  Piecewise regression is a versatile approach used in various disciplines to
approximate complex functions from limited, potentially noisy data points. In
control, piecewise regression is, e.g., used to approximate the optimal control
law of model predictive control (MPC), the optimal value function, or unknown
system dynamics. Neural networks are a common choice to solve the piecewise
regression problem. However, due to their nonlinear structure, training is
often based on gradient-based methods, which may fail to find a global optimum
or even a solution that leads to a small approximation error. To overcome this
problem and to find a global optimal solution, methods based on mixed-integer
programming (MIP) can be used. However, the known MIP-based methods are either
limited to a special class of functions, e.g., convex piecewise affine
functions, or they lead to complex approximations in terms of the number of
regions of the piecewise defined function. Both complicate a usage in the
framework of control. We propose a new MIP-based method that is not restricted
to a particular class of piecewise defined functions and leads to functions
that are fast to evaluate and can be used within an optimization problem,
making them well suited for use in control.
