New Formulations and Branching Strategies for the GOP Algorithm

Abstract

In Floudas and Visweswaran (1990, 1993), a deterministic global optimization approach was proposed for solving certain classes of nonconvex optimization problems. A global optimization algorithm, GOP, was presented for the solution of the problem through a series of primal and relaxed dual problems that provide valid upper and lower bounds respectively on the global solution. The algorithm was proven to have finite convergence to an r-global optimum. In this paper, a branch-and-bound framework of the GOP algorithm is presented, along with several reduction tests that can be applied at each node of the branch-and-bound tree. The effect of the properties is to prune the tree and provide tighter underestimators for the relaxed dual problems. We also present a mixed-integer linear programming (MILP) formulation for the relaxed dual problem, which enables an implicit enumeration of the nodes in the branch-and-bound tree at each iteration. Finally, an alternate branching scheme is presented for the solution of the relaxed dual problem through a linear number of subproblems. Simple examples are presented to illustrate the new approaches. Detailed computational results on the implementation of both versions of the algorithm can be found in the companion paper in chapter 4.