An inexact smoothing method for SOCCPs based on a one-parametric class of smoothing function☆
Introduction
Let be the second-order cone (SOC) in , also called the Lorentz cone or ice-cream cone, defined bywhere denotes the Euclidean norm. By definition, is the set of nonnegative real .
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:where denotes the Euclidean inner product, is a continuously differentiable function, and with and . Unless otherwise specified, in the following analysis we assume that and . 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 is called an SOC complementarity function associated with the second-order cone if for any ,Then, a function is called a smoothing function for SOCCP if
- (a)
is non-differentiable on ,
- (b)
h is continuously differentiable at any , and
- (c)
is an SOC complementarity function.
In this paper, we consider a one-parametric class of vector-valued functions defined bywhere is any but fixed parameter in . It is not hard to see that when is reduced to the Fischer–Burmeister smoothing function and when is reduced to the function 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 . We will show that the function has favorable properties, which include the differentiability, the strong semismoothness and the characterization of the coerciveness.
Based on the smoothing function 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”. and 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 and are defined in usual way. Let denote the interior of . () means that ().
Section snippets
Preliminaries
In this section, we recall some background materials and preliminaries results that will be used in the subsequent sections.
For any , Jordan product associated with is defined bywith being its unit element. Moreover, if , then there exists a unique vector in , denote by , such that . We defineIt is easy to verify thatMoreover, is symmetric positive definite if and only if .
Properties of the function
This section is devoted to investigating the favorable properties of , which include the differentiability, the strong semismoothness and the characterization of the coerciveness. Theorem 3.1 Let be given in (3) and . Then For any , is continuously differentiable at any with its Jacobianwhere
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 andwhere Theorem 4.1 Let be defined by (12) and . Then is semismooth everywhere and continuously differentiable at any with its Jacobian
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 . All programs are written in Matlab code, numerical test in PC, CPU Main Frequency GHz GB run circumstance Matlab 7.1. The parameters used in algorithm were as follows:
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 . 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).