Solving Nonlinear Mixed Integer Stochastic Problems: a Global Perspective

Summary

In this paper, we present a novel approach for solving nonlinear mixed integer stochastic programming problems. In particular, we consider two stage stochastic problem with nonlinearities both in the objective function and constraints, pure integer first stage and mixed-integer second stage variables. We formulate the problem by a scenario based representation, adding linear nonanticipativity constraints coming from splitting the first stage decision variables. In the separation phase we fully exploit the partial decomposable structure of SMINLPs. This allows to deal with a separable nondifferentiable problem, which can be solved by Lagrangian dual based procedure. In particular, we propose a specialization of the Randomized Incremental Subgradient Method- proposed by Bertsekas(2001)- which takes dynamically into account the information relative to the scenarios. The coordination phase is aimed at enforcing coordination among the solutions of the scenario subproblems. More specifically, we use a branch and bound in order to enforce the feasibility of the relaxed nonanticipativity constraints. In order to make more efficient the over-all method, we embed the Lagrangian iteration in a branch and bound scheme, by avoiding the exact solution of the dual problem and we propose an early branching rule and a worm start procedure to use within the Branch and Bound tree. Although SMINLPs have many application contexts, this class of problem has not been adequately treated in the literature. We propose a stochastic formulation of the Trim Loss Problem, which is new in the literature. A formal mathematical formulation is provided in the framework of two-stage stochastic programming which explicitly takes into account the uncertainty in the demand. Preliminary computational results illustrate the ability of the proposed method to determine the global optimum significantly decreasing the solution time. Furthermore, the proposed approach is able to solve instances of the problem intractable with conventional approaches.