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Solution method for discrete double obstacle problems based on a power penalty approach
1. | Shenzhen Audencia Business School, WeBank Institute of Fintech, Shenzhen University, Shenzhen 518060, China |
2. | Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong, China |
3. | Department of Mathematics and Statistics, Curtin University, GPO Box U1987, Perth, WA 6845, Australia |
We develop a power penalty approach to a finite-dimensional double obstacle problem. This problem is first approximated by a system of nonlinear equations containing two penalty terms. We show that the solution to this penalized equation converges to that of the original obstacle problem at an exponential rate when the coefficient matrices are $ M $-matrices. Numerical examples are presented to confirm the theoretical findings and illustrate the efficiency and effectiveness of the new method.
References:
[1] |
C. Avramescu,
A generalization of Miranda's theorem, Semin. Fixed Point Theory Cluj-Napoca, 3 (2002), 121-127.
|
[2] |
O. Bokanowski, S. Maroso and H. Zidani,
Some convergence results for Howard's algorithm, SIAM J. Numer. Anal., 47 (2009), 3001-3026.
doi: 10.1137/08073041X. |
[3] |
L. A. Caffarelli and R. J. McCann,
Free boundaries in optimal transport and Monge-Ampere obstacle problems, Ann. Math., 171 (2010), 673-730.
doi: 10.4007/annals.2010.171.673. |
[4] |
M. Dai and F. Yi,
Finite-horizon optimal investment with transaction costs: A parabolic double obstacle problem, J. Differ. Equ., 246 (2009), 1445-1469.
doi: 10.1016/j.jde.2008.11.003. |
[5] |
J. E. Dennis Jr. and R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice Hall, Inc., Englewood Cliffs, NJ, 1983. |
[6] |
G. Duvaut and J.-L. Lions, Inequalities in Mechanics and Physics, Springer-Verlag, Berlin, 1976. |
[7] |
D. Han and J. W. L. Wan, Multigrid methods for second order Hamilton–Jacobi–Bellman and Hamilton–Jacobi–Bellman–Isaacs equations, SIAM J. Sci. Comput., 35 (2013), S323–S344.
doi: 10.1137/120881476. |
[8] |
T. Kärkkäinen, K. Kunisch and P. Tarvainen,
Augmented Lagrangian active set methods for obstacle problems, J. Optim. Theory Appl., 119 (2003), 499-533.
doi: 10.1023/B:JOTA.0000006687.57272.b6. |
[9] |
R. Kornhuber,
Monotone multigrid methods for elliptic variational inequalities I, Numer. Math., 69 (1994), 167-184.
doi: 10.1007/BF03325426. |
[10] |
P. Kovalov and V. Linetsky, Valuing convertible bonds with stock price, volatility, interest rate, and default risk, FDIC Center for Financial Research Working Paper Series, 2008. Google Scholar |
[11] |
Y. Peres, O. Schramm, S. Sheffield and D. B. Wilson,
Tug-of-war and the infinity Laplacian, J. Amer. Math. Soc., 22 (2009), 167-210.
doi: 10.1090/S0894-0347-08-00606-1. |
[12] |
Z. Sun, Z. Liu and X. Yang,
On power penalty methods for linear complementarity problems arising from American option pricing, J. Glob. Optim., 63 (2015), 165-180.
doi: 10.1007/s10898-015-0291-6. |
[13] |
S. Wang and X. Yang,
A power penalty method for a bounded nonlinear complementarity problem, Optimization, 64 (2015), 2377-2394.
doi: 10.1080/02331934.2014.967236. |
[14] |
S. Wang, X. Q. Yang and K. L. Teo,
Power penalty method for a linear complementarity problem arising from American option valuation, J. Optim. Theory Appl., 129 (2006), 227-254.
doi: 10.1007/s10957-006-9062-3. |
[15] |
J. H. Witte and C. Reisinger,
A penalty method for the numerical solution of Hamilton-Jacobi-Bellman (HJB) equations in finance, SIAM J. Numer. Anal., 49 (2011), 213-231.
doi: 10.1137/100797606. |
[16] |
K. Zhang and X. Yang,
A power penalty method for discrete HJB equations, Optim. Lett., 14 (2020), 1419-1433.
doi: 10.1007/s11590-019-01517-7. |
[17] |
K. Zhang, X. Q. Yang, S. Wang and K. L. Teo,
Numerical performance of penalty method for American option pricing, Optim. Methods Softw., 25 (2010), 737-752.
doi: 10.1080/10556780903051930. |
[18] |
J.-X. Zhao and S. Wang,
An interior penalty approach to a large-scale discretized obstacle problem with nonlinear constraints, Numer. Algorithms, 85 (2020), 571-589.
doi: 10.1007/s11075-019-00827-2. |
[19] |
Y. Y. Zhou, S. Wang and X. Q. Yang,
A penalty approximation method for a semilinear parabolic double obstacle problem, J. Glob. Optim., 60 (2014), 531-550.
doi: 10.1007/s10898-013-0122-6. |
show all references
References:
[1] |
C. Avramescu,
A generalization of Miranda's theorem, Semin. Fixed Point Theory Cluj-Napoca, 3 (2002), 121-127.
