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

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.

arXiv commentAccepted for publication at Automatica

References in corpus (5)

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Cited by in corpus (10)

• Optimization under Uncertainty in the Era of Big Data and Deep Learning: When Machine Learning Meets Mathematical Programming
• A scenario approach for non-convex control design
• Data-driven Decision Making with Probabilistic Guarantees (Part 1): A Schematic Overview of Chance-constrained Optimization
• From Uncertainty Data to Robust Policies for Temporal Logic Planning
• A Posteriori Probabilistic Bounds of Convex Scenario Programs with Validation Tests
• Exploiting structure of chance constrained programs via submodularity
• An Approximate Shapley-Folkman Theorem
• Near-Optimal Rapid MPC using Neural Networks: A Primal-Dual Policy Learning Framework
• Sample Truncation for Scenario Approach to Closed-loop Chance Constrained Trajectory Optimization for Linear Systems
• Convexity and monotonicity in nonlinear optimal control under uncertainty