On the sample size of random convex programs with structured dependence on the uncertainty

X. Zhang, S. Grammatico, G. Schildbach, P. J. Goulart and J. Lygeros

Automatica, vol. 60, pp. 182-188, October 2015.
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  author = {X. Zhang and S. Grammatico and G. Schildbach and P. J. Goulart and J. Lygeros},
  title = {On the sample size of random convex programs with structured dependence on the uncertainty},
  journal = {Automatica},
  year = {2015},
  volume = {60},
  pages = {182-188},
  url = {http://www.sciencedirect.com/science/article/pii/S0005109815002988},
  doi = {10.1016/j.automatica.2015.07.013}

Many control design problems subject to uncertainty can be cast as chance constrained optimization programs. The Scenario Approach provides an intuitive way to address these problems by replacing the chance constraint with a finite number of sampled constraints (scenarios). The sample size critically depends on the so-called Helly?s dimension, which is 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/violation probability. This paper derives 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 lower objective value and reduced computational complexity. The efficacy of the proposed bound is demonstrated on an inventory management example, and is in general applicable to randomized Model Predictive Control of chance constrained linear systems with additive uncertain input.