Elsevier

Applied Mathematics and Computation

Volume 241, 15 August 2014, Pages 167-182
Applied Mathematics and Computation

An inexact smoothing method for SOCCPs based on a one-parametric class of smoothing function

https://doi.org/10.1016/j.amc.2014.05.007Get rights and content

Abstract

In this paper, we introduce a one-parametric class of smoothing functions which contains the Fischer–Burmeister smoothing function as special case. Based on this class of smoothing functions, an inexact smoothing method for solving second-order cone complementarity problems (SOCCPs) is proposed. In each iteration the corresponding linear system is solved only approximately. Under mild assumptions, it is proved that the proposed method has global convergence and local superlinear convergence properties. Preliminary numerical results indicate that the method is effective for large-scale problems.

Introduction

Let Kn(n1) be the second-order cone (SOC) in Rn, also called the Lorentz cone or ice-cream cone, defined byKn={(x1,x2)R×Rn-1|x2x1},where · denotes the Euclidean norm. By definition, K1 is the set of nonnegative real R+.

In this paper, we are interested in complementarity problems involving the second-order cone in its constraints. In general, the second-order cone complementarity problem has the following form:Find anxK,such thatF(x)Kandx,F(x)=0,where ·,· denotes the Euclidean inner product, F:RnRn is a continuously differentiable function, and K=Kn1××Knp with p,n1,,np1 and n1++np=n. Unless otherwise specified, in the following analysis we assume that p=1 and n1=n. This, however, does not lose any generality because our analysis can be easily extended to the general case.

Second-order cone complementarity problems have wide range of applications and, in particular, includes a large class of quadratically constrained problems as special cases [1], [2], [3]. Recently, there have been various methods proposed for solving SOCP and SOCCP, which include interior-point methods [4], [5], [6], [7], [8], smoothing Newton methods [9], [10], [11], [12], [13], [14], smoothing regularization methods [15]. Among others, smoothing methods are all based on a smoothing function, which depends on an SOC complementarity function. A function ϕ:Rn×RnRn is called an SOC complementarity function associated with the second-order cone Kn if for any (x,y)Rn×Rn,ϕ(x,y)=0xKn,yKn,x,y=0.Then, a function h:R×Rn×RnRn is called a smoothing function for SOCCP if

  • (a)

    h(0,·,·) is non-differentiable on Rn×Rn,

  • (b)

    h is continuously differentiable at any (μ,x,y)R++×Rn×Rn, and

  • (c)

    h(0,x,y) is an SOC complementarity function.

A popular choice of smoothing function is the famous Fischer–Burmeister (FB) smoothing function, i.e., ϕFB:R×Rn×RnRn associated with second-order cone defined byϕFB(μ,x,y)=x+y-x2+y2+2μ2e,where x2=xx denoting the Jordan product between x and itself, x12 being a vector such that x122=x and xKn,x+y to mean the usual componentwise addition of vectors. In addition, by smoothing the symmetric perturbed ϕFB(0,x,y), we can obtain the vector-valued function ϕ:R×Rn×RnRn defined asϕ(μ,x,y)=(1+μ)(x+y)-(x+μy)2+(μx+y)2+2μ2e,which was firstly used by Huang [16] for P0 nonlinear complementarity problems. It is easy to show this function is a smoothing function for SOCCP [17].

In this paper, we consider a one-parametric class of vector-valued functions ϕτ:R×Rn×RnRn defined byϕτ(μ,x,y)=(1+μ-μτ)(x+y)-(1+μ2-2μ2τ+μ2τ2)(x2+y2)+4μ(1-τ)xy+2μ2e,where τ is any but fixed parameter in τ[0,1]. It is not hard to see that when τ=1,ϕτ is reduced to the Fischer–Burmeister smoothing function and when τ=0,ϕτ is reduced to the function ϕ defined by (2). Thus, the class of functions defined by (3) is a generalized class of smoothing functions which contains the Fischer–Burmeister smoothing function and the function ϕ. We will show that the function ϕτ has favorable properties, which include the differentiability, the strong semismoothness and the characterization of the coerciveness.

