Pattern-Based Modeling and Solution of Probabilistically Constrained Optimization Problems

We propose a new modeling and solution method for probabilistically constrained optimization problems.The methodology is based on the integration of the stochastic programming and combinatorialpattern recognition fields. It permits the very fast solution of stochastic optimization problems in which the random variables are represented by an extremely large number of scenarios. The methodinvolves the binarization of the probability distribution, and the generation of a consistent partially defined Boolean function (pdBf) representing the combination (F,p) of the binarized probability distributionF and the enforced probability level p. We show that the pdBf representing (F,p) can becompactly extended as a disjunctive normal form (DNF). The DNF is a collection of combinatorialp-patterns, each of which defining sufficient conditions for a probabilistic constraint to hold. We propose two linear programming formulations for the generation of p-patterns which can be subsequently used to derive a linear programming inner approximation of the original stochastic problem.A formulation allowing for the concurrent generation of a p-pattern and the solution of the deterministic equivalent of the stochastic problem is also proposed. Results show that large-scale stochastic problems, in which up to 50,000 scenarios are used to describe the stochastic variables, can be consistentlysolved to optimality within a few seconds