Continuous Optimization
The prediction–correction approach to nonlinear complementarity problems

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Abstract

This paper presents a prediction–correction approach to solving the nonlinear complementarity problem (NCP). Each iteration of the new method consists of a prediction and a correction. The predictor is produced by an inexact Logarithmic-Quadratic Proximal method; and then it is corrected by the Proximal Point Algorithm. Convergence of the new method is proved under mild assumptions. Comparison to existing methods shows the superiority of the new method. Numerical experiments including the application to traffic equilibrium problems demonstrate that the new method is attractive in practice.

Introduction

Nonlinear Complementarity Problem (NCP) is to determine a vector xRn such thatx0,F(x)0,andxTF(x)=0,where F is a mapping from Rn into itself. NCP has received much attention due to its various applications arising in operational research, transportation, engineering, economic equilibrium, mathematical programming and others, see, e. g., [10], [15]. Throughout this paper we assume that F is continuous and monotone and that the solution set of (1.1) is nonempty.

Among existing effective methods for solving (1.1) is the Proximal Point Algorithm (PPA), which was presented originally by Martinet [14] for finding roots of a maximal monotone operator. For more developments on PPA, we refer to, e.g., [7], [8], [16], [19]. Let xk be the current approximation of a solution of (1.1), then PPA generates the new iterate xk+1 by solving the following auxiliary NCP:x0,ckF(x)+(x-xk)0,andxT(ckF(x)+(x-xk))=0,where ck  [c, ∞) and c is a positive constant. Compared to the monotone NCP (1.1), (1.2) is easier to handle since it is a strong monotone NCP. Let R+n{xRnx0} and PR+n denote the projection onto R+n under Euclidean norm, then it is well known [6] that solving (1.2) is equivalent to solving the following equation:(PPA)x=PR+n[xk-ckF(x)].Note that (1.3) is a nonsmooth equation and that the new iterate xk+1 cannot be computed directly via (1.3) since it is an implicit scheme. These difficulties make straightforward applications of PPA impractical in many cases. Therefore it is of importance to develop PPA with improved practical characteristics. One effort in this sense is to find approximate solutions of (1.3) instead of solving it exactly, hence results in many appropriate PPAs, see, e.g., [5], [7], [9], [11], [12], [16], [17], [19]. The major challenges of appropriate PPAs include accelerating convergence and designing inexactness restrictions that are easy to implement and tight for convergence. Recent developments on PPA also focus on replacing the linear term x  xk with more general terms such as entropic ϕ-divergence [19], Bregman functions [1], [3], [7] and Logarithmic-Quadratic Proximal (LQP) terms [2].

The original LQP method presented in [2] improves PPA (1.3) by replacing the linear term x  xk withx-(1-μ)xk-μXk2x-1,where μ  (0, 1) is a given constant, Xk=diag(x1k,x2k,,xnk) and x−1 is a n-vector whose jth element is 1/xj. We should point out that the original LQP method in [2] does not restrict μ  (0, 1) and involves another parameter ν with the restriction ν > μ > 0. (1.4) is the special case that ν = 1 and μ  (0, 1), which is more convenient for the following analysis. In particular, at the kth iteration, solving (1.1) by the LQP method is equivalent to finding the positive solution of the following system of nonlinear equations (LQP system for convenience)ckF(x)+x-(1-μ)xk-μXk2x-1=0.A more practical version of the LQP method is the inexact LQP method presented also in [2] which solves the LQP system approximately in the following sense: Find xk+1R++n and ξkRn such thatckF(x)+x-(1-μ)xk-μXk2x-1=ξkandk=0ξk<+andk=0ξk,xk<+.It is reasonable to denoteξkxk-xk+1the relative error for solving the LQP system approximately at the kth iteration. Note that (1.7) implies that the involved LQP systems need to be solved more and more exactly. The first contribution to overcoming this drawback was due to [4], in which the authors improved (1.7) with relative errors and thus presented a meaningful modification of the inexact LQP method with the attractive characteristic that the relative errors for solving the involved LQP systems can be fixed on a constant. Recently, Xu and A. Bnouhachem [20] improved the inexact LQP method in the sense that the restriction on ξk is relaxed toξkη1-μ2xk-xk+1withη(0,1),which implies that the relative errors of solving the LQP systems approximately can be fixed at η1-μ2.

This paper presents another approach, i.e., the prediction–correction approach, to make PPA implementable for solving (1.1). At each iteration, the inexact LQP method with the inexact restriction (1.8) developed in [20] is used to produce a predictor, and then the PPA (1.3) is utilized to correct the predictor. Meanwhile, an optimal step size is used in the correction. The reduced prediction–correction method is very easy to implement in the computational sense and the total computational load is very small. Numerical applications to some classical NCP and traffic equilibrium problems show that the new method is effective in practice.

