A combined homotopy interior point method for the linear complementarity problem

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Abstract

A combined homotopy interior point method for the linear complementarity problem (LCP) is proposed in this paper. We construct a homotopy path for the linear complementarity problem and prove that the homotopy path determines a smooth interior path from an arbitrary given interior point to the solution of LCP. Furthermore, we modify a predictor corrector algorithm with the technique of the cone neighborhood to trace the homotopy path. Finally, a numerical example is provided to show the effectiveness and feasibility of this method.

Introduction

The linear complementarity problem (LCP) is to find a vector pair (x, y)  Rn × Rn such that y = Mx + h,(x, y)  (0, 0),xTy = 0, where h  Rn and M is a n × n positive semidefinite matrix.

Since Karmarkar published the first paper on interior point method [1] in 1984 and received a great deal of attention due to its polynomial complexity and excellent practicality, it has been introduced to LCP and regarded as the best method for LCP [2]. However, it is a very difficult task to find a strictly feasible initial point to start the algorithm. To overcome this difficulty, recent studies have focused on some new interior point algorithms without the need to find a strictly feasible initial point. In 1993, Kojima et al. presented the first infeasible interior point algorithm with global convergence [3], soon later Zhang [4] and Wright [5] introduced this technique to the linear complementarity problem.

More recently, a class of combined homotopy interior point methods have been proposed by Lin et al. [6], [7], [8]. The combined homotopy interior point methods presented in these papers not only improve the disadvantage of finding a feasible initial point for the interior point algorithm, but also enhance the computation performance of the algorithm.

The basic idea of homotopy methods can be explained as follows: construct a homotopy from the auxiliary mapping g to the object mapping f, and then the origin problem can be solved by following the homotopy path from the zero set of the auxiliary mapping g to the zero set of the object mapping f. So we can easily choose a start point for the homotopy path from the zero set of g since the auxiliary mapping g is set artificially. For this reason, the difficulty of finding a strictly feasible initial point for the interior point algorithm can be avoided by combining the interior point with the homotopy method. Furthermore, the global convergence of the homotopy methods can guarantee the global convergence for the combined homotopy interior point methods.

Inspired by this idea, we propose a combined homotopy interior point method for the linear complementarity problem (LCP). In this paper, a smooth homotopy path has been constructed, which can be traced from a very simple initial point and obtain the solution of the linear complementarity problem. It has been proved that from an arbitrarily given interior point, this homotopy path determines a smooth interior path and terminates at a K–K–T point which is the solution of LCP. In addition, a modified predictor–corrector algorithm to tracing the homotopy path is provided and a numerical example is given.

The homotopy is constructed in Section 2 and some propositions of it are proved in Section 3. Using the technique of the cone neighborhood, we modify a predictor corrector algorithm to trace this homotopy path in Section 4 and discuss its convergence properties. Finally, a numerical example of LCP is computed in Section 5 to show the effectiveness.

Section snippets

Homotopy for LCP

Consider the following linear complementarity problem, that is to find a vector pair (x, y)  Rn × Rn such that y = Mx + h, (x, y)  (0, 0), xTy = 0, where h  Rn and M is a n × n positive semidefinite matrix.

First, we introduce a simple convex quadratic programmingminf(x)=12xTMx+hTx,s.t.x0.where h and M are the same as described in the LCP. In the programming, x are only subject to nonnegative constraints.

Obviously, x  Rn is a solution of (1) if and only if there exist a y  Rn such that (x, y) is a solution of

Technical results for homotopy path

Lemma 1

Given an arbitrary μ  [0, 1], there exists a unique solution x(μ) > 0 for minx>0F(x, μ).

Proof

From (3), we havex2F(x,μ)=(1-μ)2f(x)+μI+μ(1-μ)1x121xn2.Since ∇2f(x) = M is a positive semidefinite matrix and diag(1x12,,1xn2) is a positively definite matrix, then for any z  Rn,zTx2F(x,μ)z=(1-μ)zT2f(x)z+μzTz+μ(1-μ)zT1x121xn2z0and the equality will hold if and only if z = 0. Hence x2F(x,μ) is a positively definite matrix and F(x, μ) is a convex function of x for all μ  [0, 1]. Therefore, there exists a unique

Predictor–corrector algorithm

Denote the solution curve of the homotopy equation H(ω, μ) = 0 as Γ. From Theorem 1, we have shown that Γ is a smooth curve which starts from (x0, y0, 0) with H(ω0, 1) = 0. Since ω(μ) is the solution of H(ω, μ) = 0 and continuously depend on μ, it is obvious that the limit of ω(μ) is the solution of K–K–T system of (1) when μ  0. So we can trace the homotopy path Γ from a given (x0, y0, 0) to obtain the solution of K–K–T system of (1), that is, the solution of the linear complementarity problem.

Using the

Numerical example

Take the following linear complementarity problem as an illustrative example: to find a vector pair (x, y)  R6 × R6 such that y = Mx + h, (x, y)  (0, 0), xTy = 0, where h  R6 and M is a 6 × 6 positive semidefinite matrix.M=704224281590103032014610801203509268,h=-7.37.9-3.410.2-5.97.2.

Let β = 0.99, δ = 0.01, α = 0.5, α1 = 0.5, α2 = 0.8, α3 = 0.9, α4 = 0.99. Set the initial point as x0 = y0 = (1, 1, 1, 1, 1, 1)T, and μ0 = 1, the error ε = 10−6.

Solve the problem by the algorithm presented above, and we get an approximate solution satisfied

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This work was supported by the National Nature Science Foundation of China by Grant No. 60274048.

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