An LP/NLP based branch and bound algorithm for convex MINLP optimization problems

Abstract

This paper is aimed at improving the solution efficiency of convex MINLP problems in which the bottleneck lies in the combinatorial search for the 0–1 variables. An LP/NLP based branch and bound algorithm is proposed in which the explicit solution of an MILP master problem is avoided at each major iteration. Instead, the master problem is defined dynamically during the tree search to reduce the number of nodes that need to be enumerated. A branch and bound search is conduced to predict lower bounds by solving LP subproblems until feasible integer solutions are found. At these nodes nonlinear programming subproblems are solved, providing upper bounds and new linear approximations which are used to tighten the linear representation of the open nodes in the search tree. To reduce the size of the LP subproblems, new types of linear approximations are proposed which exploit linear substructures in the MINLP problem. Preliminary numerical results on several test problems are reported which show that the expense of solving the MI need to be enumerated, while in most cases the number of NLP subproblems to be solved remains the same.