• Previous Article
    A self adaptive inertial algorithm for solving split variational inclusion and fixed point problems with applications
  • JIMO Home
  • This Issue
  • Next Article
    Approach to the consistency and consensus of Pythagorean fuzzy preference relations based on their partial orders in group decision making
doi: 10.3934/jimo.2021018

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

* Corresponding author: Kai Zhang

Received  July 2020 Revised  October 2020 Published  December 2020

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.

Citation: Kai Zhang, Xiaoqi Yang, Song Wang. Solution method for discrete double obstacle problems based on a power penalty approach. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021018
References:
[1]

C. Avramescu, A generalization of Miranda's theorem, Semin. Fixed Point Theory Cluj-Napoca, 3 (2002), 121-127.   Google Scholar

[2]

O. BokanowskiS. Maroso and H. Zidani, Some convergence results for Howard's algorithm, SIAM J. Numer. Anal., 47 (2009), 3001-3026.  doi: 10.1137/08073041X.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

J. E. Dennis Jr. and R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice Hall, Inc., Englewood Cliffs, NJ, 1983.  Google Scholar

[6]

G. Duvaut and J.-L. Lions, Inequalities in Mechanics and Physics, Springer-Verlag, Berlin, 1976.  Google Scholar

[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.  Google Scholar

[8]

T. KärkkäinenK. 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.  Google Scholar

[9]

R. Kornhuber, Monotone multigrid methods for elliptic variational inequalities I, Numer. Math., 69 (1994), 167-184.  doi: 10.1007/BF03325426.  Google Scholar

[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. PeresO. SchrammS. 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.  Google Scholar

[12]

Z. SunZ. 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.  Google Scholar

[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.  Google Scholar

[14]

S. WangX. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

K. ZhangX. Q. YangS. 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.  Google Scholar

[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.  Google Scholar

[19]

Y. Y. ZhouS. 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.  Google Scholar

show all references

References:
[1]

C. Avramescu, A generalization of Miranda's theorem, Semin. Fixed Point Theory Cluj-Napoca, 3 (2002), 121-127.   Google Scholar

[2]

O. BokanowskiS. Maroso and H. Zidani, Some convergence results for Howard's algorithm, SIAM J. Numer. Anal., 47 (2009), 3001-3026.  doi: 10.1137/08073041X.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

J. E. Dennis Jr. and R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice Hall, Inc., Englewood Cliffs, NJ, 1983.  Google Scholar

[6]

G. Duvaut and J.-L. Lions, Inequalities in Mechanics and Physics, Springer-Verlag, Berlin, 1976.  Google Scholar

[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.  Google Scholar

[8]

T. KärkkäinenK. 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.  Google Scholar

[9]

R. Kornhuber, Monotone multigrid methods for elliptic variational inequalities I, Numer. Math., 69 (1994), 167-184.  doi: 10.1007/BF03325426.  Google Scholar

[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. PeresO. SchrammS. 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.  Google Scholar

[12]

Z. SunZ. 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.  Google Scholar

[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.  Google Scholar

[14]

S. WangX. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

K. ZhangX. Q. YangS. 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.  Google Scholar

[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.  Google Scholar

[19]

