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

Dayou Luo, Kazuya Echigo, Behçet Açıkmeşe

TL;DR

This work extends Lossless Convexification (LCvx) from continuous-time to discrete-time optimal control, establishing a rigorous framework that classifies discrete problems as normal or long-horizon. It proves that, after arbitrarily small perturbations to the dynamics, the LCvx solution satisfies the original nonconvex control constraints at all but at most $n_x-1$ grid points, with a probabilistic guarantee that this bound holds in the perturbation setting. For long-horizon cases, the paper introduces a two-phase strategy coupled with a bisection on the switching time $t_s$, leveraging the continuity of the value function to recover LCvx guarantees. The theoretical contributions are complemented by numerical examples (Moon-landing and artificial cases) illustrating perturbation-based violation control and the bisection method, thereby broadening LCvx applicability to practical discrete-time problems.

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.

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

TL;DR

This work extends Lossless Convexification (LCvx) from continuous-time to discrete-time optimal control, establishing a rigorous framework that classifies discrete problems as normal or long-horizon. It proves that, after arbitrarily small perturbations to the dynamics, the LCvx solution satisfies the original nonconvex control constraints at all but at most grid points, with a probabilistic guarantee that this bound holds in the perturbation setting. For long-horizon cases, the paper introduces a two-phase strategy coupled with a bisection on the switching time , leveraging the continuity of the value function to recover LCvx guarantees. The theoretical contributions are complemented by numerical examples (Moon-landing and artificial cases) illustrating perturbation-based violation control and the bisection method, thereby broadening LCvx applicability to practical discrete-time problems.

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 temporal grid points, where 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.

Paper Structure

This paper contains 20 sections, 9 theorems, 88 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notation
  3. Background
  4. Mechanism of LCvx
  5. LCvx on Normal cases
  6. Pertubation
  7. Quantify the LCvx invalidation
  8. LCvx on Long-horizon Cases
  9. Numerical Examples
  10. Well-Conditioned Example
  11. Artificial normal example
  12. Long-horizon example
  13. Conclusion
  14. Appendix
  15. Proof of theorem \ref{['thm: vt continuity']}
  16. ...and 5 more sections

Key Result

Theorem 8

Assume that the function $F$ and an optimal solution $(y^*, p_0)$ to Problem equ: general pertube satisfy all the assumptions in Lemma thm:clarkeexistance, $\Omega$ is compact, and $M$ is a locally Lipschitz function with respect to $(y, p)$. Then, The optimal value function ${M^*} \colon \mathbb{R}

Figures (3)

  • Figure 1: State trajectory and control magnitude for the Moon landing problem over a 60-second flight. (a) Position and velocity evolution, with velocity plotted on an inverted axis. (b) Control norm, thrust bounds, and $\sigma$ evolution. One LCvx invalid point occurs where the control norm falls below $\rho_{\min}$.
  • Figure 2: System dynamics and control response before and after perturbation for an artificial three-dimensional problem with $N = 10$ temporal grid points. The top figures display the state trajectory and control magnitude prior to perturbation, revealing LCvx invalid at three points. The bottom figures show the system's response to a perturbation designed as per Theorem \ref{['theorem: main_theorem']}, which reduces the number of nonconvex constraint violations to two. The state variable remains essentially unchanged.
  • Figure 3: State trajectory and control magnitude for the Moon landing problem over a 200-second flight. (a,c) Position and velocity evolution before and after the bisection search, with velocity plotted on an inverted axis. (b,d) Control norm, thrust bounds, and $\sigma$ evolution before and after the bisection search. Figures (a,b) indicate a long-horizon state where the control magnitudes remain below the threshold $\rho_{\text{min}}$. Figures (c,d) demonstrate the effectiveness of the bisection search method described in Algorithm \ref{['algo']}, resulting in a trajectory that exhibits an LCvx violation at one point. The information for $\sigma$ is not available before the transition point.

Theorems & Definitions (31)

  • Theorem 8
  • proof
  • Theorem 9
  • proof
  • Theorem 10: Sufficient Condition for LCvx
  • proof
  • proof
  • proof
  • proof
  • proof
  • ...and 21 more