Linear Programming with Probability Constraints - Part 2

In many relevant situations, chance constrained linear programs can be explicitly converted into efficiently solvable convex second order cone programs (SOCP), provided some information about the family of data distributions (for instance, the first two moments) is known. These issues have been discussed in the first part paper [3]. In this companion paper, we consider chance constrained linear programs where the moments of the data are unknown and need be estimated from samples. A key result is that given a finite and explicit number N of sampled data, one can construct a SOCP such that any feasible solution to the SOCP is with high probability also feasible for the original chance constrained problem. To conclude this two parts work, we present examples of application of probability constrained linear programs to constraint reduction in large-scale LP, and to problems in portfolio optimization theory and in model predictive control.