  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.
