We present a generalization of a homogeneous self-dual linear programming (LP) algorithm to solving the monotone complementarity problem (MCP). The algorithm does not need to use any "big-M" parameter or two-phase method, and it generates either a solution converging towards feasibility and complementarity simultaneously or a certificate proving infeasibility. Moreover, if the MCP is polynomially solvable with an interior feasible starting point, then it can be polynomially solved without using or knowing such information at all. To our knowledge, this is the first interior-point and infeasible-starting algorithm for solving the MCP that possesses these desired features. Preliminary computational results are presented. Key words: Monotone complementarity problem, homogeneous and self-dual, infeasible-starting algorithm. Running head: A homogeneous algorithm for MCP. Department of Management, Odense University, Campusvej 55, DK-5230 Odense M, Denmark, email: eda@busieco.ou.dk. y De...

How to initialize an algorithm to solve an optimization problem is of great theoretical and practical importance. In the simplex method for linear programming this issue is resolved by either the two-phase approach or using the so-called big M technique. In the interior point method, there is a more elegant way to deal with the initialization problem, viz. the self-dual embedding technique proposed by Ye, Todd and Mizuno [30]. For linear programming this technique makes it possible to identify an optimal solution or conclude the problem to be infeasible/unbounded by solving its embedded self-dual problem. The embedded self-dual problem has a trivial initial solution and has the same structure as the original problem. Hence, it eliminates the need to consider the initialization problem at all. In this paper, we extend this approach to solve general conic convex programming, including semidefinite programming. Since a nonlinear conic convex programming problem may lack the so-called stri...

A new predictor--corrector interior point algorithm for solving monotone linear complementarity problems (LCP) is proposed, and it is shown to be superlinearly convergent with at least order 1.5, even if the LCP has no strictly complementary solution. Unlike Mizuno's recent algorithm [16], the fast local convergence is attained without any need for estimating the optimal partition. In the special case that a strictly complementary solution does exist, the order of convergence becomes quadratic. The proof relies on an investigation of the asymptotic behavior of first and second order derivatives that are associated with trajectories of weighted centers for LCP. AMS 1991 subject classification: 90C33. Key words. monotone linear complementarity problem, primal-dual interior point method, superlinear convergence, central path. 1 1. Introduction Given n \Theta n real matrices Q and R and a real vector b of order n, the horizontal linear complementarity problem (LCP) is the problem of fin...

In this paper, we propose a Big-M smoothing method for solving the P 0 matrix linear complementarity problem. We study the trajectory defined by the augmented smoothing equations and global convergence of the method under an assumption that the original P 0 matrix linear complementarity problem has a solution. Key words. linear complementarity problem, P 0 matrix, smoothing algorithm. AMS subject classifications. 65H10, 90C30, 90C33. This author's work was supported by the Australian Research Council. y This author is supported in part by NSF Grants DMI-9522507 and DMS-9703490. 2 1 Introduction In this paper we consider the linear complementarity problem (LCP) t T s = 0; s = Mt + q; and t; s 0; where M is an n \Theta n P 0 matrix and q is an n dimensional vector. A matrix M 2 R n\Thetan is called a P 0 matrix if max i:t i 6=0 t i (Mt) i 0; for all t 2 R n ; t 6= 0: A linear complementarity problem is called a P 0 matrix LCP if the matrix M is a P 0 matrix. The clas...

h is sufficient for the boundedness of the solution set F , see Mangasarian [3]. Corollary 3 in Ye [6] states that the existence of such x and s is also necessary in the case of monotone LCPs with a so--called negative q--value. To see that it is necessary in general, suppose that there exist no such positive solution pair (x; s). Then, by Carver's theorem [1], there exists a direction u 6= 0 such that M T u 0; u 0; h T u 0: If h T u ! 0 then F = ;, by Farkas' lemma. Suppose now that there exist (x ; s ) 2 F , and consequently h T u = 0. Since u 0 and M T u 0, w