Concave programming in control theory

We show in the present paper that many open and challenging problems in control theory belong the the class of concave minimization programs. More precisely, these problems can be recast as the minimization of a concave objective function over convex LMI (Linear Matrix Inequality) constraints. As concave programming is the best studied class of problems in global optimization, several concave programs such as simplicial and conical partitioning algorithms can be used for the resolution. Moreover, these global techniques can be combined with a local Frank and Wolfe feasible direction algorithm and improved by the use of specialized stopping criteria, hence reducing the overall computational overhead. In this respect, the proposed hybrid optimization scheme can be considered as a new line of attack for solving hard control problems. Computational experiments indicate the viability of our algorithms, and that in the worst case they require the solution of a few LMI programs. Power and efficiency of the algorithms are demonstrated for a realistic inverted-pendulum control problem. Overall, this dedication reflects the key role that concavity and LMIs play in difficult control problems.