Piecewise regression via mixed-integer programming for MPC

TL;DR

This paper introduces a novel mixed-integer programming approach for piecewise regression that produces globally optimal, fast-evaluating functions suitable for control applications, overcoming limitations of existing methods.

Contribution

The paper presents a new MIP-based method for piecewise regression that is not limited to specific function classes and is efficient for control use.

Findings

Provides globally optimal piecewise functions

Functions are fast to evaluate and suitable for control

Overcomes limitations of neural network training and existing MIP methods

Abstract

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.