Y. Y. ZhouS. 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.  Google Scholar

Figure 1.  Solution $ U(s) $ with $ N = 99 $. The solution is obtained with the lower order penalty method ($ k = 2 $). Tolerance $ \varepsilon = 10^{-6} $ is chosen for the Newton iteration. The smoothing interval is chosen to be $ (0,10^{-3}) $. $ \lambda = 10^{3} $
Figure 2.  Solution $ u(x,y) $ with the mesh grid $ 60 \times 60 $, obtained with the lower order penalty method ($ k = 2 $). Tolerance is set to be $ 10^{-6} $. $ \epsilon = 10^{-3} $. $ \lambda = 10^{3} $
Figure 3.  The lower coincidence sets (dot '$ \cdot $') and the upper coincidence sets (star '$ \ast $') of solution $ u(x,y) $ on the mesh grid $ 60 \times 60 $. The solution is obtained with the lower order penalty method ($ k = 2 $). Tolerance $ \varepsilon = 10^{-6} $ is chosen for the Newton iteration. The smoothing interval is chosen to be $ (0,10^{-3}) $. $ \lambda = 10^{3} $
Table 1.  Results computed by the power penalty method. Tolerance $ \varepsilon = 10^{-6} $ is chosen for the Newton iteration. The smoothing parameter $ \epsilon = 10^{-3} $. 'Ra' stands for ratio
$k=1$ $k=2$
$\lambda_{i}$ $[x_{\lambda_{i}}]_{1}$ $[x_{\lambda_{i}}]_{4}$ $\Vert x-x_{\lambda_{i}}\Vert_{\infty}$ Ra $\lambda_{i}$ $[x_{\lambda_{i}}]_{1}$ $[x_{\lambda_{i}}]_{4}$ $\Vert x-x_{\lambda_{i}}\Vert_{\infty}$ Ra
$10^{2}$ $0.5119$ $5.3052$ $6.25\times10^{-1}$ $10^{2}$ $0.9985$ $5.0011$ $2.81\times10^{-1}$
$10^{3}$ $0.9430$ $5.0327$ $6.49\times10^{-2}$ $0.98$ $10^{3}$ $0.9998$ $5.0002$ $2.85\times10^{-3}$ $1.99$
$10^{4}$ $0.9942$ $5.0033$ $6.59\times10^{-3}$ $0.99$ $10^{4}$ $0.9999$ $5.0001$ $4.93\times10^{-5}$ $1.76$
$10^{5}$ $0.9994$ $5.0003$ $6.60\times10^{-4}$ $1.00$ $10^{5}$ $1.0000$ $5.0000$ $6.90\times10^{-6}$ $1.69$
$k=1$ $k=2$
$\lambda_{i}$ $[x_{\lambda_{i}}]_{1}$ $[x_{\lambda_{i}}]_{4}$ $\Vert x-x_{\lambda_{i}}\Vert_{\infty}$ Ra $\lambda_{i}$ $[x_{\lambda_{i}}]_{1}$ $[x_{\lambda_{i}}]_{4}$ $\Vert x-x_{\lambda_{i}}\Vert_{\infty}$ Ra
$10^{2}$ $0.5119$ $5.3052$ $6.25\times10^{-1}$ $10^{2}$ $0.9985$ $5.0011$ $2.81\times10^{-1}$
$10^{3}$ $0.9430$ $5.0327$ $6.49\times10^{-2}$ $0.98$ $10^{3}$ $0.9998$ $5.0002$ $2.85\times10^{-3}$ $1.99$
$10^{4}$ $0.9942$ $5.0033$ $6.59\times10^{-3}$ $0.99$ $10^{4}$ $0.9999$ $5.0001$ $4.93\times10^{-5}$ $1.76$
$10^{5}$ $0.9994$ $5.0003$ $6.60\times10^{-4}$ $1.00$ $10^{5}$ $1.0000$ $5.0000$ $6.90\times10^{-6}$ $1.69$
Table 2.  Total number of iterations for the linear penalty ($ k = 1 $), lower order penalty ($ k = 2 $), and modified policy methods with $ N = 99 $. Tolerance is set to be $ 10^{-6} $. $ \epsilon = 10^{-3} $. $ \lambda $ is set to be $ 10^{6} $ and $ 10^{3} $ for $ k = 1 $ and $ k = 2 $, respectively
$ k=1 $ $ k=2 $ Modified policy iter.
Iterations 9 12 88
$ k=1 $ $ k=2 $ Modified policy iter.
Iterations 9 12 88
Table 3.  Comparison of computational efficiency among power penalty methods, Gauss-Seidel iteration method and active set method. Tolerance is set to be $ 10^{-6} $. $ \epsilon = 10^{-3} $. $ \lambda $ is set to be $ 10^{6} $ and $ 10^{3} $ for $ k = 1 $ and $ k = 2 $, respectively
$k=1$ $k=2$ Gauss-Seidel Active set
$N=49$ Iterations 11 16 426 Failed
Time (s) 1.26 1.89 32.81 Failed
$N=59$ Iterations 15 17 740 17
Time (s) 7.78 8.96 198.7 6.43
$k=1$ $k=2$ Gauss-Seidel Active set
$N=49$ Iterations 11 16 426 Failed
Time (s) 1.26 1.89 32.81 Failed
$N=59$ Iterations 15 17 740 17
Time (s) 7.78 8.96 198.7 6.43
[1]

