A new one-step smoothing newton method for the second-order cone complementarity problem

Abstract

A new one-step smoothing Newton method for the second-order cone complementarity problem Liang Fang a * † and Congying Han b Communicated by J. CashIn this paper, we present a new one-step smoothing Newton method for solving the second-order cone complementarity problem (SOCCP). Based on a new smoothing function, the SOCCP is approximated by a family of parameterized smooth equations. At each iteration, the proposed algorithm only need to solve one system of linear equations and perform only one Armijo-type line search. The algorithm is proved to be convergent globally and superlinearly without requiring strict complementarity at the SOCCP solution. Moreover, the algorithm has locally quadratic convergence under mild conditions. Numerical experiments demonstrate the feasibility and efficiency of the new algorithm. Copyright © 2010 John Wiley & Sons, Ltd.Keywords: second-order cone complementarity; smoothing Newton method; Jordan product; coerciveness; global convergence IntroductionConsider the following second-order cone complementarity problem (SOCCP) of finding a vector x ∈ R n such thatwhere ·, · represents the Euclidean inner product, f : R n → R n is a continuously differentiable P 0 -function, Q is the Cartesian product of second-order cones (SOCs), i.e.with n 1 +n 2 +···+n m = n, and the n i -dimensional SOC Q n i is defined bywherex = (x 2 ,x 3 , ···,x n ) ∈ R n i −1 . For simplicity, the problem defined by (1), (2) and (3) is written by P 0 −SOCCP(f ). It is well known that Q n i is a closed, convex, self-dual cone with nonempty interior given by int(Q n i ):={(x 1 ,x) ∈ R×R n i −1 |x 1 > x } and the boundary given by:bd(Q n i ):={(x 1 ,x) ∈ R×R n i −1 |x 1 = x }.We may often drop the Subscripts if the dimension is evident from the context. Here and below, · refers to the Euclidean norm, n i is the dimension of Q n i , Q 1 denoting the set of nonnegative reals R + , and for abbreviation, we write x = (x 1 ,x) ∈ R n−1 instead of (x 1 ,x T ) T . It can be easily seen that P 0 -SOCCP(f ) is a natural extension of the nonlinear complementarity problem (NCP) over the nonnegative orthant cone R n + with n 1 = n 2 =···=n m = 1. Without loss of generality, in this paper, we assume that m = 1 and n 1 = n in the subsequent analysis, since our analysis can be easily extended to general cases. Recently there has been strong interests in smoothing Newton methods for solving the SOCCPs [1--6]. One motivation is that SOCCPs have wide applications in many fields [7,8]. The SOCCPs have been utilized as a general framework for quadratic programming, linear complementarity, and many other mathematical programming problems. Different concepts have been developed to treat SOCCPs. Some approaches employ a reformulation of P 0 −SOCCP(f ) as an unconstrained smooth minimization problem or a system of nonlinear equations, and different methods have been developed to treat them [1,2,5,6,9]. The idea of smoothing Newton method is to use a smooth function to reformulate the problem concerned as a family of parameterized smooth equ...

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