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 forvarious mathematical programming problems such as linearprogramming, convex quadratic programming and convexprogramming in general. This monograph presents a study ofinterior-point algorithms for the linear complementarityproblem (LCP) which is known as a mathematical model forprimal-dual pairs of linear programs and convex quadraticprograms. A large family of potential reduction algorithmsis presented in a unified way for the class of LCPs wherethe underlying matrix has nonnegative principal minors(P0-matrix). This class includes various importantsubclasses such as positive semi-definite matrices,P-matrices, P*-matrices introduced in this monograph, andcolumn sufficient matrices. The family contains not only theusual potential reduction algorithms but also path followingalgorithms and a damped Newton method for the LCP. The maintopics are global convergence, global linear convergence,and the polynomial-time convergence of potential reductionalgorithms included in the family.