that lim_𝑘→∞Ψ_FB(𝑣^(𝑘))=+∞, contradicting the boundedness of {Ψ_FB(𝑣^(𝑘))}. This proves the level-boundedness of Ψ_FB.
We next assume that Ψ_FB is level-bounded. Let {𝑝^(𝑘)}⊂𝐑^𝑛 be an arbitrary sequence such that ‖𝑝^(𝑘)‖→∞ and let 𝑥^(𝑘)∶=𝑓(𝑝^(𝑘)) and 𝑦^(𝑘)∶=𝑔(𝑝^(𝑘)). Then we have 2Ψ_FB𝑥^(𝑘),𝑦^(𝑘),𝑝^(𝑘)=𝜙_FB𝑥^(𝑘),𝑦^(𝑘)2+𝑓𝑝^(𝑘)−𝑥^(𝑘)2+𝑔𝑝^(𝑘)−𝑦^(𝑘)2=𝜙_FB𝑓𝑝^(𝑘),𝑔𝑝^(𝑘)2=2Ψ_FB𝑝^(𝑘). (58) Thus, from ‖(𝑥^(𝑘),𝑦^(𝑘),𝑝^(𝑘))‖≥‖𝑝^(𝑘)‖→∞ and the level-boundedness of Ψ_FB, we have lim_𝑘→∞Ψ_FB(𝑝^(𝑘))=+∞. Therefore, Ψ_FB is level-bounded.

Remark 17. Consider SOCCP (3). Let Ψ_FB∶𝐑^𝑛→𝐑 be a function such that Ψ_FB(𝑥)∶=(1/2)‖𝜙_FB(𝑥,𝑓(𝑥))‖² for any 𝑥∈𝐑^𝑛. Then, in the same way as in Lemma 16, we can show that Ψ_FB is level-bounded if and only if Ψ_FB is level-bounded (we have only to consider 𝑔(𝑝)≡𝑝).

We now provide some sufficient conditions under which Assumption (A2) holds.

Proposition 18. Consider SOCCP (5). Assume that 𝑓 and 𝑔 have linear growth. Assume further that 𝑓 and 𝑔 satisfy one of the following statements: (a)𝑓 and 𝑔 have the joint uniform Jordan 𝑃-property;(b)𝑓 and 𝑔 have the joint Cartesian weak coerciveness. Then Ψ_FB is level-bounded.

Proof. It follows from [19] that Ψ_NR(𝑝)∶=(1/2)‖𝜙_NR(𝑓(𝑝),𝑔(𝑝))‖² is level-bounded for each case. Thus from Lemma 16 and (29), we have desired results.

The following condition was given by Pan and Chen [13] to establish the level-boundedness property of the merit function Ψ_FB(𝑥) defined in Remark 17.

Condition A. Consider SOCCP (3). For any sequence {𝜉_𝑘}⊂𝐑^𝑛 satisfying ‖𝜉_𝑘‖→∞ with 𝜉^𝑖_𝑘∈𝐑^𝑛𝑖, if there exists an index 𝜈∈{1,…,𝑚} such that {𝜆₁(𝜉^𝜈_𝑘)} and {𝜆₁(𝑓^𝜈(𝜉_𝑘))} are bounded below, and 𝜆₂(𝜉^𝜈_𝑘),𝜆₂(𝑓^𝜈(𝜉_𝑘))→∞, then limsup_𝑘→∞𝜉^𝜈_𝑘𝜉^𝜈_𝑘,𝑓^𝜈𝜉_𝑘𝑓^𝜈𝜉_𝑘>0. (59) Under Condition A, we have the following proposition, which corresponds to Proposition 5.2 of [13].

Proposition 19. Consider SOCCP (3). Assume that 𝑓 has the uniform Cartesian 𝑃-property and satisfies Condition A. Then Ψ_FB is level-bounded.

4.2. Global Convergence

In this section, we show the global convergence of Algorithm 13. We first give the well-definedness of the algorithm.

Lemma 20. Suppose that Assumption (A3) holds. Let 𝑡 be any fixed positive number. Every stationary point 𝑣^∗_𝑡 of Ψ_𝑡 satisfies Ψ_𝑡(𝑣^∗_𝑡)=0.

Proof. For each stationary point 𝑣^∗_𝑡 of Ψ_𝑡, ∇Ψ_𝑡(𝑣^∗_𝑡)=∇𝐻_𝑡(𝑣^∗_𝑡)𝐻(𝑣^∗_𝑡)=0 holds. Since, from 𝑡>0 and Assumption (A3), ∇𝐻_𝑡(𝑣^∗_𝑡) is nonsingular, we have 𝐻_𝑡(𝑣^∗_𝑡)=0, and thus, Ψ_𝑡(𝑣^∗_𝑡)=0.

It follows from (35) that 2Ψ_FB(𝑣)̂≤𝐻_𝑡(𝑣)̂+𝐻_FB(𝑣)̂−𝐻_𝑡(𝑣̂≤)2Ψ_𝑡(𝑣)̂+√2𝑚𝑡 (60) for any 𝑣̂∈𝐑^2𝑛+ℓ, and hence we have, from Assumption (A2) and (55), that Ψ_𝑡 is level-bounded for any fixed 𝑡>0. Therefore, there exists at least one stationary point of Ψ_𝑡. Thus from Lemma 20, the system 𝐻_𝑡(𝑣)=0 has at least one solution, and hence, there exists a point 𝑣 satisfying ‖𝐻_𝑡𝑘(𝑣)‖<𝛽_𝑘 in Step 2 at each iteration.

We are now ready to show the well-definedness of the algorithm.

Proposition 21. Suppose that Assumptions (A2) and (A3) hold. Then Algorithm 13 is well-defined.

