On the Sample Size of Random Convex Programs with Structured Dependence on the Uncertainty (Extended Version)

TL;DR

This paper improves the sample size bounds for randomized convex programs with structured dependence on uncertainty, reducing computational costs and conservativeness in chance constrained control problems.

Contribution

It derives tighter Helly's dimension bounds for structured chance constraints, enhancing efficiency and solution quality in randomized control design.

Findings

Reduced scenario requirements lead to lower computational complexity.

Improved bounds enhance objective value and reduce conservativeness.

Demonstrated effectiveness on an inventory management example.

Abstract

The "scenario approach" provides an intuitive method to address chance constrained problems arising in control design for uncertain systems. It addresses these problems by replacing the chance constraint with a finite number of sampled constraints (scenarios). The sample size critically depends on Helly's dimension, a quantity always upper bounded by the number of decision variables. However, this standard bound can lead to computationally expensive programs whose solutions are conservative in terms of cost and violation probability. We derive improved bounds of Helly's dimension for problems where the chance constraint has certain structural properties. The improved bounds lower the number of scenarios required for these problems, leading both to improved objective value and reduced computational complexity. Our results are generally applicable to Randomized Model Predictive Control of chance constrained linear systems with additive uncertainty and affine disturbance feedback. The efficacy of the proposed bound is demonstrated on an inventory management example.