Hui Gao, Jian Lv, Xiaoliang Wang, Liping Pang. An alternating linearization bundle method for a class of nonconvex optimization problem with inexact information. Journal of Industrial & Management Optimization, 2021, 17 (2) : 805-825. doi: 10.3934/jimo.2019135

[2]

Lekbir Afraites, Chorouk Masnaoui, Mourad Nachaoui. Shape optimization method for an inverse geometric source problem and stability at critical shape. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021006

[3]

Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056

[4]

Gloria Paoli, Gianpaolo Piscitelli, Rossanno Sannipoli. A stability result for the Steklov Laplacian Eigenvalue Problem with a spherical obstacle. Communications on Pure & Applied Analysis, 2021, 20 (1) : 145-158. doi: 10.3934/cpaa.2020261

[5]

Li Cai, Fubao Zhang. The Brezis-Nirenberg type double critical problem for a class of Schrödinger-Poisson equations. Electronic Research Archive, , () : -. doi: 10.3934/era.2020125

[6]

George W. Patrick. The geometry of convergence in numerical analysis. Journal of Computational Dynamics, 2021, 8 (1) : 33-58. doi: 10.3934/jcd.2021003

[7]

Alberto Bressan, Sondre Tesdal Galtung. A 2-dimensional shape optimization problem for tree branches. Networks & Heterogeneous Media, 2020  doi: 10.3934/nhm.2020031

[8]

Yasmine Cherfaoui, Mustapha Moulaï. Biobjective optimization over the efficient set of multiobjective integer programming problem. Journal of Industrial & Management Optimization, 2021, 17 (1) : 117-131. doi: 10.3934/jimo.2019102

[9]

Claudia Lederman, Noemi Wolanski. An optimization problem with volume constraint for an inhomogeneous operator with nonstandard growth. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020391

[10]

Ying Liu, Yanping Chen, Yunqing Huang, Yang Wang. Two-grid method for semiconductor device problem by mixed finite element method and characteristics finite element method. Electronic Research Archive, 2021, 29 (1) : 1859-1880. doi: 10.3934/era.2020095

[11]

Pierluigi Colli, Gianni Gilardi, Jürgen Sprekels. Deep quench approximation and optimal control of general Cahn–Hilliard systems with fractional operators and double obstacle potentials. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 243-271. doi: 10.3934/dcdss.2020213

[12]

Guo Zhou, Yongquan Zhou, Ruxin Zhao. Hybrid social spider optimization algorithm with differential mutation operator for the job-shop scheduling problem. Journal of Industrial & Management Optimization, 2021, 17 (2) : 533-548. doi: 10.3934/jimo.2019122

[13]

Gang Bao, Mingming Zhang, Bin Hu, Peijun Li. An adaptive finite element DtN method for the three-dimensional acoustic scattering problem. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 61-79. doi: 10.3934/dcdsb.2020351

[14]

Lan Luo, Zhe Zhang, Yong Yin. Simulated annealing and genetic algorithm based method for a bi-level seru loading problem with worker assignment in seru production systems. Journal of Industrial & Management Optimization, 2021, 17 (2) : 779-803. doi: 10.3934/jimo.2019134

[15]

Gang Luo, Qingzhi Yang. The point-wise convergence of shifted symmetric higher order power method. Journal of Industrial & Management Optimization, 2021, 17 (1) : 357-368. doi: 10.3934/jimo.2019115

[16]

Zexuan Liu, Zhiyuan Sun, Jerry Zhijian Yang. A numerical study of superconvergence of the discontinuous Galerkin method by patch reconstruction. Electronic Research Archive, 2020, 28 (4) : 1487-1501. doi: 10.3934/era.2020078

[17]

Bing Yu, Lei Zhang. Global optimization-based dimer method for finding saddle points. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 741-753. doi: 10.3934/dcdsb.2020139

[18]

C. J. Price. A modified Nelder-Mead barrier method for constrained optimization. Numerical Algebra, Control & Optimization, 2020  doi: 10.3934/naco.2020058

[19]

Min Chen, Olivier Goubet, Shenghao Li. Mathematical analysis of bump to bucket problem. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5567-5580. doi: 10.3934/cpaa.2020251

[20]

Qingfang Wang, Hua Yang. Solutions of nonlocal problem with critical exponent. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5591-5608. doi: 10.3934/cpaa.2020253

2019 Impact Factor: 1.366

Article outline

Figures and Tables

[Back to Top]