3 research outputs found
Adjustable Robust Two-Stage Polynomial Optimization with Application to AC Optimal Power Flow
In this work, we consider two-stage polynomial optimization problems under
uncertainty. In the first stage, one needs to decide upon the values of a
subset of optimization variables (control variables). In the second stage, the
uncertainty is revealed and the rest of optimization variables (state
variables) are set up as a solution to a known system of possibly non-linear
equations. This type of problem occurs, for instance, in optimization for
dynamical systems, such as electric power systems. We combine tools from
polynomial and robust optimization to provide a framework for general
adjustable robust polynomial optimization problems. In particular, we propose
an iterative algorithm to build a sequence of (approximately) robustly feasible
solutions with an improving objective value and verify robust feasibility or
infeasibility of the resulting solution under a semialgebraic uncertainty set.
At each iteration, the algorithm optimizes over a subset of the feasible set
and uses affine approximations of the second-stage equations while preserving
the non-linearity of other constraints. The algorithm allows for additional
simplifications in case of possibly non-convex quadratic problems under
ellipsoidal uncertainty. We implement our approach for AC Optimal Power Flow
and demonstrate the performance of our proposed method on Matpower instances.Comment: 28 pages, 3 table
Inner and outer approximations of existentially quantified equality constraints
Abstract. We propose a branch and prune algorithm that is able to compute inner and outer approximations of the solution set of an existentially quantified constraint where existential parameters are shared between several equations. While other techniques that handle such constraints need some preliminary formal simplification of the problem or only work on simpler special cases, our algorithm is the first pure numerical algorithm that can approximate the solution set of such constraints in the general case. Hence this new algorithm allows computing inner approximations that were out of reach until today.