Summary: A Branch-Reduce-Cut Algorithm for the Global Optimization of
Probabilistically Constrained Linear Programs
Myun-Seok Cheon, Shabbir Ahmed
and Faiz Al-Khayyal
School of Industrial & Systems Engineering
Georgia Institute of Technology, 765 Ferst Drive, Atlanta, GA 30332.
April 25, 2005
We consider probabilistically constrained linear programs with general distributions for the uncertain
parameters. These problems involve non-convex feasible sets. We develop a branch-and-bound algorithm
that searches for a global optimal solution to this problem by successively partitioning the non-convex
feasible region and by using bounds on the objective function to fathom inferior partition elements. This
basic algorithm is enhanced by domain reduction and cutting plane strategies to reduce the size of the
partition elements and hence tighten bounds. The proposed branch-reduce-cut algorithm exploits the
monotonicity properties inherent in the problem, and requires solving linear programming subproblems.
We provide convergence proofs for the algorithm. Some illustrative numerical results involving problems
with discrete distributions are presented.
Various applications in reliability and risk management (cf.) give rise to probabilistically-constrained
linear programs (PCLP) of the following form