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研究生: 伊爾凡
Irfan Nurhidayat
論文名稱: An ordinary differential equation approach for nonlinear programming and nonlinear complementarity problem
An ordinary differential equation approach for nonlinear programming and nonlinear complementarity problem
指導教授: 陳界山
Chen, Jein-Shan
學位類別: 碩士
Master
系所名稱: 數學系
Department of Mathematics
論文出版年: 2019
畢業學年度: 107
語文別: 英文
論文頁數: 58
中文關鍵詞: FTIMNLPNCPODE
英文關鍵詞: FTIM, NLP, NCP, ODE
DOI URL: http://doi.org/10.6345/THE.NTNU.DM.008.2019.B01
論文種類: 學術論文
相關次數: 點閱:106下載:20
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  • In this thesis, we consider an ordinary differential equation (ODE) approach for solving nonlinear programming (NLP) and nonlinear complementarity problem (NCP). The Karush-Kuhn Tucker (KKT) optimality conditions of NLP and NCP are used to get the new NCP-functions. A special technique is employed to reformulate of the NCP as the system of nonlinear algebraic equations (NAEs) later on reformulated once more by force of an original time-like function into an ordinary differential equation (ODE).
    Afterwards, a group preserving scheme (GPS) is a package to reformulate an ODE into the new numerical equation in a way the ODEs system is designed into a nonlinear dynamical system (NDS) and is continued to discover the new numerical equation through activating the Lorentz group SO 0 (n, 1) and its Lie algebra so(n, 1). Lastly, the fictitious time integration method (FTIM) is utilized into this new numerical equation to determine an approximation solution in numerical experiments area.

    Contents List of Tables .......................................... 3 List of Figures ......................................... 4 1 Motivation and Introduction ......................... 10 1.1 Motivation .......................................... 10 1.2 Introduction ........................................ 12 2 Preliminaries ....................................... 17 2.1 Original time-like function ......................... 17 2.2 The Lorentz group SO0(n,1) and the Lie algebra so(n,1) ......................................................... 18 2.3 Performance profile .................................. 19 3 ODE reformulation ................................... 20 3.1 Transformation into an ODEs system .................. 20 3.2 GPS for differential equations system ................ 22 3.2.1 Group preserving scheme ........................... 23 3.2.2 One-step GPS ...................................... 26 4 Numerical experiments ............................... 28 4.1 Example 1 ........................................... 29 4.2 Example 2 ........................................... 42 4.3 Example 3 ........................................... 48 5 Conclusions ........................................... 54 Bibliography ........................................... 55

    [1] Burden, R. L.; Faires, J. D. (2011), Numerical analysis, ninth edition, Boston, Brooks/Cole, Cengage Learning.

    [2] Chen, J-S., On some NCP-functions based on the generalized Fischer-Burmeister function, Asia-Pacific J. Oper. Res., vol.24, pp. 401-420, 2007.

    [3] De Luca, T.; Facchinei, F.; Kanzow, C., A semismooth equation approach to the solution of nonlinear complementarity problems, Math. Program., vol.75, pp. 407-439, 1996.

    [4] Dolan, E. D.; More, J. J., Benchmarking optimization software with performance profiles, Math. Program., vol.91, pp. 201-213, 2002.

    [5] Ferris, M. C.; Pang, J-S. , Engineering and economic applications of complementarity problems, SIAM Review, vol.39, pp. 669-713, 1997.

    [6] Gallier, J.; Quaintance, J. (2017), Notes on differential geometry and Lie groups, Department of Computer and Information Science, University of Pennsylvania, Philadelphia, USA.

    [7] Harker, PT.; Pang, J-S., Finite-dimensional variational inequality and nonlinear complementarity problem: A survey of theory, algorithms, and applications, Mathematical Programming, vol.48, pp. 161-220, 1990.

    [8] Huang, C-H.; Weng, K-J.; Chen, J-S.; Chu, H-W.; Li, M-Y., On four discrete-type families of NCP-functions, Journal of Nonlinear and Convex Analysis, pp. 1-27, 2017.

    [9] Kanzow, C.; Yamashita, N.; Fukushima, M., New NCP-Functions and their properties, Journal of Optimization Theory and Applications, vol.94, pp. 115-135, 1997.

