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Douglas-Rachford Algorithm for Control- and State-constrained Optimal Control Problems

Regina S. Burachik, Bethany I. Caldwell, C. Yalçın Kaya

TL;DR

This work develops a Douglas--Rachford framework to solve state- and control-constrained linear-quadratic optimal control problems in continuous time by splitting constraints into an affine ODE set $\cal A$ and a box set $\cal B$ and deriving proximal mappings $\operatorname{Prox}_f$ and $\operatorname{Prox}_g$. A numerically computed projector onto the affine set $\cal A$ is introduced to realize $\operatorname{Prox}_g$, requiring a TPBVP solved via shooting, while $\operatorname{Prox}_f$ uses simple bound-clamping for the state and control components. A conjecture relates DR-based costates $\lambda^{DR}$ to the true costate $\lambda$, enabling practical verification of optimality, particularly at junction times where constraints switch activity. Numerical experiments on a harmonic oscillator and a mass–spring system show that DR can yield higher accuracy and faster computation than direct discretization with Ipopt in many scenarios, albeit with limitations in problems featuring discontinuous controls. The results suggest a versatile projection-based approach for convex continuous-time OCPs and point toward extensions to more general convex objectives and alternative projection schemes.

Abstract

We consider the application of the Douglas-Rachford (DR) algorithm to solve linear-quadratic (LQ) control problems with box constraints on the state and control variables. We split the constraints of the optimal control problem into two sets: one involving the ODE with boundary conditions, which is affine, and the other a box. We rewrite the LQ control problems as the minimization of the sum of two convex functions. We find the proximal mappings of these functions which we then employ for the projections in the DR iterations. We propose a numerical algorithm for computing the projection onto the affine set. We present a conjecture for finding the costates and the state constraint multipliers of the optimal control problem, which can in turn be used in verifying the optimality conditions. We carry out numerical experiments with two constrained optimal control problems to illustrate the working and the efficiency of the DR algorithm compared to the traditional approach of direct discretization.

Douglas-Rachford Algorithm for Control- and State-constrained Optimal Control Problems

TL;DR

This work develops a Douglas--Rachford framework to solve state- and control-constrained linear-quadratic optimal control problems in continuous time by splitting constraints into an affine ODE set and a box set and deriving proximal mappings and . A numerically computed projector onto the affine set is introduced to realize , requiring a TPBVP solved via shooting, while uses simple bound-clamping for the state and control components. A conjecture relates DR-based costates to the true costate , enabling practical verification of optimality, particularly at junction times where constraints switch activity. Numerical experiments on a harmonic oscillator and a mass–spring system show that DR can yield higher accuracy and faster computation than direct discretization with Ipopt in many scenarios, albeit with limitations in problems featuring discontinuous controls. The results suggest a versatile projection-based approach for convex continuous-time OCPs and point toward extensions to more general convex objectives and alternative projection schemes.

Abstract

We consider the application of the Douglas-Rachford (DR) algorithm to solve linear-quadratic (LQ) control problems with box constraints on the state and control variables. We split the constraints of the optimal control problem into two sets: one involving the ODE with boundary conditions, which is affine, and the other a box. We rewrite the LQ control problems as the minimization of the sum of two convex functions. We find the proximal mappings of these functions which we then employ for the projections in the DR iterations. We propose a numerical algorithm for computing the projection onto the affine set. We present a conjecture for finding the costates and the state constraint multipliers of the optimal control problem, which can in turn be used in verifying the optimality conditions. We carry out numerical experiments with two constrained optimal control problems to illustrate the working and the efficiency of the DR algorithm compared to the traditional approach of direct discretization.
Paper Structure (12 sections, 2 theorems, 47 equations, 3 figures, 3 tables, 2 algorithms)

This paper contains 12 sections, 2 theorems, 47 equations, 3 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Optimal Control Problem
  3. Optimality conditions
  4. Splitting and Proximal Mappings
  5. Douglas--Rachford Algorithm
  6. Algorithm for projector onto $\cal A$
  7. A conjecture for the costates for Problem (P)
  8. Numerical Experiments
  9. Harmonic oscillator
  10. Simple spring--mass system
  11. Numerical discussion and comparisons
  12. Conclusion

Key Result

Theorem 1

The proximal mapping of $f$ is given as $\operatorname{Prox}_f(x^-,u^-) = (y,v)$ such that the components of $y$ and $v$ are expressed as for all $t\in[t_0,t_f],\ i=1,\dots,n, \ j=1,\dots,m$.

Figures (3)

  • Figure 1: (PHO) Case 2 plots (see Table \ref{['tbl:probs']}) using DR with $N=10^3, -0.025\leq x_1(t)$. Vertical lines indicate the interval in which the state constraint becomes active.
  • Figure 2: (PSM) Case 2 plots (see Table \ref{['tbl:probs']}) using DR with $N=10^3, -0.2\leq x_1(t)$. Vertical lines indicate the interval in which the state constraint becomes active.
  • Figure 3: Multipliers $\mu_1^2,\mu_2^2$ for Case 2 (PHO) with $N=10^3$ using DR (solid lines) and Ipopt (dotted lines). Note that $\mu_1^2(t)$ found by Ipopt attains a maximum value of 33, which is not shown in the graph.

Theorems & Definitions (9)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Remark 1
  • Remark 2
  • Remark 3
  • Conjecture 1
  • Remark 4