# American Institute of Mathematical Sciences

• Previous Article
A novel quality prediction method based on feature selection considering high dimensional product quality data
• JIMO Home
• This Issue
• Next Article
Human resources optimization with MARS and ANN: Innovation geolocation model for generation Z
doi: 10.3934/jimo.2021018
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

## 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 Early access 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:

show all references

##### References:
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 $(0,10^{-3})$. $\lambda = 10^{3}$
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 $10^{-6}$. $\epsilon = 10^{-3}$. $\lambda = 10^{3}$
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 $(0,10^{-3})$. $\lambda = 10^{3}$
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 $10^{2}$ $0.5119$ $5.3052$ $6.25\times10^{-1}$ $10^{2}$ $0.9985$ $5.0011$ $2.81\times10^{-1}$ $10^{3}$ $0.9430$ $5.0327$ $6.49\times10^{-2}$ $0.98$ $10^{3}$ $0.9998$ $5.0002$ $2.85\times10^{-3}$ $1.99$ $10^{4}$ $0.9942$ $5.0033$ $6.59\times10^{-3}$ $0.99$ $10^{4}$ $0.9999$ $5.0001$ $4.93\times10^{-5}$ $1.76$ $10^{5}$ $0.9994$ $5.0003$ $6.60\times10^{-4}$ $1.00$ $10^{5}$ $1.0000$ $5.0000$ $6.90\times10^{-6}$ $1.69$
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
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] Changjun Yu, Kok Lay Teo, Liansheng Zhang, Yanqin Bai. On a refinement of the convergence analysis for the new exact penalty function method for continuous inequality constrained optimization problem. Journal of Industrial & Management Optimization, 2012, 8 (2) : 485-491. doi: 10.3934/jimo.2012.8.485 [2] Yibing Lv, Zhongping Wan. Linear bilevel multiobjective optimization problem: Penalty approach. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1213-1223. doi: 10.3934/jimo.2018092 [3] Song Wang. Numerical solution of an obstacle problem with interval coefficients. Numerical Algebra, Control & Optimization, 2020, 10 (1) : 23-38. doi: 10.3934/naco.2019030 [4] Liuyang Yuan, Zhongping Wan, Jingjing Zhang, Bin Sun. A filled function method for solving nonlinear complementarity problem. Journal of Industrial & Management Optimization, 2009, 5 (4) : 911-928. doi: 10.3934/jimo.2009.5.911 [5] L’ubomír Baňas, Amy Novick-Cohen, Robert Nürnberg. The degenerate and non-degenerate deep quench obstacle problem: A numerical comparison. Networks & Heterogeneous Media, 2013, 8 (1) : 37-64. doi: 10.3934/nhm.2013.8.37 [6] Ming Chen, Chongchao Huang. A power penalty method for the general traffic assignment problem with elastic demand. Journal of Industrial & Management Optimization, 2014, 10 (4) : 1019-1030. doi: 10.3934/jimo.2014.10.1019 [7] Fengming Ma, Yiju Wang, Hongge Zhao. A potential reduction method for the generalized linear complementarity problem over a polyhedral cone. Journal of Industrial & Management Optimization, 2010, 6 (1) : 259-267. doi: 10.3934/jimo.2010.6.259 [8] X. X. Huang, D. Li, Xiaoqi Yang. Convergence of optimal values of quadratic penalty problems for mathematical programs with complementarity constraints. Journal of Industrial & Management Optimization, 2006, 2 (3) : 287-296. doi: 10.3934/jimo.2006.2.287 [9] Lekbir Afraites. A new coupled complex boundary method (CCBM) for an inverse obstacle problem. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021069 [10] Ming-Zheng Wang, M. Montaz Ali. Penalty-based SAA method of stochastic nonlinear complementarity problems. Journal of Industrial & Management Optimization, 2010, 6 (1) : 241-257. doi: 10.3934/jimo.2010.6.241 [11] Regina S. Burachik, C. Yalçın Kaya. An update rule and a convergence result for a penalty function method. Journal of Industrial & Management Optimization, 2007, 3 (2) : 381-398. doi: 10.3934/jimo.2007.3.381 [12] Zhongwen Chen, Songqiang Qiu, Yujie Jiao. A penalty-free method for equality constrained optimization. Journal of Industrial & Management Optimization, 2013, 9 (2) : 391-409. doi: 10.3934/jimo.2013.9.391 [13] Radouen Ghanem, Billel Zireg. Numerical solution of bilateral obstacle optimal control problem, where the controls and the obstacles coincide. Numerical Algebra, Control & Optimization, 2020, 10 (3) : 275-300. doi: 10.3934/naco.2020002 [14] Yibing Lv, Tiesong Hu, Jianlin Jiang. Penalty method-based equilibrium point approach for solving the linear bilevel multiobjective programming problem. Discrete & Continuous Dynamical Systems - S, 2020, 13 (6) : 1743-1755. doi: 10.3934/dcdss.2020102 [15] Pierre Fabrie, Elodie Jaumouillé, Iraj Mortazavi, Olivier Piller. Numerical approximation of an optimization problem to reduce leakage in water distribution systems. Mathematical Control & Related Fields, 2012, 2 (2) : 101-120. doi: 10.3934/mcrf.2012.2.101 [16] Ali Fuat Alkaya, Dindar Oz. An optimal algorithm for the obstacle neutralization problem. Journal of Industrial & Management Optimization, 2017, 13 (2) : 835-856. doi: 10.3934/jimo.2016049 [17] Daniel Morales-Silva, David Yang Gao. Complete solutions and triality theory to a nonconvex optimization problem with double-well potential in $\mathbb{R}^n$. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 271-282. doi: 10.3934/naco.2013.3.271 [18] Jun Lai, Ming Li, Peijun Li, Wei Li. A fast direct imaging method for the inverse obstacle scattering problem with nonlinear point scatterers. Inverse Problems & Imaging, 2018, 12 (3) : 635-665. doi: 10.3934/ipi.2018027 [19] Xiaofei Liu, Yong Wang. Weakening convergence conditions of a potential reduction method for tensor complementarity problems. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021080 [20] Wei-Zhe Gu, Li-Yong Lu. The linear convergence of a derivative-free descent method for nonlinear complementarity problems. Journal of Industrial & Management Optimization, 2017, 13 (2) : 531-548. doi: 10.3934/jimo.2016030

2020 Impact Factor: 1.801