A hybrid parametric/stochastic programming approach for mixed-integer nonlinear problems under uncertainty

This paper presents a hybrid parametric stochastic programming approach for mixed-integer convex nonlinear optimization problems under uncertainty. This approach is based upon an iterative two-stage stochastic optimization framework, where in the first stage design and integer variables are fixed and an expected profit is calculated by solving nonlinear programs (NLPs) at the integration points, in the space of uncertain parameters. Then, in the second stage, a master problem is formulated, based upon the dual information obtained by solving NLPs in the first stage, and solved to identify a new-vector of design and integer variables. The solution procedure terminates when the solution of the first- and second-stage problems is within a certain tolerance. The basic idea of the hybrid approach, proposed in this work, is that in the first stage parametric programming techniques are used to obtain profit as a function of uncertain parameters. This reduces the computation of the expected profit to a function evaluation of the profit function at the integration points. Therefore, the solution of the NLPS at the integration points is avoided. The use of efficient integration techniques, which require a smaller number of integration points, in light of the hybrid approach is also discussed.