A projected–gradient interior–point algorithm for complementarity problems J. J. J´ udice * J. M. Martinez † J. Patr´ıcio ‡ R. Andreani § Abstract Interior–point algorithms are nowadays among the most efficient techniques for processing monotone complementarity problems. In this paper, a procedure for globalizing interior– point methods by using the maximum stepsize is introduced. The algorithm combines exact or inexact interior–point and projected–gradient search techniques and employs a line–search procedure for the natural merit function associated to the complementarity problem. Fur- thermore for linear complementarity problems, the maximum stepsize is shown to be accepted in all iterations employing the exact interior–point search direction. A number of classes of complementarity and optimization problems are discussed, which the algorithm is able to process by either finding a solution or showing that no solution exists. A modification of the algorithm for dealing with infeasible linear complementarity problems is introduced, which solely employs interior–point search directions in practice. Computational experience on the solution of complementarity problems and linear and convex quadratic programs by the new algorithm is included, which illustrates the efficiency of this methodology. 1 Introduction In this paper we discuss the solution of the Complementarity Problem (CP) by interior–point methods. The CP consists of finding vectors z, w ∈ n and y ∈ m such that H (x, y, w)= 0, x ⊤ w =0, x, w ≥ 0 where H : n+m+n −→ n+m is continuously differentiable in Ω=  (x, y, w) ∈ n+m+n : x, w ≥ 0  An interesting particular case is the so–called Linear Complementarity Problem (LCP), where the function H is affine of the form H (x, w)= Mx − w + q with M ∈ n×n and q ∈ n are given. Many applications of the LCP have been proposed in science, engineering and economics. We suggest [6, 7, 21] for good studies of the CP, including its properties, algorithms and applications. In particular Variational Inequalities and the computation of minima of continuously differentiable functions reduce to CP [6, 7, 21]. * Departamento de Matem´ atica, Universidade de Coimbra, and Instituto de Telecomunica¸ c˜ oes. E-mail: Joaquim.Judice@co.it.pt † Instituto de Matem´ atica, Estat´ıstica e Computa¸ c˜ ao Cient´ıfica, Universidade de Campinas, Brazil. E-mail: martinez@ime.unicamp.br ‡ ´ Area Interdepartamental de Matem´ atica, Escola Superior de Tecnologia de Tomar, and Instituto de Telecomu- nica¸ c˜ oes. E-mail: Joao.Patricio@aim.estt.ipt.pt § Instituto de Matem´ atica, Estat´ıstica e Computa¸ c˜ ao Cient´ıfica, Universidade de Campinas, Brazil. E-mail: andreani@ime.unicamp.br 1