Verification and Validation of Convex Optimization Algorithms for Model Predictive Control

Abstract

Advanced embedded algorithms are growing in complexity and they are an essential contributor to the growth of autonomy in many areas. However, the promise held by these algorithms cannot be kept without proper attention to the considerably stronger design constraints that arise when the applications of interest, such as aerospace systems, are safety-critical. Formal verification is the process of proving or disproving the "correctness" of an algorithm with respect to a certain mathematical description of it by means of a computer. This article discusses the formal verification of the Ellipsoid method, a convex optimization algorithm, and its code implementation as it applies to receding horizon control. Options for encoding code properties and their proofs are detailed. The applicability and limitations of those code properties and proofs are presented as well. Finally, floating-point errors are taken into account in a numerical analysis of the Ellipsoid algorithm. Modifications to the algorithm are presented which can be used to control its numerical stability.

Verified reductions for optimization

“…Narkawicz and Muñoz [34] have devised a verified numeric algorithm to find bounds and global optima. Cohen et al [10,11] have developed a framework for verifying optimization algorithms using the ANSI/ISO C Specification Language (ACSL) [5].…”

Verified reductions for optimization

“…Narkawicz and Muñoz [35] have devised a verified numeric algorithm to find bounds and global optima. Cohen et al [11,12] have developed a framework for verifying optimization algorithms using the ANSI/ISO C Specification Language (ACSL) [5].…”

A New Algorithm for Linear Programming in Critical Systems

“…Linear programming does not allow to avoid the previous discussion about binary issues. On one hand, [11,8] manage to apply formal verification to the ellipsoid algorithm [15] with simple floating point representation using prior on the inputs. This result is important to advances on the application of a linear program solver on a critical task.…”

“…Another one is to rely on standard routines and/or with the simplest possible algorithm. This explains why there is still research on old algorithm like the ellipsoid method [11,8]: even if it is the worst polynomial time algorithm for linear programming, it has some good features like the fact that it does not require matrix inversion. This also explains why, neither [20] nor [23] are fully satisfying as they require either custom routines or custom data-structure.…”