Based on the smoothing function ϕτ defined by (3), we are interested in smoothing methods for solving SOCCPs in this paper. In most smoothing methods, each iteration needs to find a solution of system of linear equations exactly, when solving a large-scale problems, which may be expensive from a computational point of view. These motivate us to proposed an inexact version of the smoothing methods. In our inexact smoothing method, the system of linear equations is solved only up to a certain degree of accuracy. The accuracy level of approximate solution is controlled by the forcing term, which links the norm of residual vector to the norm of mapping at the current iterate. We will show that the proposed method is globally and locally superlinear convergent under suitable assumptions. Numerical experiments indicate that our method is effective for large-scale problems.

The paper is organized as follows: In the next section, we introduce preliminaries which will be a basic tool for analysis. In Section 3, we study a few properties of vector-valued function (3). An inexact smoothing method for solving SOCCP is proposed and convergence results are analyzed in Section 4. In Section 5, numerical experiments are presented. Conclusions are given in Section 6.

The following notations will be used throughout this paper. means “is defined as”. R+ and R++ denote the nonnegative and positive reals. All vectors are column vectors, the superscript T denotes transpose. The symbol · stands for the 2-norm and I denotes an identity matrix of suitable dimension. Landau symbols o(·) and O(·) are defined in usual way. Let intK denote the interior of K. xy (xy) means that x-yK (x-yintK).

Section snippets

Preliminaries

In this section, we recall some background materials and preliminaries results that will be used in the subsequent sections.

For any x=(x1,x2),y=(y1,y2)R×Rn-1, Jordan product associated with Kn is defined byxy=xTyy1x2+x1y2,with e¯=(1,0,,0)Rn being its unit element. Moreover, if x0, then there exists a unique vector in Kn, denote by x12, such that x122=x12x12=x. We defineLx=x1x2Tx2x1I.It is easy to verify thatLxy=xy,yRn.Moreover, Lx is symmetric positive definite if and only if xintKn.

Properties of the function ϕτ

This section is devoted to investigating the favorable properties of ϕτ, which include the differentiability, the strong semismoothness and the characterization of the coerciveness.

Theorem 3.1

Let ϕτ be given in (3) and τ[0,1]. Then

  • (a)

    For any z=(μ,x,y)R×Rn×Rn,ϕτ(μ,x,y)=x+μ(1-τ)y+y+μ(1-τ)x-x+μ(1-τ)y2+y+μ(1-τ)x2+2μ2e.

  • (b)

    ϕτ is continuously differentiable at any z=(μ,x,y)R++×Rn×Rn with its Jacobianϕτ(μ,x,y)=(1-τ)(x+y)-Lω-1(1-τ)(Lω1y+Lω2x)+2μe(1+μ-μτ)I-Lω-1Lω1+μ(1-τ)Lω2(1+μ-μτ)I-Lω-1Lω2+μ(1-τ)Lω1,whereω=x+μ(1-τ)y2

Algorithm and convergence analysis

In this section, we present an inexact smoothing method to solve second-order cone complementarity problems based on the function (3). Under some assumptions, it is proved that the proposed method has global and local superlinear convergence properties.

Let z(μ,x,y)R×Rn×Rn andHτ(z)=μΦτ(μ,x,y),whereΦτ(μ,x,y)=y-F(x)ϕτ(μ,x,y).

Theorem 4.1

Let Hτ:R×Rn×RnR×Rn×Rn be defined by (12) and τ[0,1]. Then

  • (a)

    Hτ(z) is semismooth everywhere and continuously differentiable at any z=(μ,x,y)R++×Rn×Rn with its JacobianHτ(z)=1

Numerical results

In this section we use the inexact smoothing method to solve linear and nonlinear SOCCPs, and compare the numerical performance of the method corresponding to different τ[0,1]. All programs are written in Matlab code, numerical test in PC, CPU Main Frequency 3.20 GHz 0.99 GB run circumstance Matlab 7.1. The parameters used in algorithm were as follows:σ=0.0006,γ=0.5,δ=0.5,μ0=0.002.

The starting points of algorithm can be chosen randomly, which are from a uniform distribution in the interval

Conclusions

In this paper, we introduced a one-parametric class of smoothing function which contains the famous Fischer–Burmeister smoothing function as special cases. Base on this generalized class of smoothing function, we proposed an inexact smoothing method for solving SOCCP. We proved that the method is globally and local superlinearly convergent under some suitable assumptions. In addition, numerical results are reported, which provides some helpful advices on the choice of τ. It is worth

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    This work is supported by National Natural Science Foundations of China (No. 10971162) and Anhui Province Education Department, Natural Science Research Item (No. KJ2013A235).

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