Necessary and sufficient conditions for quasiconvexity of a class of mixed-integer quadratic programs with applications in hybrid MPC

The paper derives necessary and sufficient conditions for quasiconvexity of piecewise quadratic functions. The conditions are stated in terms of linear inequalities which can be verified efficiently. To show the relevance of the result, the paper considers a class of hybrid MPC problems where the system model is piecewise affine and the control input is subject to constraints. Minimizing a quadratic cost results in a mixed integer quadratic program where the objective function is piecewise quadratic. Quasiconvexity can be determined using the result of the paper. The results of the present paper has potential to increase the applicability of hybrid model predictive control in high-speed control applications. In high speed applications, the only option has been to solve the mixed integer program explicitly and this quickly becomes intractable because of growing complexity. However, if the problem can be shown to be quasiconvex it opens up the possibility to use an efficient on-line approach. A hybrid MPC example is considered which is shown to be quasiconvex for a subset of the initial conditions.