Exact Computation of Global Minima of a Nonconvex Portfolio Optimization Problem

Abstract

The goal of this project was to compute minimal cost solutions satisfying the demand of pre-given product portfolios and to investigate the dependence of the fix costs and investment costs on the product portfolio.

The most important parameters characterizing the production facilities are the number and the size of the reactors. The production is subject to shelf-life constraints, i.e., products cannot be stored longer than one week.

Even if we analyze this problem under the simple assumption of constant batch sizes and limit ourself to only one time period covering one week, the computation of minimum cost scenarios requires that we determine global minima of a nonconvex MINLP problem. An objective function built up by the sum of concave functions and trilinear products terms involving the variables describing the number of batches, the utilization rates and the volume of the reactor are the nonlinear features in the model.

We have successfully applied four different solution techniques to solve this problem. (1) An exact transformation allows us to represent the nonlinear constraints by MILP constraints. Using piecewise linear approximations for the objective function the problem is solved with XPress-MP, a commercial MILP solver. (2) The local MINLP Branch-and-Bound solver SBB which is part of the modeling system GAMS. (3) The Branch&Reduce Optimization Navigator (BARON) also called from GAMS. (4) A taylorized Branch&Bound approach based on the construction of a lower bounding problem by underestimating the concave objective function with piecewise linear approximations described in a forecoming paper.

Our overall conclusion from a detailed analysis of specific portfolio cases is that the problem, for some cases, can be solved with nowadays standard solvers’ capacities but it requires a lot of CPU time. Therefore, in order not to cover only special cases and also to cope with the scaling properties of this problem suffering from weak lower bounds, we recommend to use taylorized approaches in addition.