# Approximate Multiparametric Mixed-integer Convex Programming

## TL;DR

This paper introduces a massively parallel algorithm for generating explicit, near-optimal solutions to multiparametric mixed-integer convex programs, significantly speeding up hybrid model predictive control applications.

## Contribution

It presents a novel algorithm with a convergence metric and a static solution map, enabling faster, deterministic solutions for complex optimization problems.

## Key findings

- Achieves up to 1000x speedup over online optimization
- Provides a convergent solution with a new optimal cost overlap metric
- Supports massively parallel implementation on hundreds of processors

## Abstract

We propose an algorithm for generating explicit solutions of multiparametric mixed-integer convex programs to within a given suboptimality tolerance. The algorithm is applicable to a very general class of optimization problems, but is most useful for hybrid model predictive control, where on-line implementation is hampered by the worst-case exponential complexity of mixed-integer solvers. The output is a simplicial partition which defines a static map from the current state to a suboptimal solution. The primary theoretical contribution of this paper is to introduce a non-zero optimal cost overlap metric which is necessary and sufficient for convergence. The overlap size is also linked to partition complexity. The algorithm is massively parallelizable and our implementation, which is publicly available, is run on a cluster of several hundred processors. Not only does our solution have a deterministic runtime, simulations show that our approach is faster than on-line optimization by up to three orders of magnitude.