Comparison of Norm Minimizations in Scenario Approach and Its Application to Robust Optimal Control

In this paper, we investigate the effect of norm selection in the scenario approach to robust feasibility problems. A robust feasibility problem can be converted into a robust optimization problem by minimizing a fictitious cost function, and the scenario approach is a method to find a stochastic approximation of the optimal solution in an affordable computational time. When considering a norm of the decision variable as the fictitious cost function, the choice of the norm may affect the robustness of the solution obtained by the scenario approach. To demonstrate the effect of norm selection, we compare the 2-, ∞-, and 1-norms by simulating two illustrative examples. Moreover, the effectiveness of the ∞- and 1-norms compared to the 2-norm is demonstrated in a numerical example of a finite-horizon optimal control of uncertain linear systems.