Proof. To establish the well-definedness of Algorithm 13, we only need to prove the well-definedness and the finite termination property of Step 2 at each iteration. Now we fix 𝑘 and 𝑡_𝑘>0. Since ∇𝐻_𝑡𝑘(𝑣̂^(𝑗)) is nonsingular for any 𝑣̂^(𝑗)∈𝐑^2𝑛+ℓ by 𝑡_𝑘>0 and Assumption (A3), 𝑑^(𝑗) is uniquely determined for any 𝑗≥0. In addition, we have ∇Ψ_𝑡𝑘𝑣̂^(𝑗)⊤𝑑^(𝑗)=−𝐻_𝑡𝑘𝑣̂^(𝑗)⊤∇𝐻_𝑡𝑘𝑣̂^(𝑗)⊤∇𝐻_𝑡𝑘𝑣̂^(𝑗)−⊤𝐻_𝑡𝑘𝑣̂^(𝑗)=−𝐻_𝑡𝑘𝑣̂^(𝑗)2≤0. (61) If ‖𝐻_𝑡𝑘(𝑣̂^(𝑗)+𝑑^(𝑗))‖≤𝛽_𝑘, then Step 2 terminates in Step 2.2. If ‖𝐻_𝑡𝑘(𝑣̂^(𝑗)+𝑑^(𝑗))‖>𝛽_𝑘, then integer 𝑙 satisfying (50) can be found at Step 2.3, because 𝑑^(𝑗)≠0 and ∇Ψ_𝑡𝑘(𝑣̂^(𝑗))^⊤𝑑^(𝑗)<0. Thus, Step 2 is well-defined at each iteration.
Next we prove the finite termination property of Step 2. To prove by contradiction, we assume that Step 2 never stops and then 𝐻_𝑡𝑘𝑣̂^(𝑗)>𝛽_𝑘>0 (62) holds for all 𝑗≥0. We consider two cases. (i) The case where there exists a subsequence 𝐽 such that lim_{𝑗∈𝐽,𝑗→∞}𝜏_𝑗=0. From the boundedness of the level set of Ψ_𝑡𝑘 at 𝑣̂⁽⁰⁾ and the line search rule (50), {𝑣̂^(𝑗)} is bounded. In addition, from the continuous differentiability of Ψ_𝑡𝑘 and Assumption (A3), {𝑑^(𝑗)} is also bounded. Thus, there exists a subsequence 𝐽^′⊂𝐽 such that lim_{𝑗∈𝐽′,𝑗→∞}𝑣̂^(𝑗)=𝑣_∗̂,lim_{𝑗∈𝐽′,𝑗→∞}𝑑^(𝑗)=𝑑_∗. (63) Now 𝜏_𝑗≠1 holds for all sufficiently large 𝑗∈𝐽^′, and hence, we have Ψ_𝑡𝑘𝑣̂^(𝑗)+𝜌^𝑙𝑗−1𝑑^(𝑗)−Ψ_𝑡𝑘𝑣̂^(𝑗)𝜌^𝑙𝑗−1>−2𝜎Ψ_𝑡𝑘𝑣̂^(𝑗). (64) Passing to the limit 𝑗→∞ with 𝑗∈𝐽^′ on the above inequality and taking (62) into account, we have ∇Ψ_𝑡𝑘𝑣̂_∗^⊤𝑑_∗≥−2𝜎Ψ_𝑡𝑘𝑣_∗̂>−2Ψ_𝑡𝑘𝑣_∗̂. (65) On the other hand, it follows from (61) that ∇Ψ_𝑡𝑘(𝑣_∗̂)^⊤𝑑_∗=−2Ψ_𝑡𝑘(𝑣_∗̂), which contradicts (65). (ii) The case where there exists 𝜏>0 such that 𝜏_𝑗>𝜏 for all 𝑗. It follows from (50) that Ψ_𝑡𝑘𝑣̂^(𝑗)≤(1−2𝜎𝜏)^𝑗Ψ_𝑡𝑘𝑣̂⁽⁰⁾, (66) which implies ‖𝐻_𝑡𝑘(𝑣̂^(𝑗))‖≤𝛽_𝑘 holds for sufficiently large 𝑗. This contradicts (62). Therefore, the proof is complete.

In order to show the global convergence of the proposed method, we recall the mountain pass theorem (see [24], e.g.), which is as follows.

Lemma 22. Let 𝜑∶𝐑^𝑛→𝐑 be a continuously differentiable and level-bounded function. Let 𝒞⊂𝐑^𝑛 be a nonempty and compact set and let 𝜑 be the minimum value of 𝜑 on bd 𝒞, that is, 𝜑∶=min_{𝑥∈bd𝒞}𝜑(𝑥). (67) Assume that there exist vectors 𝜉∈𝒞 and 𝜂∉𝒞 such that 𝜑(𝜉)<𝜑 and 𝜑(𝜂)<𝜑. Then, there exists a vector 𝜁∈𝐑^𝑛 such that ∇𝜑(𝜁)=0 and 𝜑(𝜁)≥𝜑.

By using the mountain pass theorem, we can show the following global convergence property.

Theorem 23. Suppose that Assumptions (A1)–(A3) hold. Then, any accumulation point of the sequence {𝑣^(𝑘)} generated by Algorithm 13 is bounded, and hence, at least one accumulation point exists, and any such point is a solution of SOCCP (1).