|
[2] |
O. Bokanowski, S. Maroso and H. Zidani,
Some convergence results for Howard's algorithm, SIAM J. Numer. Anal., 47 (2009), 3001-3026.
doi: 10.1137/08073041X. |
[3] |
L. A. Caffarelli and R. J. McCann,
Free boundaries in optimal transport and Monge-Ampere obstacle problems, Ann. Math., 171 (2010), 673-730.
doi: 10.4007/annals.2010.171.673. |
[4] |
M. Dai and F. Yi,
Finite-horizon optimal investment with transaction costs: A parabolic double obstacle problem, J. Differ. Equ., 246 (2009), 1445-1469.
doi: 10.1016/j.jde.2008.11.003. |
[5] |
J. E. Dennis Jr. and R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice Hall, Inc., Englewood Cliffs, NJ, 1983. |
[6] |
G. Duvaut and J.-L. Lions, Inequalities in Mechanics and Physics, Springer-Verlag, Berlin, 1976. |
[7] |
D. Han and J. W. L. Wan, Multigrid methods for second order Hamilton–Jacobi–Bellman and Hamilton–Jacobi–Bellman–Isaacs equations, SIAM J. Sci. Comput., 35 (2013), S323–S344.
doi: 10.1137/120881476. |
[8] |
T. Kärkkäinen, K. Kunisch and P. Tarvainen,
Augmented Lagrangian active set methods for obstacle problems, J. Optim. Theory Appl., 119 (2003), 499-533.
doi: 10.1023/B:JOTA.0000006687.57272.b6. |
[9] |
R. Kornhuber,
Monotone multigrid methods for elliptic variational inequalities I, Numer. Math., 69 (1994), 167-184.
doi: 10.1007/BF03325426. |
[10] |
P. Kovalov and V. Linetsky, Valuing convertible bonds with stock price, volatility, interest rate, and default risk, FDIC Center for Financial Research Working Paper Series, 2008. Google Scholar |
[11] |
Y. Peres, O. Schramm, S. Sheffield and D. B. Wilson,
Tug-of-war and the infinity Laplacian, J. Amer. Math. Soc., 22 (2009), 167-210.
doi: 10.1090/S0894-0347-08-00606-1. |
[12] |
Z. Sun, Z. Liu and X. Yang,
On power penalty methods for linear complementarity problems arising from American option pricing, J. Glob. Optim., 63 (2015), 165-180.
doi: 10.1007/s10898-015-0291-6. |
[13] |
S. Wang and X. Yang,
A power penalty method for a bounded nonlinear complementarity problem, Optimization, 64 (2015), 2377-2394.
doi: 10.1080/02331934.2014.967236. |
[14] |
S. Wang, X. Q. Yang and K. L. Teo,
Power penalty method for a linear complementarity problem arising from American option valuation, J. Optim. Theory Appl., 129 (2006), 227-254.
doi: 10.1007/s10957-006-9062-3. |
[15] |
J. H. Witte and C. Reisinger,
A penalty method for the numerical solution of Hamilton-Jacobi-Bellman (HJB) equations in finance, SIAM J. Numer. Anal., 49 (2011), 213-231.
doi: 10.1137/100797606. |
[16] |
K. Zhang and X. Yang,
A power penalty method for discrete HJB equations, Optim. Lett., 14 (2020), 1419-1433.
doi: 10.1007/s11590-019-01517-7. |
[17] |
K. Zhang, X. Q. Yang, S. Wang and K. L. Teo,
Numerical performance of penalty method for American option pricing, Optim. Methods Softw., 25 (2010), 737-752.
doi: 10.1080/10556780903051930. |
[18] |
J.-X. Zhao and S. Wang,
An interior penalty approach to a large-scale discretized obstacle problem with nonlinear constraints, Numer. Algorithms, 85 (2020), 571-589.
doi: 10.1007/s11075-019-00827-2. |
[19] |
Y. Y. Zhou, S. Wang and X. Q. Yang,
A penalty approximation method for a semilinear parabolic double obstacle problem, J. Glob. Optim., 60 (2014), 531-550.
doi: 10.1007/s10898-013-0122-6. |



Ra | Ra | ||||||||
Ra | Ra | ||||||||
Modified policy iter. | |||
Iterations | 9 | 12 | 88 |
Modified policy iter. | |||
Iterations | 9 | 12 | 88 |
Gauss-Seidel | Active set | ||||
Iterations | 11 | 16 | 426 | Failed | |
Time (s) | 1.26 | 1.89 | 32.81 | Failed | |
Iterations | 15 | 17 | 740 | 17 | |
Time (s) | 7.78 | 8.96 | 198.7 | 6.43 |
Gauss-Seidel | Active set | ||||
Iterations | 11 | 16 | 426 | Failed | |
Time (s) | 1.26 | 1.89 | 32.81 | Failed | |
Iterations | 15 | 17 | 740 | 17 | |
Time (s) | 7.78 | 8.96 | 198.7 | 6.43 |
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