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NONLINEAR LAGRANGIANS FOR NONLINEAR PROGRAMMING BASED ON MODIFIED FISCHER-BURMEISTER NCP FUNCTIONS

Yonghong Ren1, Fangfang Guo2, Yang Li3   

  1. 1 School of Mathematics, Liaoning Normal University, Dalian, China;
    2 School of Mathematics Sciences, Dalian University of Technology, Dalian, China;
    3 School of Science, Dalian Nationalities University, Dalian, China
  • Received:2014-08-26 Revised:2015-03-24 Online:2015-07-15 Published:2015-07-15
  • Supported by:

    The work of the first author was supported by the National Natural Science Foundation of China (11171138) and General Project of Scientific Research of the Education Department of Liaoning Province in 2015 (L2015291). The work of the third author was supported by the Fundamental Research Dalian Nationalities University (DC201502050406).

Yonghong Ren, Fangfang Guo, Yang Li. NONLINEAR LAGRANGIANS FOR NONLINEAR PROGRAMMING BASED ON MODIFIED FISCHER-BURMEISTER NCP FUNCTIONS[J]. Journal of Computational Mathematics, 2015, 33(4): 396-414.

This paper proposes nonlinear Lagrangians based on modified Fischer-Burmeister NCP functions for solving nonlinear programming problems with inequality constraints. The convergence theorem shows that the sequence of points generated by this nonlinear Lagrange algorithm is locally convergent when the penalty parameter is less than a threshold under a set of suitable conditions on problem functions, and the error bound of solution, depending on the penalty parameter, is also established. It is shown that the condition number of the nonlinear Lagrangian Hessian at the optimal solution is proportional to the controlling penalty parameter. Moreover, the paper develops the dual algorithm associated with the proposed nonlinear Lagrangians. Numerical results reported suggest that the dual algorithm based on proposed nonlinear Lagrangians is effective for solving some nonlinear optimization problems.

CLC Number: 

[1] A. Auslender, R. Cominetti and M. Haddou, Asymptotic analysis of penalty and barrier methods in convex and linear programming, Mathematics of Operations Research, 22(1997), 43-62.

[2] A. Ben-Tel and M. Zibulevsky, Penalty/barrier multiplier methods for convex programming problems, SIAM Journal on Optimization, 7(2)(1997), 347-366.

[3] D.P. Bertsekas, Constrained Optimization and Lagrange Multiplier Methods, Academic Press, New York, 1982.

[4] D.P. Bertsekas, Nonlinear Programming, 2nd ed., Athena Scientific, Belmont, Massachusetts, 1999.

[5] C. Charalambous, Nonliear least pth optimization and nonlinear programming, Mathematical Programming, 12(1977), 195-225.

[6] D. Goldfarb, R. Polyak, K. Scheinberg and I. Yuzeforich, A modified barrier-augmented Lagrangian method for constrained minimization, Computational Optimization and Applications, 14(1999), 55-74.

[7] J.P. Dussault, Augmented non-quadratic penalty algorithms, Mathematical Programming, 99(2004), 467-486.

[8] I. Griva and R.A. Polyak, Primal-dual nonlinear rescaling method with dynamic scaling parameter update, Mathematical Programming, 106(2)(2006), 237C259 .

[9] M.R. Hestenes, Multiplier and gradient method, Journal on Optimization Theory and Applications, 4(1969), 303-320.

[10] C. Kanzow and H. Kleinmichel, A new class of semismooth Newton-type methods for nonlinear complementarity Problems, Computational Optimization and Applications, 11(1998), 227-251.

[11] D. Li, Zero duality gap for a class of nonconvex optimization problems, Journal of Optimization Theory and Applications, 85(1995), 309-323.

[12] L. Lukšan and J. Vlcek, Test problems for nonsmooth unconstrained and linearly constrained optimization, Technical Report 798(2000), Institute of Computer Science, Academy of Sciences of the Czech Republic, Prague.

[13] O.L. Mangasarian, Unconstrained Lagrangians in nonlinear programming, SIAM Journal on Control, 13(1975), 772-791.

[14] R.A. Polyak, Modified barrier function: theory and mehtods, Mathematical Programming, 54(1992), 177-222.

[15] R.A. Polyak, Log-Sigmoid multipliers method in constrained optimization, Annals of operations Research, 101(2001), 427-460.

[16] R.A. Polyak, Nonlinear rescaling vs. smoothing technique in convex optimization, Mathematical Programming, 92(2002), 197-235.

[17] R.A. Polyak and I. Griva, Primal-Dual nonlinear rescaling method for convex optimization, Journal of Optimization Theory and Applications, 122(2004), 111-156.

[18] R.A. Polyak and M. Teboulle, Nonlinear rescaling and classical-like methods in convex optimization, Mathematical Programming , 76(1997), 265-284.

[19] M.J.D. Powell, A method for nonlinear constraints in minimization problems, In Optimization (R. Fletcher, ed.), Academic Press, New York, 283-298, 1969.

[20] Y.H. Ren, L.W. Zhang and X.T. Xiao, A nonlinear Lagrangian based on Fischer-Burmeister NCP function. Applied Mathematics and Computation, 188(2007), 1344-1363.

[21] R.T. Rockafellar, A dual approach to solving nonlinear programming problems by unconstrained optimization, Mathematical programming, 5(1973), 354-373.

[22] R.T. Rockafellar, The multiplier method of Hestenes and Powell applied to convex programming, Journal of Optimization Theory and Applications, 12(1973), 555-562.

[23] R.T. Rockafellar, Augmented Lagrange multiplier functions and duality in nonconvex programming, SIAM Journal on Control, 12(1974), 268-285.

[24] L.W. Zhang, Y.H. Ren, Y. Wu and X.T. Xiao, A class of nonlinear Lagrangians: theory and algorithm, Asia-Pacific Journal of Operational Research, 25:3 (2008), 327-371.
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