Proof. From the choices of 𝑡_𝑘 and 𝛽_𝑘 in Step 3 of Algorithm 13, 𝑡_𝑘 and 𝛽_𝑘 converge to zero. Since 𝛽_𝑘→0, we have lim_𝑘→∞‖𝐻_𝑡𝑘(𝑣^(𝑘))‖=0. Thus, it follows from 𝑡_𝑘→0 and (35) that lim_𝑘→∞‖𝐻_FB(𝑣^(𝑘))‖=0. It implies from the continuity of 𝐻_FB that any accumulation point of the sequence {𝑣^(𝑘)} is a solution of 𝐻_FB(𝑣)=0, and hence, it suffices to prove the boundedness of {𝑣^(𝑘)}. To the contrary, we assume {𝑣^(𝑘)} is unbounded. Then there exist an index set 𝐾 and a subsequence {𝑣^(𝑘)}_𝑘∈𝐾 such that lim_{𝑘∈𝐾,𝑘→∞}‖𝑣^(𝑘)‖=∞. Since, by Assumption (A1), the solution set 𝒮 is bounded, there exists a compact neighborhood 𝒞 of 𝒮 such that 𝒮⊂int𝒞. From the boundedness of 𝒞, 𝑣^(𝑘)∉𝒞 for all 𝑘 sufficiently large 𝑘∈𝐾. In addition, from 𝒮⊂int𝒞, we have 𝜑∶=min_{𝑣∈bd𝒞}Ψ_FB(𝑣)>0, (68) for otherwise, there would exist 𝑣∈bd𝒞 with Ψ_FB(𝑣)=0, that is, 𝑣∈𝒮∩bd𝒞, which contradicts 𝒮⊂int𝒞. Since 𝑡_𝑘 is small enough for all sufficiently large 𝑘, it follows from (35) that −𝜑2<Ψ_𝑡𝑘(𝑣)−Ψ_FB(𝑣)<𝜑2 (69) holds for any 𝑣∈𝒞. Now we take 𝑣∈𝒮⊂𝒞. Then (69) yields Ψ_𝑡𝑘(𝑣)−Ψ_FB(𝑣)=Ψ_𝑡𝑘(𝑣)<𝜑2. (70) Letting 𝑣̃^(𝑘)∈argmin_{𝑣∈bd𝒞}Ψ_𝑡𝑘(𝑣), (71) we have from (68) and 𝑣̃^(𝑘)∈bd𝒞 that 𝜑≤Ψ_FB(𝑣̃^(𝑘)). Therefore, it follows from (69) and (70) that, for all sufficiently large 𝑘, min_{𝑣∈bd𝒞}Ψ_𝑡𝑘(𝑣)=Ψ_𝑡𝑘𝑣̃^(𝑘)𝜑>−2+Ψ_FB𝑣̃^(𝑘)𝜑≥−2+𝜑𝜑=2. (72) On the other hand, since 0≤‖𝐻_𝑡𝑘(𝑣^(𝑘+1))‖≤𝛽_𝑘 and 𝛽_𝑘→0, we get Ψ_𝑡𝑘𝑣^(𝑘+1)<𝜑2 (73) for all 𝑘 sufficiently large.
Now we choose sufficiently large 𝑘 satisfying all the above arguments with 𝑘+1∈𝐾 and apply Lemma 22 with 𝜑=Ψ_𝑡𝑘,𝜉=𝑣∈𝒞,𝜂=𝑣⁽𝑘₊₁₎∉𝒞,𝜑=min_{𝑣∈bd𝒞}Ψ_𝑡𝑘(𝑣)>𝜑2. (74) Then there exists 𝜁∈𝐑^2𝑛+ℓ satisfying ∇Ψ_𝑡𝑘(𝜁)=0,Ψ_𝑡𝑘(𝜁)≥𝜑𝜑>2>0, (75) which contradicts Lemma 20, and therefore the proof is complete.

4.3. Local 𝑄-Superlinear and 𝑄-Quadratic Convergence

In Section 4.2, we have shown that the sequence {𝑣^(𝑘)} is bounded, and any accumulation point of {𝑣^(𝑘)} is a solution of SOCCP (1). In this section, we prove that {𝑣^(𝑘)} is superlinearly convergent, or more strongly, quadratically convergent if ∇𝐹 is locally Lipschitzian. In order to establish the superlinear (quadratic) convergence of the algorithm, we need an assumption that every accumulation point of {∇𝐻_𝑡𝑘(𝑣^(𝑘))} is nonsingular. We first consider a sufficient condition for this assumption to hold.

Let {𝑣^(𝑘)} and {𝑡_𝑘} be the sequences generated by Algorithm 13, and let 𝑣^∗=(𝑥^∗,𝑦^∗,𝑝^∗) be any accumulation point of {𝑣^(𝑘)}. Then, by Theorem 23, 𝑣^∗ is a solution of SOCCP (1). We call the following condition nondegeneracy of a solution of the SOCCP (see also [13, 25]).

Definition 24. Let 𝑣^∗=(𝑥^∗,𝑦^∗,𝑝^∗)∈𝐑^2𝑛+ℓ be a solution of SOCCP (1) with 𝑥^∗=(𝑥^∗1,…,𝑥^∗𝑚), 𝑦^∗=(𝑦^∗1,…,𝑦^∗𝑚)∈𝐑^𝑛1×⋯×𝐑^𝑛𝑚. Then we say that 𝑣^∗ is nondegenerate if 𝑥^∗+𝑦^∗≻0, or equivalently, 𝑥^∗𝑖+𝑦^∗𝑖≻0 for all 𝑖=1,…,𝑚.

For a nondegenerate solution, we have the next lemma.

Lemma 25. Let 𝑣^∗=(𝑥^∗,𝑦^∗,𝑝^∗)∈𝐑^2𝑛+ℓ be a nondegenerate solution of SOCCP (1), and put 𝑧^𝑖=(𝑥^𝑖,𝑦^𝑖), 𝑧^∗𝑖=(𝑥^∗𝑖,𝑦^∗𝑖)∈𝐑^2𝑛𝑖 for 𝑖=1,…,𝑚. Let 𝑡∈𝐑 be a nonzero number. Then, for each 𝑖, the following holds: lim_{𝑡,𝑧𝑖→0,𝑧∗𝑖}∇_𝑥𝑖𝜙^𝑖_𝑡𝑥^𝑖,𝑦^𝑖=𝐼−𝐿_𝑥∗𝑖𝐿_𝑢⁻¹_∗𝑖,lim_{𝑡,𝑧𝑖→0,𝑧∗𝑖}∇_𝑦𝑖𝜙^𝑖_𝑡𝑥^𝑖,𝑦^𝑖=𝐼−𝐿_𝑦∗𝑖𝐿_𝑢⁻¹_∗𝑖, (76) where 𝐿_𝑢⁻¹_∗𝑖=𝑏^∗𝑖−𝑐^∗𝑖𝑤^∗𝑖⊤₂−𝑐^∗𝑖𝑤^∗𝑖₂𝑎^∗𝑖𝐼+𝑏^∗𝑖−𝑎^∗𝑖𝑤^∗𝑖₂𝑤^∗𝑖⊤₂ (77) with 𝑎^∗𝑖2∶=𝜆₁𝑤^∗𝑖+𝜆₂𝑤^∗𝑖,𝑏^∗𝑖1∶=21𝜆₁𝑤^∗𝑖+1𝜆₂𝑤^∗𝑖,𝑐^∗𝑖1∶=21𝜆₁𝑤^∗𝑖−1𝜆₂𝑤^∗𝑖. (78) Here 𝑤^∗𝑖 and 𝑢^∗𝑖 are defined by (37) with 𝑥^∗𝑖 and 𝑦^∗𝑖. We also write 𝑤^∗𝑖=((𝑤^∗𝑖)₁,(𝑤^∗𝑖)₂)∈𝐑×𝐑^𝑛𝑖−1, and set (𝑤^∗𝑖)₂∶=(𝑤^∗𝑖)₂/‖(𝑤^∗𝑖)₂‖ if (𝑤^∗𝑖)₂≠0, and otherwise, set (𝑤^∗𝑖)₂ to any vector in 𝐑^𝑛𝑖−1 satisfying ‖(𝑤^∗𝑖)₂‖=1.

