An Ellipsoid Algorithm for linear optimization with uncertain LMI constraints

In this paper, an efficient algorithm based on the ellipsoid method is proposed to solve a linear optimization problem over a set of uncertain Linear Matrix Inequalities (LMIs). First, an Ellipsoid Algorithm (EA) with deep cuts is introduced for solving the set of uncertain LMIs. The proposed ellipsoid algorithm is shown to converge to a probabilistically feasible point with high confidence level and in fewer iterations compared to other EA methods. Then, through a set of new cuts, the objective function is minimized while maintaining the probabilistic feasibility of the solution.