Implementation and acceleration of optimal control for systems biology

Abstract

Optimal control theory provides insight into complex resource allocation decisions. The forward–backward sweep method (FBSM) is an iterative technique commonly implemented to solve two-point boundary value problems arising from the application of Pontryagin’s maximum principle (PMP) in optimal control. The FBSM is popular in systems biology as it scales well with system size and is straightforward to implement. In this review, we discuss the PMP approach to optimal control and the implementation of the FBSM. By conceptualizing the FBSM as a fixed point iteration process, we leverage and adapt existing acceleration techniques to improve its rate of convergence. We show that convergence improvement is attainable without prohibitively costly tuning of the acceleration techniques. Furthermore, we demonstrate that these methods can induce convergence where the underlying FBSM fails to converge. All code used in this work to implement the FBSM and acceleration techniques is available on GitHub at https://github.com/Jesse-Sharp/Sharp2021 .

Optimal vaccine allocation for the control of sexually transmitted infections

“…is commonly used to improve convergence properties (Lenhart and Workman, 2007). The choice of µ significantly accelerates convergence relative to the direct update (Sharp et al, 2021), particularly if µ is updated between iterations. However, since the optimal choice for µ is problem-dependent, here we use a constant µ = 0.9 for all iterations.…”

“…However, since the optimal choice for µ is problem-dependent, here we use a constant µ = 0.9 for all iterations. Convergence properties for the FBSM for this case are discussed in (Lenhart and Workman, 2007;Sharp et al, 2021). As a convergence criterion, we require the relative error to be negligibly small so that…”

Understanding cellular growth strategies via optimal control

“…Incorporating more complex and general environmental feedback and resource dynamics to our modelling framework will certainly allow for important insights. In these cases, it may be necessary to adopt other mathematical tools than the Bellman equations [ 21 ], if computational burden becomes otherwise too large. Therapies against cancer cell, microbial, pest and parasite populations can all be seen via the lens of systematic growth manipulation.…”

“…Earlier applications can be found from the field of behavioural ecology, including studies to solve optimal age- and size-dependent life-history strategies [ 16 – 18 ]. Control theory has also been used to study how evolving populations use information and possible modes of transmitting genetic material [ 19 , 20 ], and there is a rising interest on the systems biology side as well [ 21 , 22 ].…”

On the numerical stability of discretised Optimal Control Problems

“…Based on the sequence of optimisation and time discretisation, solution procedures for Optimal Control Problems (OCPs) can be classified as indirect and direct approaches. The indirect method first derives the necessary optimality conditions and forms the Differential-Algebraic Equations (DAEs) with the twopoint boundary conditions, also known as the Two-Point Boundary Value Problem (TPBVP) or Hamiltonian Boundary-Value Problem (HBVP), which are then discretisated in time, yielding a discrete system [9,10]. Conversely, the direct method first introduces the time discretisation of the continuous system and then the necessary optimality conditions, derived from the resulting discrete system [1].…”