Proof. Since 𝑣^∗ is a solution of SOCCP (1), it follows from [5, Propsition 4.2] that 𝑥^∗𝑖+𝑦^∗𝑖−((𝑥^∗𝑖)²+(𝑦^∗𝑖)²)^1/2=0 for all 𝑖=1,…,𝑚. Hence, from the nondegeneracy of 𝑣^∗, we have 𝑢^∗𝑖=𝑥^∗𝑖2+𝑦^∗𝑖21/2=𝑥^∗𝑖+𝑦^∗𝑖≻0. (79) By Property 1(c), this implies that 𝐿_𝑢∗𝑖 is nonsingular. In order to prove this lemma, it suffices to show lim_{𝑡,𝑧𝑖→0,𝑧∗𝑖}𝐿_𝑢⁻¹_𝑖𝑡=𝐿_𝑢⁻¹_∗𝑖. (80) Since (36) yields lim_{(𝑡,𝑧𝑖)→(0,𝑧∗𝑖)}𝑢^𝑖_𝑡=𝑢^∗𝑖, (80) follows from Property 1(c).

The following proposition gives a sufficient condition for the nonsingularity of accumulation points of {∇𝐻_𝑡𝑘(𝑣^(𝑘))}.

Proposition 26. Suppose that Assumptions (A1)–(A3) hold. Let {𝑣^(𝑘)} be a sequence generated by Algorithm 13 and let 𝑣^∗ be any accumulation point of it. Moreover, assume that 𝑣^∗ is nondegenerate and the Jacobian matrix ∇𝐹(𝑣^∗)^⊤ has the Cartesian mixed Jordan 𝑃-property, that is, rank∇_𝑝𝐹(𝑣^∗)=ℓ and ∇𝐹𝑣^∗⊤𝜉,𝜂,𝜑=0,𝜉,𝜂≠0,𝜑∈𝐑^ℓ𝜉=𝜉¹,…,𝜉^𝑚,𝜂=𝜂¹,…,𝜂^𝑚∈𝐑^𝑛1×⋯×𝐑^𝑛𝑚⟹𝑡ℎ𝑒𝑟𝑒𝑒𝑥𝑖𝑠𝑡𝑠𝑎𝑛𝑖𝑛𝑑𝑒𝑥𝜈𝑠𝑢𝑐ℎ𝑡ℎ𝑎𝑡𝜉^𝜈∘𝜂^𝜈≻0. (81)Then, every accumulation point of {∇𝐻_𝑡𝑘(𝑣^(𝑘))} is nonsingular.