The rest of the paper is organized as follows. The algorithm is given in Section 2. In Section 3, we provide some fundamental theoretical results. In particular, we focus on how to choose the optimal step size in the correction. Section 4 mainly analyzes convergence of the new method. Comparison to the work of Xu is given in Section 5. Then Section 6 reports some numerical results so as to demonstrate that the new method is effective. Finally, some conclusions are drawn in Section 7.

Section snippets

Algorithm

Given constants c > 0, η  (0, 1), μ  (0, 1) and γ  [1, 2); Starting from x0R++n, each iteration consists of the following two steps:

  • Step 1.

    Prediction Step

    Use the inexact LQP method in [20] to find x˜kR++n and ξkRn such thatckF(x˜k)+x˜k-(1-μ)xk-μXk2(x˜k)-1=ξkck(F(x˜k)-F(xk))with the inexact criterionξkη1-μ2xk-x˜k;

  • Step 2.

    Correction Step

    Use the PPA (1.3) to correct the predictor and so generate the new iterate xk+1:xk+1(αk)=PR+nxk-αkck1+μF(x˜k),whereαk=γϕ(xk,x˜k,ξk,μ)d(xk,x˜k,ξk,μ)2;ϕ(xk,x˜k,ξk,μ)=11+μxk-x˜k

Some theoretical results

In this section we first list some fundamental lemmas that are useful in the consequent analysis and then investigate the strategy of how to choose the step size αk in the correction (2.3). The first lemma provides some basic inequalities of projection onto R+n without proof.

Lemma 3.1

Let PR+n denote the projection onto R+n under Euclidean norm. Then we have the following fundamental inequalities:{z-PR+n(z)}T{x-PR+n(z)}0,xR+n,zRn;PR+n(y)-PR+n(z)y-z,y,zRn;PR+n(y)-x2y-x2-y-PR+n(y)2,xR+n

Convergence

In this section convergence of the new method is proved under mild assumptions. The following theorem plays an important role in the convergence analysis.

Theorem 4.1

Let x be an arbitrary point of the solution set of (1.1); let x˜k and xk+1k) be generated by the algorithm. Then we havexk+1(αk)-x2xk-x2-γ(2-γ)(1-η2+μ2)4(1+μ)xk-x˜k2.

Proof

First we observe that2ϕ(xk,x˜k,ξk,μ)=(2.5)21+μxk-x˜k2+21+μ(xk-x˜k)Tξk=xk-x˜k2+21+μ(xk-x˜k)Tξk+1-μ1+μxk-x˜k2>xk-x˜k2+21+μ(xk-x˜k)Tξk+1(1+μ)2ξk2=(2.6)(xk,x˜k,ξ

Comparison to Xu’s method

In this section we compare, from the theoretical point of view, our new prediction–correction method to Xu’s algorithm in [20].

As proved in [20] (see also (4.2)), it is easy to verify that(xk-x)Td(xk,x˜k,ξk,μ)ϕ(xk,x˜k,ξk,μ)>12d(xk,x˜k,ξk,μ)2,which means that -d(xk,x˜k,ξk,μ) is a descent direction of 12x-x2 at the point xk. Based on this fact, Xu and A. Bnouhachem [20] searched the “optimal” step size along the decent direction and thus presented a hybrid algorithm for NCP. In particular,

Application to classical NCP

This subsection reports the numerical results when the new algorithm is applied to some classical NCP studied by many researchers, see, e.g., [10], [13], [18], [20]. To verify the theoretical conclusions made in the last section, the numerical comparison to Xu’s method is also reported.

The NCP we tested takes the following form:F(x)=D(x)+Mx+qwhere D(x) and Mx + q are the nonlinear part and the linear part of F(x), respectively. In particular, M = ATA + B, where A and B are n × n skew-symmetric matrices

Conclusion

This paper provides an effective approach to make the well-known Proximal Point Algorithm (PPA) implementable; and thus presents a prediction–correction method to solve the nonlinear complementarity problem (NCP). The inexact variant of the LQP method developed in [20] is used to produce a predictor, and then the predictor is corrected by PPA to generate the new iterate. We compare the new algorithm with the hybrid algorithm in [20] in both theoretical and numerical senses. Applications to some

Acknowledgements

The author would thank sincerely Professors Bingsheng He and Jason Jianzhong Zhang for their valuable helps and discussions. Special gratitude is expressed to Professors Jane J Ye and Julie Zhou in University of Victoria for their kind supports.

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