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Center for Nonlinear Studies |
This talk focuses on the solution of continuous optimization problems where uncertainty is present in the data for constraints. An example is the optimal power flow problem with stochastic demand and limits on the power flow through each line. One might seek to find solutions that are robust in the sense that they are feasible for every possible scenario. However, since this can lead to overly conservative solutions, it is often more desirable to find solutions that satisfies the stochastic constraints only with a high probability instead.We propose a new approximation of these chance constraints that results in differentiable functions. They can be included directly in nonlinear optimization problem formulations and handled by standard nonlinear optimization packages. For problems with joint chance constraints, we propose a new trust-region sequential quadratic programming algorithm with provable convergence guarantees.Our formulation is based on sample average approximation and considers a set of random realizations of the constraints. Conventionally, a mixed-integer programming formulation is used in this setting to select the optimal subset of enforced constraints. However, this approach typically results in large computation times due to the implicit enumeration in the branch-and-bound search. In contrast, our compact reformulation combines all constraint samples into a single differentiable constraint, avoiding any combinatorial complexity. This is achieved by means of a smoothed cumulative distribution function for the random constraint values.We will present numerical experiments that showcase the efficiency of this approach. In particular, we consider DC optimal power flow problems with stochastic demand and joint chance constraints that require the satisfaction of all line limits.This is work in collaboration with Alejandra Pena-Ordieres at Northwestern University, James Luedtke and Line Roald at the University of Wisconsin-Madison, and Dan Molzahn at Argonne National Laboratories.
Host: Hassan Hijazi