Proof. By Theorem 23, the sequence {𝑣^(𝑘)} is bounded and has at least one accumulation point 𝑣^∗. Hence, we may assume that {𝑣^(𝑘)} converges to 𝑣^∗ without loss of generality. It follows from Lemma 25 and 𝑡_𝑘→0 that any accumulation point of {∇𝐻_𝑡𝑘(𝑣^(𝑘))}, say 𝐽₀, is given in the following form: 𝐽₀∶=diag𝐼−𝐿_𝑥∗𝑖𝐿_𝑢⁻¹_∗𝑖^𝑚_𝑖=1∇_𝑥𝐹𝑣^∗diag𝐼−𝐿_𝑦∗𝑖𝐿_𝑢⁻¹_∗𝑖^𝑚_𝑖=1∇_𝑦𝐹𝑣^∗𝑂∇_𝑝𝐹𝑣^∗. (82) In order to prove that 𝐽₀ is nonsingular, suppose that 𝐽^⊤₀(𝜉,𝜂,𝜑)=0, where (𝜉,𝜂,𝜑)∈𝐑^2𝑛+ℓ. We will show that (𝜉,𝜂,𝜑)=0. It follows from (82) that ∇𝐹𝑣^∗⊤𝜉,𝜂,𝜑=0,𝐼−𝐿_𝑢⁻¹_∗𝑖𝐿_𝑥∗𝑖𝜉^𝑖+𝐼−𝐿_𝑢⁻¹_∗𝑖𝐿_𝑦∗𝑖𝜂^𝑖=0,𝑖=1,…,𝑚, (83)(84) where 𝜉=(𝜉¹,…,𝜉^𝑚), 𝜂=(𝜂¹,…,𝜂^𝑚)∈𝐑^𝑛1×⋯×𝐑^𝑛𝑚. Multiplying both sides of the above equations by 𝐿_𝑢∗𝑖 from the left-hand side, we get 𝐿_𝑢∗𝑖−𝐿_𝑥∗𝑖𝜉^𝑖+𝐿_𝑢∗𝑖−𝐿_𝑦∗𝑖𝜂^𝑖=𝐿_{𝑢∗𝑖−𝑥∗𝑖}𝜉^𝑖+𝐿_{𝑢∗𝑖−𝑦∗𝑖}𝜂^𝑖=𝐿_𝑦∗𝑖𝜉^𝑖+𝐿_𝑥∗𝑖𝜂^𝑖=0 (85) for all 𝑖=1,…,𝑚, where the second equality uses the fact 𝑢^∗𝑖=𝑥^∗𝑖+𝑦^∗𝑖 (see (79)). Suppose on the contrary that (𝜉,𝜂)≠0. Then from (83) and the assumption (81), we have that 𝜉^𝜈∘𝜂^𝜈≻0 (86) for some 𝜈∈{1,…,𝑚}. Since 𝑥^∗𝜈, 𝑦^∗𝜈≽0, by Property 1(b), we have 𝐿_𝑥∗𝜈≽𝑂 and 𝐿_𝑦∗𝜈≽𝑂. By using Property 1(a), (85) can be rewritten equivalently as 𝐿_𝑦∗𝜈𝜉^𝜈+𝑥^∗𝜈∘𝜂^𝜈=0or𝑦^∗𝜈∘𝜉^𝜈+𝐿_𝑥∗𝜈𝜂^𝜈=0. (87) Multiplying both sides of the first equation in (87) by (𝜉^𝜈)^⊤ from the left, we have 𝜉^𝜈⊤𝐿_𝑦∗𝜈𝜉^𝜈+𝜉^𝜈,𝑥^∗𝜈∘𝜂^𝜈=𝜉^𝜈⊤𝐿_𝑦∗𝜈𝜉^𝜈+𝑥^∗𝜈,𝜉^𝜈∘𝜂^𝜈=0. (88) Since 𝐿_𝑦∗𝜈 is positive semidefinite, we have ⟨𝑥^∗𝜈,𝜉^𝜈∘𝜂^𝜈⟩≤0. Similarly, multiplying both sides of the second equation in (87) by (𝜂^𝜈)^⊤ from the left, we have ⟨𝑦^∗𝜈,𝜉^𝜈∘𝜂^𝜈⟩≤0. Adding these two inequalities yields 𝑥^∗𝜈+𝑦^∗𝜈,𝜉^𝜈∘𝜂^𝜈≤0. (89) On the other hand, by the nondegeneracy of 𝑣^∗, we have 𝑥^∗𝜈+𝑦^∗𝜈≻0. This together with (86) yields 𝑥^∗𝜈+𝑦^∗𝜈,𝜉^𝜈∘𝜂^𝜈>0, (90) which contradicts (89), and hence, we must have (𝜉,𝜂)=0. Then, since the matrix ∇_𝑝𝐹(𝑣^∗)^⊤∈𝐑^{(𝑛+ℓ)×ℓ} has full column rank, we also have from (83) that 𝜑=0. Therefore, 𝐽₀ is nonsingular.

Next, we show the local convergence properties of the sequence {𝑣^(𝑘)} generated by Algorithm 13. The following lemma plays a key role in proving such properties.

Lemma 27. Suppose that Assumptions (A1)–(A3) hold. Let {𝑣^(𝑘)} be a sequence generated by Algorithm 13 and let 𝑣^∗ be any accumulation point of it. In addition, assume that every accumulation point of {∇𝐻_𝑡𝑘(𝑣^(𝑘))} is nonsingular. Then, for 𝑣^(𝑘) sufficiently close to 𝑣^∗, 𝑣^(𝑘)+𝑑^(𝑘)−𝑣^∗=𝑜𝑣^(𝑘)−𝑣^∗ (91) holds, where 𝑑^(𝑘) is defined by −∇𝐻_𝑡𝑘(𝑣^(𝑘))^−⊤𝐻_𝑡𝑘(𝑣^(𝑘)). Moreover, if ∇𝐹 is locally Lipschitzian and 𝑟≥2, then 𝑣^(𝑘)+𝑑^(𝑘)−𝑣^∗=𝑂𝑣^(𝑘)−𝑣^∗2 (92) holds.

Proof. From Theorem 23, {𝑣^(𝑘)} is bounded, and hence, there exists at least one accumulation point, and any such point 𝑣^∗ satisfies 𝐻_FB(𝑣^∗)=0. Since every accumulation point of {∇𝐻_𝑡𝑘(𝑣^(𝑘))} is nonsingular, we have, from Assumption (A3), that there exists a constant 𝑐₁>0 such that ∇𝐻_𝑡𝑘𝑣^(𝑘)−⊤≤𝑐₁ (93) for all 𝑘. Let 𝑉_𝑘∈𝜕𝐻_FB(𝑣^(𝑘)) such that 𝑉_𝑘∈argmin_{𝑉∈𝜕𝐻FB𝑣(𝑘)}∇𝐻_𝑡𝑘𝑣^(𝑘)−𝑉. (94) Note that 𝑉_𝑘 exists for all 𝑘, because 𝜕𝐻_FB(𝑣^(𝑘)) is compact [17, page 70]. We have from (93) and 𝐻_FB(𝑣^∗)=0 that 𝑣^(𝑘)+𝑑^(𝑘)−𝑣^∗=𝑣^(𝑘)−∇𝐻_𝑡𝑘𝑣^(𝑘)−⊤𝐻_𝑡𝑘𝑣^(𝑘)−𝑣^∗≤∇𝐻_𝑡𝑘𝑣^(𝑘)−⊤∇𝐻_𝑡𝑘𝑣^(𝑘)⊤𝑣^(𝑘)−𝑣^∗−𝐻_𝑡𝑘𝑣^(𝑘)≤𝑐₁∇𝐻_𝑡𝑘𝑣^(𝑘)−𝑉_𝑘^⊤𝑣^(𝑘)−𝑣^∗+𝑉^⊤_𝑘𝑣^(𝑘)−𝑣^∗−𝐻^′_FB𝑣^∗;𝑣^(𝑘)−𝑣^∗+𝐻^′_FB𝑣^∗;𝑣^(𝑘)−𝑣^∗−𝐻_FB𝑣^(𝑘)−𝐻_FB𝑣^∗+𝐻_FB𝑣^(𝑘)−𝐻_𝑡𝑘𝑣^(𝑘). (95) It follows from (54) that, for 𝑣^(𝑘) sufficiently close to 𝑣^∗, ∇𝐻_𝑡𝑘𝑣^(𝑘)−𝑉_𝑘^⊤𝑣^(𝑘)−𝑣^∗≤∇𝐻_𝑡𝑘𝑣^(𝑘)−𝑉_𝑘𝑣^(𝑘)−𝑣^∗=𝑂𝑣^(𝑘)−𝑣^∗2. (96) From (35), the inequality 𝑡_𝑘≤𝜅𝛿^𝑟_𝑘=𝜅‖𝐻_FB(𝑣^(𝑘))‖^𝑟 by Step 3 of Algorithm 13, and the local Lipschitz continuity of 𝐻_FB, we get 𝐻_FB𝑣^(𝑘)−𝐻_𝑡𝑘𝑣^(𝑘)≤√2𝑚𝑡_𝑘≤√2𝑚𝜅𝐻_FB𝑣^(𝑘)𝑟=√2𝑚𝜅𝐻_FB𝑣^(𝑘)−𝐻_FB𝑣^∗𝑟=𝑂𝑣^(𝑘)−𝑣^∗𝑟. (97) Since, by Proposition 11, 𝐻_FB is semismooth, we have from (15) and (17) that 𝑉^⊤_𝑘𝑣^(𝑘)−𝑣^∗−𝐻^′_FB𝑣^∗;𝑣^(𝑘)−𝑣^∗+𝐻^′_FB𝑣^∗;𝑣^(𝑘)−𝑣^∗−𝐻_FB𝑣^(𝑘)−𝐻_FB𝑣^∗=𝑜𝑣^(𝑘)−𝑣^∗. (98) Therefore, from (95)–(97) and 𝑟>1, we obtain (91). Moreover, if ∇𝐹 is locally Lipschitzian, then, by Proposition 11, 𝐻_FB is strongly semismooth, and hence, we have 𝑉^⊤_𝑘𝑣^(𝑘)−𝑣^∗−𝐻^′_FB𝑣^∗;𝑣^(𝑘)−𝑣^∗+𝐻^′_FB𝑣^∗;𝑣^(𝑘)−𝑣^∗−𝐻_FB𝑣^(𝑘)−𝐻_FB𝑣^∗=𝑂𝑣^(𝑘)−𝑣^∗2. (99) Therefore, from (95)–(97) and 𝑟≥2, we obtain (92).

