Solution of Monotone Complementarity and General Convex Programming Problems Using a Modified Potential Reduction Interior Point Method

We present a homogeneous algorithm equipped with a modified potential function for the monotone complementarity problem. We show that this potential function is reduced by at least a constant amount if a scaled Lipschitz condition is satisfied. A practical algorithm based on this potential function is implemented in a software package named iOptimize. The implementation in iOptimize maintains global linear polynomial-time convergence properties while achieving practical performance. When compared with a mature software package MOSEK (barrier solver version 6.0.0.106), iOptimize solves convex quadratic programming problems, convex quadratically constrained quadratic programming problems, and general convex programming problems in fewer iterations. Moreover, several problems for which MOSEK fails are solved to optimality. We also find that iOptimize seems to detect infeasibility more reliably than general nonlinear solvers Ipopt (version 3.9.2) and Knitro (version 8.0).