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A New Lagrangian Multiplier Method on Constrained Optimization
Department of Mathematics, Henan University of Science and Technology, Luoyang, China
Department of Mathematics, Henan University of Science and Technology, Luoyang, China
Department of Mathematics, Henan University of Science and Technology, Luoyang, China
- 1 Department of Mathematics, Henan University of Science and Technology, Luoyang, China
- 2 Department of Mathematics, Henan University of Science and Technology, Luoyang, China
- 3 Department of Mathematics, Henan University of Science and Technology, Luoyang, China
Applied Mathematics·Volume 03 (2012)·Pages 1409–1414·Published 31 October 2012·DOI10.4236/am.2012.330198
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Abstract
In this paper, a new augmented Lagrangian function with 4-piecewise linear NCP function is introduced for solving nonlinear programming problems with equality constrained and inequality constrained. It is proved that a solution of the original constrained problem and corresponding values of Lagrange multipliers can be found by solving an unconstrained minimization of the augmented Lagrange function. Meanwhile, a new Lagrangian multiplier method corresponding with new augmented Lagrangian function is proposed. And this method is implementable and convergent.
KeywordsNonlinear ProgrammingNCP FunctionLagrange FunctionMultiplierConvergence
- G. Pillo and L. Grippo, “A New Class of Augmented Lagrangian in Nonlinear Programming,” SIAM Journal on Control and Optimization, Vol. 17, No. 5, 1979, pp. 617-628.
- G. Pillo and L. Grippo, “An Augmented Lageangian function for Inequlity Constrains in Nonlinear Programming Problems,” Optimization Theorem and Application, Vol. 36, No. 4, 1982, pp. 495-520. doi:10.1007/BF00940544
- G. Pillo and S. Lucidi, “An augmented Lagrange Function with Improved Exactness Properties,” SIAM Journal on Control and Optimization, Vol. 12, No. 2, 2001, pp. 376-406.
- D. G. Pu and Z. Jin, “A New Lagrange Multiplier Method,” Tongji University Journal, Vol. 9, No. 38, 2010, pp. 1384-1391.
- K. D. Li and D. G. Pu, “A class of New Lagrange Multiplier Methods,” OR Transaction, Vol. 10, No. 4, 2006, pp. 9-24.
- X.-W. Du and Y.-L. Shang, “Exact Augmented Lagrange Function for Nonlinear Programming Problems with Inequality and Equality Constraints,” Applied Mathematics and Mechanics, Vol. 26, No. 12, 2005, pp. 1651-1656.
- K. Su, “Trust-Region Filter Method with NCP Function,” Journal of Systems Science and Mathematical Science, Vol. 28, No. 12, 2008, pp. 1525-1534
- D. G. Pu, “A New Filter Method,” OR Transaction, Vol. 1, No. 15, 2011, pp. 46-58.
- W. Hua and D. G. Pu, “A Modified QP-Free Feasible Method,” Applied Mathmatical Modelling, Vol. 35, No. 4, 2011, pp. 1696-1708. doi:10.1016/j.apm.2010.10.002
- K. Li and D. G. Pu, “Filter QP-Free Method with 3-Piece Wise Linear NCP Function,” OR Transaction, Vol. 12, No. 2, 2008, pp. 49-57.
- Y. X. Yuan, “Numerical Method for Nonlinear Programming,” Shanghai Science and Technology Press, Shanghai, 1993.
- B. L. Chen, “The Theory of Optimization and Calculate,” 2nd Edition, Tsinghua University Press, Beijing, 2005.