Using Lemma 27, we obtain the local 𝑄-superlinear (𝑄-quadratic) convergence result.

Theorem 28. Suppose that Assumptions (A1)–(A3) hold. Let {𝑣^(𝑘)} be a sequence generated by Algorithm 13. If every accumulation point of {∇𝐻_𝑡𝑘(𝑣^(𝑘))} is nonsingular, then the following statements hold. (a)For all 𝑘 sufficiently large, ‖𝐻_FB(𝑣̂^(𝑗)+𝑑^(𝑗))‖≤𝛽_𝑘 is satisfied at 𝑗=0 in Step 2.2 of Algorithm 13. Moreover, for all 𝑘 sufficiently large, 𝑣^(𝑘+1)=𝑣^(𝑘)+𝑑^(𝑘) holds, where 𝑑^(𝑘)=−∇𝐻_𝑡𝑘(𝑣^(𝑘))^−⊤𝐻_𝑡𝑘(𝑣^(𝑘)).(b)The whole sequence {𝑣^(𝑘)} converges 𝑄-superlinearly to a solution 𝑣^∗ of SOCCP (1). Moreover, if ∇𝐹 is locally Lipschitz continuous and 𝑟≥2, then the sequence {𝑣^(𝑘)} converges 𝑄-quadratically.

Proof. Since (b) is directly obtained from (a) and Lemma 27, it suffices to prove (a). Namely, we prove that ‖𝐻_𝑡𝑘(𝑣^(𝑘)+𝑑^(𝑘))‖≤𝛽_𝑘=𝛽₀𝜂^𝑘 for all 𝑘 sufficiently large. We have from (93) that 𝑑^(𝑘)=−∇𝐻_𝑡𝑘𝑣^(𝑘)−⊤𝐻_𝑡𝑘𝑣^(𝑘)≤𝑐₁𝐻_𝑡𝑘𝑣^(𝑘), (100) and hence, it follows from the boundedness of {𝑣^(𝑘)} and the continuity of 𝐻_𝑡𝑘 that {𝑑^(𝑘)} is bounded. Let 𝑣^∗ be any accumulation point of 𝑣^(𝑘), and let 𝑣^(𝑘) be sufficiently close to 𝑣^∗. By Theorem 23, 𝑣^∗ is a solution of SOCCP (1), and thus, 𝐻_FB(𝑣^∗)=0. From the local Lipschitz continuity of 𝐻_FB, we may assume that ‖𝐻_FB(𝑣^(𝑘))‖ is small enough. By Lemma 27 and (100), there exists a constant 𝑐₂∈(0,1) such that 𝑣^(𝑘)−𝑣^∗≤−𝑑^(𝑘)+𝑣^(𝑘)+𝑑^(𝑘)−𝑣^∗≤𝑐₁𝐻_𝑡𝑘𝑣^(𝑘)+𝑜𝑣^(𝑘)−𝑣^∗≤𝑐₁𝐻_𝑡𝑘𝑣^(𝑘)+𝑐₂𝑣^(𝑘)−𝑣^∗. (101) Therefore, we have from (35) with 𝑡_𝑘≤𝜅‖𝐻_FB(𝑣^(𝑘))‖^𝑟≤𝜅‖𝐻_FB(𝑣^(𝑘))‖ that 𝑣^(𝑘)−𝑣^∗≤𝑐₁1−𝑐₂𝐻_𝑡𝑘𝑣^(𝑘)≤𝑐₁1−𝑐₂𝐻_𝑡𝑘𝑣^(𝑘)−𝐻_FB𝑣^(𝑘)+𝐻_FB𝑣^(𝑘)≤𝑐₁1−𝑐₂√2𝑚𝑡_𝑘+𝐻_FB𝑣^(𝑘)=𝑂𝐻_FB𝑣^(𝑘). (102) This together with Lemma 27 and the local Lipschitz continuity of 𝐻_FB yields that 𝐻_FB𝑣^(𝑘)+𝑑^(𝑘)=𝐻_FB𝑣^(𝑘)+𝑑^(𝑘)−𝐻_FB𝑣^∗=𝑂𝑣^(𝑘)+𝑑^(𝑘)−𝑣^∗=𝑜𝑣^(𝑘)−𝑣^∗=𝑜𝐻_FB𝑣^(𝑘). (103) Therefore, it follows again from (35) with 𝑡_𝑘≤𝜅‖𝐻_FB(𝑣^(𝑘))‖^𝑟 that 𝐻_𝑡𝑘𝑣^(𝑘)+𝑑^(𝑘)≤𝐻_𝑡𝑘𝑣^(𝑘)+𝑑^(𝑘)−𝐻_FB𝑣^(𝑘)+𝑑^(𝑘)+𝐻_FB𝑣^(𝑘)+𝑑^(𝑘)≤√2𝑚𝑡_𝑘+𝐻_FB𝑣^(𝑘)+𝑑^(𝑘)=𝑜𝐻_FB𝑣^(𝑘). (104) From (35), the choices of 𝑡_𝑘 and 𝛽_𝑘 in Step 3 of Algorithm 13, and 0<𝜂≤𝜂<1, we get 𝐻_FB𝑣^(𝑘)≤𝐻_FB𝑣^(𝑘)−𝐻_𝑡𝑘−1𝑣^(𝑘)+𝐻_𝑡𝑘−1𝑣^(𝑘)≤√2𝑚𝑡_𝑘−1+𝛽_𝑘−1≤√2𝑚𝑡₀𝜂^𝑘−1+𝛽₀𝜂^𝑘−1=𝑐₃𝜂^𝑘, (105) where 𝑐₃=𝜂⁻¹(√2𝑚𝑡₀+𝛽₀), and hence, (104) yields ‖𝐻_𝑡𝑘(𝑣^(𝑘)+𝑑^(𝑘))‖=𝑜(𝜂^𝑘). This implies that ‖𝐻_𝑡𝑘(𝑣^(𝑘)+𝑑^(𝑘))‖≤𝛽_𝑘=𝛽₀𝜂^𝑘. Taking into account 𝑣̂⁽⁰⁾=𝑣^(𝑘) and 𝑑⁽⁰⁾=𝑑^(𝑘), we have ‖𝐻_FB(𝑣̂⁽⁰⁾+𝑑⁽⁰⁾)‖≤𝛽_𝑘 and 𝑣^(𝑘+1)=𝑣^(𝑘)+𝑑^(𝑘). Since, by Lemma 27, 𝑣^(𝑘+1) remains in the neighborhood of 𝑣^∗, the desired results are obtained.

