A logarithmic-time solution to the point location problem for parametric linear programming

The optimiser of a (multi) parametric linear program (pLP) is a piecewise affine function defined over a polyhedral subdivision of the set of feasible states. Once this affine function has been pre-calculated, the optimal solution can be computed for a particular parameter by determining the region that contains it. This is the so-called point location problem. In this paper, we show that this problem can be written as an additively weighted nearest neighbour search that can be solved in time linear in the dimension of the state space and logarithmic in the number of regions. It is well-known that linear model predictive control (MPC) problems based on linear control objectives (e.g., 1- or -norm) can be posed as pLPs, and on-line calculation of the control law involves the solution to the point location problem. Several orders of magnitude sampling speed improvement are demonstrated over traditional MPC and closed-form MPC schemes using the proposed scheme.