Revisiting Lossless Convexification: Theoretical Guarantees for Discrete-time Optimal Control Problems

TL;DR

This paper extends Lossless Convexification (LCvx) to discrete-time optimal control problems, providing theoretical guarantees and methods for normal and long-horizon cases, thus broadening its practical applicability.

Contribution

It generalizes LCvx to discrete-time problems, introduces a classification into normal and long-horizon cases, and proposes solutions including a bisection method for the latter.

Findings

LCvx applied to discrete-time problems yields near-constraint satisfaction at most n_x - 1 points.

A bisection search combined with LCvx handles long-horizon cases effectively.

Theoretical guarantees are established for the extended LCvx method.

Abstract

Lossless Convexification (LCvx) is a modeling approach that transforms a class of nonconvex optimal control problems, where nonconvexity primarily arises from control constraints, into convex problems through convex relaxations. These convex problems can be solved using polynomial-time numerical methods after discretization, which converts the original infinite-dimensional problem into a finite-dimensional one. However, existing LCvx theory is limited to continuous-time optimal control problems, as the equivalence between the relaxed convex problem and the original nonconvex problem holds only in continuous time. This paper extends LCvx to discrete-time optimal control problems by classifying them into normal and long-horizon cases. For normal cases, after an arbitrarily small perturbation to the system dynamics (recursive equality constraints), applying the existing LCvx method to discrete-time problems results in optimal controls that meet the original nonconvex constraints at all but no more than $n_x - 1$ temporal grid points, where $n_x$ is the state dimension. For long-horizon cases, the existing LCvx method fails, but we resolve this issue by integrating it with a bisection search, leveraging the continuity of the value function from the relaxed convex problem to achieve similar results as in normal cases. This paper improves the theoretical foundation of LCvx, expanding its applicability to real-world discrete-time optimal control problems.