Solving Chance-Constrained Optimization Under Nonparametric Uncertainty Through Hilbert Space Embedding

In this article, we present an efficient algorithm for solving a class of chance-constrained optimization under nonparametric uncertainty. Our algorithm is built on the possibility of representing arbitrary distributions as functions in Reproducing Kernel Hilbert Space (RKHS). We use this foundation to formulate chance-constrained optimization as one of minimizing the distance between a desired distribution and the distribution of the constraint functions in the RKHS. We provide a systematic way of constructing the desired distribution based on the notion of scenario approximation. Furthermore, we use the kernel trick to show that the computational complexity of our reformulated optimization problem is comparable to solving a deterministic variant of the chance-constrained optimization. We validate our formulation on two important robotic applications: 1) reactive collision avoidance of mobile robots in uncertain dynamic environments and 2) inverse-dynamics-based path-tracking of manipulators under perception uncertainty. In both these applications, the underlying chance constraints are defined over nonlinear and nonconvex functions of uncertain parameters and possibly also decision variables. We also benchmark our formulation with the existing approaches in terms of sample complexity and the achieved optimal cost highlighting significant improvements in both these metrics.