5. Numerical Experiments

In this section, we show some numerical results for Algorithm 13. The program was coded in MATLAB 7, and computations were carried out on a machine with Intel Core i7-3770 K CPU (3.50 GHz×2) and 8.0 GB RAM. We set the parameters 𝜂=𝜂=0.1, 𝜌=0.66, 𝜎=0.1, 𝑟=2, 𝜅=1, 𝜅=0.2, and 𝛽₀=2. We also set the function 𝑡_𝑖 as follows (see [22] for details): 𝑡_𝑖𝑧^𝑖𝑧,𝛿∶=^𝑖𝛿21−𝛿²if𝑧^𝑖≠(0,0)∈𝐑²,𝛿<1,10¹⁰otherwise (106) for 𝑛_𝑖=1, and 𝑡_𝑖𝑧^𝑖𝜆,𝛿∶=¹𝑤^𝑖𝛿22𝑤^𝑖₁−𝜆₁𝑤^𝑖𝛿²if𝑧^𝑖∈𝒵^𝑖₁,𝛿<2𝑤^𝑖₁𝜆₁𝑤^𝑖,𝛿𝑤^𝑖₁22(2−𝛿)if𝑧^𝑖∈𝒵^𝑖₂,𝛿<2,10¹⁰otherwise (107) for 𝑛_𝑖≥2. For all problems, we randomly chose the initial point (𝑥⁽⁰⁾,𝑦⁽⁰⁾,𝑝⁽⁰⁾)∈𝐑^2𝑛+ℓ whose components were distributed on the interval [0,1], by using rand command of MATLAB. The stopping criterion in Step 1 is relaxed to 𝐻_FB𝑣^(𝑘)≤10⁻⁸. (108)

We first solve the following second-order cone programming (SOCP) problem: minimize𝑐^⊤𝑥subjectto𝐴𝑥+𝑏=0,𝑥∈𝒦, (109) which is reformulated as SOCCP (1) with 𝐹𝑥,𝑦,𝑝∶=𝑂−𝐼𝐴^⊤𝑥𝑦𝑝+𝑐𝐴𝑂𝑂𝑏 (110) equivalently. We generate one hundred test problems randomly such that there exist primal and dual strictly feasible solutions. Specifically, we first choose matrix 𝐴∈𝐑^ℓ×𝑛 whose components are distributed on [−100,100], vectors 𝑥,𝑦∈int𝒦 and 𝑝∈𝐑^ℓ randomly, and then set 𝑏∶=−𝐴𝑥 and 𝑐∶=−𝐴^⊤𝑝+𝑦. Here, each component of (𝑥^𝑖)₂ is distributed on [−100,100] and (𝑥^𝑖)₁ is set to (𝑥^𝑖)₁∶=‖(𝑥^𝑖)₂‖+𝛼, where 𝛼∈(0,100] is chosen randomly, and 𝑦 is also generated similarly, while each component of 𝑝 is distributed on [0,1]. In order to compare our method with another method, we solve SOCP (109) by SDPT3 [26, 27], which is the software of interior point methods for solving semidefinite, second-order cone, and linear programming problems. We use SDPT3 with the default parameter and option settings. The obtained results are shown in Table 1, in which “Iter” and “CPU” denote the average values of the number of iterations and the CPU time in seconds, respectively. In particular, the value of “Iter” in “our method” denotes the number of times that the Newton equations (49) have been solved. In the column of 𝒦, (𝒦⁵)³ denotes 𝒦⁵×𝒦⁵×𝒦⁵, for example. Since SDPT3 failed to solve some test problems when 𝒦=(𝒦⁵)³×(𝒦²)²×𝐑₊, the average values in this case were taken over the successful trials only. We see from Table 1 that our method is superior to or at least comparable with SDPT3 from the viewpoint of the number of iterations. On the other hand, from the viewpoint of CPU time, our method is also superior to SDPT3 for small-scale problems. However, SDPT3 outperforms our method for middle- or large-scale problems. We believe that this is because SDPT3 is coded to reduce the computational costs by means of some fundamental techniques on matrix computation and so forth. For further development of our method, we will need more appropriate tuning of our code. However, it is not the purpose of this paper.

