**Reference:**

J. Xu,
T. van den Boom, and
B. De Schutter,
"Optimistic optimization for continuous nonconvex piecewise affine
functions," *Automatica*, Mar. 2021. Article 109476.

**Abstract:**

This paper considers the global optimization problem of a continuous
nonconvex piecewise affine (PWA) function over a polytope. This type
of optimization problem often arises in the context of control of
continuous PWA systems. In literature, it has been shown that the
given problem can be formulated as a mixed integer linear programming
problem, the worst-case complexity of which grows exponentially with
the number of polyhedral subregions in the domain of the PWA function.
In this paper, we propose a solution approach that is more efficient
for continuous PWA functions with a large number of polyhedral
subregions. The proposed approach is founded on optimistic
optimization, which is based on hierarchical partitioning of the
feasible set and which focuses on the most promising region when
searching for the global optimum. The advantage of optimistic
optimization is that one can guarantee bounds on the suboptimality
with respect to the global optimum given a finite computational budget
(e.g. the number of iterations). In particular, the gap between the
best value returned by the proposed algorithm and the real optimum can
be made arbitrarily small as the computational budget increases. We
derive the analytic expressions for the core parameters required by
optimistic optimization for continuous PWA functions. The efficiency
of the resulting algorithm is illustrated with numerical examples.

Online version of the paper

@article{Xuvan:15-030,

author={J. Xu and T. van den Boom and B. {D}e Schutter},

title={Optimistic optimization for continuous nonconvex piecewise affine functions},

journal={Automatica},

month=mar,

year={2021},

note={Article 109476},

doi={10.1016/j.automatica.2020.109476}

}

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