Convex hull of two quadratic constraints is an LMI set

Abstract

In this work, we are interested in the convex hull of the region determined by two quadratic polynomial constraints. Our main result is that if this region is nonempty, its convex hull is either IR or the feasible set of another pair of quadratic constraints which are, in fact, positive linear combinations of the original ones. We propose an algorithm to find these positive combinations efficiently and convert them into linear matrix inequalities (LMI). This result can be utilized to find the optimal value of a linear objective function over a set determined by two general quadratic inequality constraints. As a result, we show that such problems can be solved in polynomial time. To the best of our knowledge, this was not known until now. When the literature is investigated it can be seen that one of the most suitable approaches that can be utilized to obtain the solution of this problem are relaxation methods. However, using relaxations one can efficiently compute bounds on the global optimal solution which may not turn out to be exact. We show that our approach always finds the exact solution more efficiently (asymptotically) when compared with the available relaxation. Moreover, by an example, we also show that our method may perform way better these approaches for some choice of quadratic constraints. Lastly, we show how the results developed can be applied to a certain class of control problems.