Table 1 

Numerical comparison of our method with SDPT3.

In order to confirm the local behaviors of the sequence generated by Algorithm 13, we list the value of ‖𝐻_FB(𝑣̂^(𝑗))‖ at each outer iteration 𝑘 in Table 2. In addition, to investigate how the parameter 𝑟 affects the rate of convergence, we performed the algorithm with 𝑟=1,1.5,2. We also investigate the relation between the choices of 𝑟 and the behavior of {𝑡_𝑘}; we list the behaviors of {𝑡_𝑘}. We chose one of the above test problems in the case 𝒦=𝒦⁵⁰⁰×𝒦²⁰⁰×(𝒦¹⁰⁰)³. We note that 2.66e−09 means 2.66×10⁻⁹, for example. We see from Table 2 that the sequence generated by Algorithm 13 seems to converge 𝑄-quadratically and the parameter 𝑟 does not affect the convergence of the sequence. On the other hand, we find that the choices of 𝑟 affect the behavior of {𝑡_𝑘}.

Table 2 

Numerical behaviors of ‖𝐻_FB(𝑣̂^(𝑗))‖.

The next experiment is an application of Algorithm 13 to the robust Nash equilibrium problem in the game theory. The robust Nash equilibrium [2, 3, 28, 29] is a new solution concept for noncooperative games with uncertain information. In this model, it is assumed that each player’s cost (pay-off) function and/or the opponents’ strategies are uncertain, but they belong to some uncertainty sets and each player chooses his strategy by taking the worst possible case into consideration. In other words, each player makes decision according to the robust optimization policy. In this experiment, we focus on the following 2-person robust Nash game with quadratic cost functions: Player1:minimize_𝑥1max_{𝛿𝑥2∈𝐷2}12𝑥^1⊤𝐴₁₁𝑥¹+𝑥^1⊤𝐴₁₂𝑥²+𝛿𝑥²subjectto𝑥¹≥0,𝑒^⊤̂𝑥¹=1;Player2:minimize_𝑥2max_{𝛿𝑥1∈𝐷1}12𝑥^2⊤𝐴₂₂𝑥²+𝑥^2⊤𝐴₂₁𝑥¹+𝛿𝑥¹subjectto𝑥²≥0,𝑒^⊤̂𝑥²=1, (111)(112) where 𝐴_𝑖𝑗∈𝐑^{𝑚𝑖×𝑚𝑗} for (𝑖,𝑗)∈{1,2}×{1,2} are given matrices, 𝑒̂ is the vector of ones of appropriate dimension, and 𝑥¹∈𝐑^𝑚1 and 𝑥²∈𝐑^𝑚2 denote the mixed strategies for Players 1 and 2, respectively. Moreover, 𝛿𝑥¹ and 𝛿𝑥² mean the estimation error or noise, and each player knows that they belong to the uncertainty sets 𝐷₁ and 𝐷₂, respectively. Under this situation, the tuple (𝑥¹,𝑥²) is called a robust Nash equilibrium when 𝑥¹ and 𝑥² solve (111) and (112) simultaneously. In this experiment, we set 𝐴₁₁∶=30−12−3−513151923,𝐴₁₂,𝐴∶=−113−10−8−15−23−1−6₂₁∶=3−111006−166,𝐴₂₂,𝐷∶=319612464829₁∶=𝛿𝑥₁∣𝛿𝑥₁≤𝜌₁,𝑒^⊤̂𝛿𝑥₁,𝐷=0₂∶=𝛿𝑥₂∣𝛿𝑥₂≤𝜌₂,𝑒^⊤̂𝛿𝑥₂=0 (113) and change the values of 𝜌₁ and 𝜌₂ variously. Since 𝐷₁ and 𝐷₂ are defined by means of Euclidean norm, the robust Nash equilibrium problem can be reformulated as an SOCCP equivalently (the reformulated SOCCP is explicitly written in Section  5.1.1 of [3]. We thus omit the details here). Here, we emphasize that the reformulated SOCCP cannot be expressed as any SOCP, and hence existing software such as SDPT3 cannot be applied. Moreover, if the reformulated SOCCP is rewritten of the form (1), then it satisfies neither (4) nor (6). The obtained results are summarized in Table 3, in which 𝑥¹ and 𝑥² denote the obtained robust Nash equilibria for various choices of uncertainty radiuses 𝜌₁ and 𝜌₂. For all problems, we could calculate the robust Nash equilibria correctly. Moreover, as is discussed in the existing papers, we can observe that the robust Nash equilibria move smoothly as the values of 𝜌₁ and 𝜌₂ change gradually.

Table 3 

Robust Nash equilibria with various choices of uncertainty radiuses.

6. Conclusion

In this paper, we have proposed a smoothing Newton method with appropriate parameter control based on the Fischer-Burmeister function for solving the SOCCP. We have shown its global and 𝑄-quadratic convergence properties under some assumptions. In addition, we have considered some sufficient conditions for the assumptions. In numerical experiments, we have confirmed the effectiveness of the proposed methods.

Acknowledgments

The authors would like to thank the academic editor Prof. Jein-Shan Chen and the anonymous reviewers who helped them improve the original paper. The first author is supported in part by the Grant-in-Aid for Scientific Research (C) 25330030 and Young Scientists (B) 25870239 of Japan Society for the Promotion of Science. The third author is supported in part by the Grant-in-Aid for Young Scientists (B) 22760064 of Japan Society for the Promotion of Science.