    [10] Ku, C-Y.; Yeih, W.; Liu, C-S.; Chi, C-C., Applications of the fictitious time integration method using a new time-like function, CMES: Computer Modeling in Engineering and Sciences, vol.43, no.2, pp. 173-190, 2009.

    [11] Liu, C-S.; Chang, J-R.; Chang, K-H.; Chen, Y-W. , Simultaneously estimating the time-dependent damping and stiffness coefficients with the aid of vibrational data,
    CMC: Computer, Materials, and Continua, vol.7, pp. 97-107, 2008.

    [12] Liu, C-S.; Atluri, S. N., A fictitious time integration method (FTIM) for solving mixed complementarity problems with applications to non-linear optimization, CMES: Computer Modeling in Engineering and Sciences, vol.34, no.2, pp. 155-178, 2008a.

    [13] Liu, C-S.; Atluri, S. N., A novel time integration method for solving a large system of non-linear algebraic equations, CMES: Computer Modeling in Engineering and Sciences, vol.31, pp. 71-83, 2008b.

    [14] Liu, C-S.; Atluri, S. N., A fictitious time integration method for the numerical solution of the Fredholm integral equation and for numerical differentiation of noisy data, and its relation to the filter theory, CMES: Computer Modeling in Engineering and Sciences, vol.41, pp. 243-261, 2009.

    [15] Liu, C-S., Cone of non-linear dynamical system and group preserving schemes, Int. J. Non-Linear Mech., vol.36, pp. 1047-1068, 2001.

    [16] Liu, C-S., A Lie-group shooting method for simultaneously estimating the time-dependent damping and stiffness coefficients, CMES: Computer Modeling in Engineering and Sciences, vol.27, pp. 137-149, 2008a.

    [17] Liu, C-S., A fictitious time integration method for solving the discretized inverse Sturm-Liouville problems, for specified eigenvalues, CMES: Computer Modeling in Engineering and Sciences, vol.36, pp. 261-286, 2008b.

    [18] Liu, C-S., A fictitious time integration method for two-dimensional quasilinear elliptic boundary value problems, CMES: Computer Modeling in Engineering and Sciences, vol.33, pp. 179-198, 2008c.

    [19] Liu, C-S., A time-marching algorithm for solving nonlinear obstacle problems with the aid of an NCP-function, CMC: Computer, Materials, and Continua, vol.8, pp. 53-65, 2008d.

    [20] Liu, C-S., Solving an inverse Sturm-Liouville problem by a Lie-group method, Boundary Value Problems, Article ID 749865, 2008e.

    [21] Liu, C-S., Identifying time-dependent damping and stiffness functions by a simple and yet accurate method, J. Sound Vib., vol.318, pp. 148-165, 2008f.

    [22] Liu, C-S. , A fictitious time integration method for solving m-point boundary value problems, CMES: Computer Modeling in Engineering and Sciences, vol.39, pp. 125-154, 2009.

    [23] Mangasarian, O. L., Equivalence of the complementarity problem to a system of nonlinear equations, SIAM Journal on Applied mathematics, vol.31, pp. 89-92, 1976.

    [24] Pang, J-S. (1994), Complementarity problems, Handbook of Global Optimization, R Horst and P Pardalos (eds.), pp. 271-338, Boston, MA: Kluwer Academic Publishers.

    [25] Pourrajabian, A.; Ebrahimi, R.; Mirzaei, M.; Shams, M., Applying genetic algorithms for solving nonlinear algebraic equations, Applied Mathematics and Computation, vol.219, pp. 11483-11494, 2013.

    [26] Sun, D.; Qi, L., On NCP-functions, Computational Optimization and Applications,vol. 13, pp. 201-220, 1999.

    [27] Tseng, P., Global behaviour of a class of merit functions for the nonlinear complementarity problem, Journal of Optimization Theory and Applications, vol.89, pp. 17-37, 1996.

    [28] Yamashita, N.; Fukushima, M., Modified Newton methods for solving a semismooth reformulation of monotone complementarity problems, Mathematical Programming, vol.76, pp. 469-491, 1997.

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