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This paper provides a zeroth-order optimisation framework for non-smooth andpossibly non-convex cost functions with matrix parameters that are real andsymmetric. We provide complexity bounds on the number of iterations required toensure a given accuracy level for both the convex and non-convex case. Thederived complexity bounds for the convex case are less conservative thanavailable bounds in the literature since we exploit the symmetric structure ofthe underlying matrix space. Moreover, the non-convex complexity bounds arenovel for the class of optimisation problems we consider. The utility of theframework is evident in the suite of applications that use symmetric matricesas tuning parameters. Of primary interest here is the challenge of tuning thegain matrices in model predictive controllers, as this is a challenge known tobe inhibiting industrial implementation of these architectures. To demonstratethe framework we consider the problem of MIMO diesel air-path control, andconsider implementing the framework iteratively ``in-the-loop'' to reducetracking error on the output channels. Both simulations and experimentalresults are included to illustrate the effectiveness of the proposed frameworkover different engine drive cycles.

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