Inverse Parametric Quadratic Programming and an Application to Hybrid Control

A. Hempel, P. J. Goulart and J. Lygeros

@inproceedings{HGL:2012,
  author = {A. Hempel and P. J. Goulart and J. Lygeros},
  title = {Inverse Parametric Quadratic Programming and an Application to Hybrid Control},
  booktitle = {IFAC Workshop on Nonlinear Model Predictive Control},
  year = {2012},
  pages = {68-73},
  url = {http://dx.doi.org/10.3182/20120823-5-NL-3013.00033},
  doi = {10.3182/20120823-5-NL-3013.00033}
}

We present the complete solution to the inverse parametric quadratic programming problem: from a given continuous piecewise affine function we construct both the constraints and objective function of a parametric quadratic program, such that the supplied function is the unique parametric minimizer for the constructed problem data. In contrast to past approaches to this problem, our method does not rely on prior knowledge of the constraint set or sufficient sampling of the optimizer function, and is guaranteed to solve the inverse optimization problem exactly if a solution exists. We then apply this inverse optimization technique to the control of piecewise affine systems. By recasting the hybrid system dynamics as the parametric solution to a quadratic program obtained from our inverse optimization technique, we derive an equivalent linear complementarity model via the Karush-Kuhn-Tucker conditions of the identified optimization problem. This approach allows one to solve an optimal control problem for a piecewise affine system by solving a mathematical program with equilibrium constraints. Simulation results suggest that the computational effort required to solve such problems can be significantly smaller than that required for conventional mixed-integer quadratic programming approaches for systems with piecewise affine dynamics. We demonstrate via two numerical examples that globally optimal points can be identified using this approach.