Semismooth Newton methods with a shooting-like technique for solving a constrained free-boundary HJB equation.

*(English)*Zbl 1459.91160Summary: This paper describes novel numerical methods for solving a constrained free-boundary Hamilton-Jacobi-Bellman (HJB) equation. The equation comes from an optimal dividend problem within financial insurance and has a classical solution under strict assumptions. However the numerical approach is significant since the problem can be classified as a constrained variational inequality with free boundary, for which numerical methods are few. We combine the semismooth Newton method with a shooting-like method to treat the non-smoothness and free boundary of the problem, respectively. We introduce two algorithms, differing only in their treatment of the inequality constraints; the first applies a sweeping method, the second utilizes projected Newton. For the latter, the associated convergence analysis is discussed. In the end, we show that the local convergence rate for this method is at least superlinear. The contribution of this paper is a numerical approach that can be applied to more general free boundary problems with first or second derivative constraints for which analytical solutions are not known.

##### Keywords:

Hamilton-Jacobi-Bellman equation; free boundary; projected semismooth Newton; inequality state constraints; stochastic control##### Software:

KELLEY
PDF
BibTeX
XML
Cite

\textit{H. Jiang} and \textit{N. L. Gibson}, J. Comput. Appl. Math. 391, Article ID 113428, 13 p. (2021; Zbl 1459.91160)

Full Text:
DOI

##### References:

[1] | Fleming, W. H.; Soner, H. M., Controlled Markov Processes and Viscosity Solutions, Vol. 25 (2006), Springer Science & Business Media · Zbl 1105.60005 |

[2] | Kushner, H. J.; Dupuis, P. G., Numerical Methods for Stochastic Control Problems in Continuous Time, Vol. 24 (1992), Springer US |

[3] | Asmussen, S.; Taksar, M., Controlled diffusion models for optimal dividend pay-out, Insur. Math. Econ., 20, 1, 1-15 (1997) · Zbl 1065.91529 |

[4] | Jeanblanc-Picqué, M.; Shiryaev, A. N., Optimization of the flow of dividends, Russ. Math. Surv., 50, 2, 257-277 (1995) · Zbl 0878.90014 |

[5] | Ulbrich, M., Semismooth Newton Methods for Variational Inequalities and Constrained Optimization Problems in Function Spaces, Vol. 11 (2011), MOS-SIAM Series on Optimization · Zbl 1235.49001 |

[6] | Bertsekas, D. P., Projected Newton methods for optimization problems with simple constraints, SIAM J. Control Optim., 20, 2, 221-246 (1982) · Zbl 0507.49018 |

[7] | Qi, L.; Tong, X.; Li, D., Active-set projected trust-region algorithm for box-constrained nonsmooth equations, J. Optim. Theory Appl., 120, 3, 601-625 (2004) · Zbl 1140.65331 |

[8] | Ulbrich, M., Nonmonotone trust-region methods for bound-constrained semismooth equations with applications to nonlinear mixed complementarity problems, SIAM J. Optim., 11, 4, 889-917 (2001) · Zbl 1010.90085 |

[9] | Lenhart, S.; Workman, J. T., Optimal Control Applied to Biological Models (2007), Chapman and Hall/CRC · Zbl 1291.92010 |

[10] | Jiang, H., Stochastic and Numerical Analysis on Optimization Problems (2019), Oregon State University |

[11] | Clarke, F. H., Optimization and Nonsmooth Analysis, Vol. 5 (1990), Society for Industrial and Applied Mathematics |

[12] | Qi, L.; Sun, J., A nonsmooth version of Newton’s method, Math. Program., 58, 1-3, 353-367 (1993) · Zbl 0780.90090 |

[13] | Kelley, C., Iterative Methods for Optimization, Vol. 18 (1999), Society for Industrial and Applied Mathematics · Zbl 0934.90082 |

[14] | Isaacson, E.; Keller, H. B., Analysis of Numerical Methods (2012), Courier Corporation · Zbl 0168.13101 |

[15] | Kelley, C.; Sachs, E., Solution of optimal control problems by a pointwise projected Newton method, SIAM J. Control Optim., 33, 6, 1731-1757 (1995) · Zbl 0841.49013 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.