Distribution-Dependent Robust Linear Optimization with Asymmetric Uncertainty and Application to Optimal Control

We consider a linear programming problem in which the constraint matrix is uncertain. Each element of the constraint matrix is modeled as a random variable whose range is asymmetrically bounded around its mean. We construct a formulation that yields a solution with a better objective value, compared to the classical robust optimization approach, while taking the risk that the solution may become infeasible to the original problem. We address the risk by establishing upper bounds on the probability that it violates the constraints of the problem. These bounds exploit full distributional information on the random elements or limited distributional information such as the true means or sample means of the random elements. We explore the application of our methodology to the optimal control of linear uncertain systems with constraints.