Combination of Direct Methods and Homotopy in Numerical Optimal Control: Application to the Optimization of Chemotherapy in Cancer

Antoine Olivier () and Camille Pouchol ()
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Antoine Olivier: Sorbonne Universités
Camille Pouchol: Sorbonne Universités

Journal of Optimization Theory and Applications, 2019, vol. 181, issue 2, No 7, 479-503

Abstract: Abstract We consider a state-constrained optimal control problem of a system of two non-local partial differential equations, which is an extension of the one introduced in a previous work in mathematical oncology. The aim is to control the tumor size through chemotherapy while avoiding the emergence of resistance to the drugs. The numerical approach to solve the problem was the combination of direct methods and continuation on discretization parameters, which happen to be insufficient for the more complicated model, where diffusion is added to account for mutations. In the present paper, we propose an approach relying on changing the problem so that it can theoretically be solved thanks to a Pontryagin’s maximum principle in infinite dimension. This provides an excellent starting point for a much more reliable and efficient algorithm combining direct methods and continuations. The global idea is new and can be thought of as an alternative to other numerical optimal control techniques.

Keywords: Optimal control; Direct methods; Homotopy; Mathematical oncology; Pontryagin’s Maximum Principle; 49M05; 49M25; 92C50 (search for similar items in EconPapers)
Date: 2019
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DOI: 10.1007/s10957-018-01461-z

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