A Unified Approach to Interior Point Algorithms for Linear Complementarity Problems
Springer Science & Business Media, Sep 25, 1991 - Language Arts & Disciplines - 108 pages
Following Karmarkar's 1984 linear programming algorithm, numerous interior-point algorithms have been proposed for various mathematical programming problems such as linear programming, convex quadratic programming and convex programming in general. This monograph presents a study of interior-point algorithms for the linear complementarity problem (LCP) which is known as a mathematical model for primal-dual pairs of linear programs and convex quadratic programs. A large family of potential reduction algorithms is presented in a unified way for the class of LCPs where the underlying matrix has nonnegative principal minors (P0-matrix). This class includes various important subclasses such as positive semi-definite matrices, P-matrices, P*-matrices introduced in this monograph, and column sufficient matrices. The family contains not only the usual potential reduction algorithms but also path following algorithms and a damped Newton method for the LCP. The main topics are global convergence, global linear convergence, and the polynomial-time convergence of potential reduction algorithms included in the family.
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acen affine scaling algorithm for linear arithmetic operations artificial problem LCP column sufficient matrices compute an exact Construct the artificial convergence results convex quadratic programming defined denotes diag diagonal matrix direction dx,dy direction parameter feasible region follows function f G S++ global convergence Hence inequality initial point interior point algorithms interior point method iterations Lemma linear complementarity problems linear programming Mathematical Programming matrix M belongs Megiddo minimizer Mizuno and Yoshise n x n neighborhood Newton equations Noma NP-complete P-matrix path of centers path-following algorithm Po-matrices point x,y polynomial-time convergence positive number positive semi-definite positive semi-definite matrix potential function potential reduction algorithm primal-dual interior point primal-dual pair Proceedings quadratic programming satisfies Condition 2.1 Scen search direction Section sequence x*,y solution x*,y solves the LCP step size parameter stopping criterion Subseries LNAI Suppose Technical report trajectory UIP method vector q xk,yk Yoshise 34