A Unified Approach to Interior Point Algorithms for Linear Complementarity Problems

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.