Zeroth-order optimisation on subsets of symmetric matrices with application to MPC tuning

This paper provides a zeroth-order optimisation framework for non-smooth and possibly non-convex cost functions with matrix parameters that are real and symmetric. We provide complexity bounds on the number of iterations required to ensure a given accuracy level for both the convex and non-convex case. The derived complexity bounds for the convex case are less conservative than available bounds in the literature since we exploit the symmetric structure of the underlying matrix space. Moreover, the non-convex complexity bounds are novel for the class of optimisation problems we consider. The utility of the framework is evident in the suite of applications that use symmetric matrices as tuning parameters. Of primary interest here is the challenge of tuning the gain matrices in model predictive controllers, as this is a challenge known to be inhibiting industrial implementation of these architectures. To demonstrate the framework we consider the problem of MIMO diesel air-path control, and consider implementing the framework iteratively ``in-the-loop'' to reduce tracking error on the output channels. Both simulations and experimental results are included to illustrate the effectiveness of the proposed framework over different engine drive cycles.