Global methods for nonlinear complementarity problems formulate the problem as a system of nonsmooth nonlinear equations approach, or use continuation to trace a path defined by a smooth system of nonlinear equations. We formulate the nonlinear complementarity problem as a bound-constrained nonlinear least squares problem. Algorithms based on this formulation are applicable to general nonlinear complementarity problems, can be started from any nonnegative starting point, and each iteration only requires the solution of systems of linear equations. Convergence to a solution of the nonlinear complementarity problem is guaranteed under reasonable regularity assumptions. The converge rate is Q-linear, Q- superlinear, or Q-quadratic, depending on the tolerances used to solve the subproblems. Global Methods for Nonlinear Complementarity Problems Jorge J. Mor'e 1 Introduction The solution of economic equilibria has been an important motivation for the development of algorithms for nonlin...

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...

A new class of continuation methods is presented which, in particular, solve linear complementarity problems with copositive-plus and L -matrices. Let a# b 2 R be nonnegativevectors. Weembed the complementarity problem with a continuously differentiable mapping f : R in an artificial system of F (x# y)=(a#ib) and (x# y) 0 # () where F : R is defined by F (x# y)=(x 1 y 1 # ...#x n y n # y ; f(x)) and 0 and i 0 are parameters. A pair (x# y) is a solution of the complementarity problem if and only if it solves ()for = 0 and i = 0. A general idea of continuation methods founded on the system () is as follows.

... In this paper, we survey the various theoretical and practical issues related to degeneracy in IPM's for linear programming. We survey results which for the most part already appeared in the literature. Roughly speaking, we shall deal with four topics: the effect of degeneracy on the convergence of IPM's, on the trajectories followed by the algorithms, the effect of degeneracy in numerical performance, and on finding basic solutions.

We study the problem of solving a constrained system of nonlinear equations by a combination of the classical damped Newton method for (unconstrained) smooth equations and the recent interior point potential reduction methods for linear programs, linear and nonlinear complementarity problems. In general, constrained equations provide a unified formulation for many mathematical programming problems, including complementarity problems of various kinds and the Karush-Kuhn-Tucker systems of variational inequalities and nonlinear programs. Combining ideas from the damped Newton and interior point methods, we present an iterative algorithm for solving a constrained system of equations and investigate its convergence properties. Specialization of the algorithm and its convergence analysis to complementarity problems of various kinds and the Karush-Kuhn-Tucker systems of variational inequalities are discussed in detail. We also report the computational results of the implementation of the algo...

. In this paper, we study some basic properties of the monotone semidefinite nonlinear complementarity problem (SDCP). We show that the trajectory continuously accumulates into the solution set of the SDCP passing through the set of the infeasible but positive definite matrices under certain conditions. Especially, for the monotone semidefinite linear complementarity problem, the trajectory converges to an analytic center of the solution set, provided that there exists a strictly complementary solution. Finally, we propose the globally convergent infeasible-interior-point algorithm for the SDCP. Key words Monotone Semidefinite Complementarity Problem, Trajectory, Interior Point Algorithm Research Report B-312 on Mathematical and Computing Sciences, Department of Mathematical and Computing Sciences, Tokyo Institute of Technology. 1 Introduction. Let M(n) and S(n) denote the class of n2n real matrices and the class of n2n symmetric real matrices, respectively. Assume that A; B 2 M(n)....

. Let C be a full dimensional, closed, pointed and convex cone in a finite dimensional real vector space E with an inner product hx; yi of x; y 2 E , and M a maximal monotone subset of E 2 E . This paper studies the existence and continuity of centers of the monotone generalized complementarity problem associated with C and M: Find (x; y) 2 M " (C 2C 3 ) such that hx; yi = 0. Here C 3 = fy 2 E : hx; yi 0 for all x 2 Cg denotes the dual cone of C. The main result of the paper unifies and extends some results established for monotone complementarity problems in Euclidean space and monotone semidefinite linear complementarity problems in symmetric matrices. Key words Central Trajectory, Path of Centers, Complementarity Problem, Interior Point Algorithm, Linear Program Research Report B-303 on Mathematical and Computing Sciences, Department of Mathematical and Computing Sciences, Tokyo Institute of Technology. 1 Introduction. The central trajectory or the path of centers is known ...

Extending our previous work [11], this paper presents a general potential reduction Newton method for solving a constrained system of nonlinear equations. A main convergence result for the method is established. Specializations of the method to a convex semidefinite program and a monotone complementarity problem in symmetric matrices are discussed. Strong convergence results are established in these specializations. 1 Introduction In the paper [11], we have introduced the problem of solving a system of nonlinear equations subject to additional constraints on the variables, i.e., a constrained system of equations. We have demonstrated that constrained equations (CEs) provide a unifying framework for the study of complementarity problems of various types, including the standard nonlinear complementarity problem and the Karush-Kuhn-Tucker system of a variational inequality. Postulating a partitioning property of the CE, we have introduced an interior point potential reduction algorithm f...

. Most known continuation methods for P 0 complementarity problems require some restrictive assumptions, such as the strictly feasible condition and a properness condition, to guarantee the existence and the boundedness of certain homotopy continuation trajectory. To relax such restrictions, we propose in this paper a new homotopy formulation for the complementarity problem based on which a new homotopy continuation trajectory is generated. For P 0 complementarity problems, the most promising feature of this trajectory is the assurance of the existence and the boundedness of the trajectory under a condition that is strictly weaker than the standard ones used widely in the literature of continuation methods. Particularly, the often-assumed strictly feasible condition is not required here. When applied to P complementarity problems, the boundedness of the proposed trajectory turns out to be equivalent to the solvability of the problem, and the entire trajectory converges to the (unique)...

. We study several properties of a homotopy solution mapping, based on Zhang and Zhang's homotopy formulation, for continuous nonlinear complementarity problems with such nonmonotone maps as quasi-monotone, E 0 -, and exceptional regular functions. For these classes of complementarity problems, we establish several sufficient conditions to assure the nonemptyness and boundedness of the homotopy solution mapping. Under the P 0 property, all these sufficient conditions also guarantee the uniqueness and continuity of the homotopy solution mapping, and hence the range of this mapping forms a continuous path passing through an arbitrary point in R n ++ \Theta R n ++ to a solution of complementarity problem. Our analysis is very different from Zhang and Zhang's. We use homotopy invariance theorem of degree to develop a general sufficient condition for the nonemptyness of the homotopy solution mapping instead of using the parameterized Sard's theorem. As a result of this, we only require...