Abstract.
Auslender, Cominetti and Haddou have studied, in the convex case, a new family of penalty/barrier functions. In this paper, we analyze the asymptotic behavior of augmented penalty algorithms using those penalty functions under the usual second order sufficient optimality conditions, and present order of convergence results (superlinear convergence with order of convergence 4/3). Those results are related to the analysis of ‘‘pure’’ penalty algorithms, as well as ‘‘augmented’’ penalty using a quadratic penalty function. Limited numerical examples are presented to appreciate the practical impact of this local asymptotic analysis.
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References
Auslender, A., Cominetti, R., Haddou, M.: Asymptotic analysis of penalty and barrier methods in convex and linear programming. Math. Oper. Res. 22(1), Feb. 1997
Ben-Tal, A., Zibulevsky, M.: Penalty/barrier multiplier methods for convex programming problems. SIAM J. Optim. 7(2) 347–366, May 1997
Breitfeld, M., Shanno, D.: Preliminary computational experience with modified log-barrier functions for large-scale nonlinear programming. Rutcor Research Report RRR 08-93, Rutgers University, New Jersey, 1993
Broyden, C., Attia, N.: A smooth sequential penalty function method for solving nonlinear programming problems. In: P. Thoft-Christenses, (ed.), System Modelling and Optimization, Springler-Verlag, Berlin, 1983, pp. 237–245
Broyden, C., Attia., N.: Penalty functions, newton’s method, and quadratic programming. J. Optim. Theor. Appl. 58, (1988)
Cominetti, R., Dussault, J.-P.: A stable exponential-penalty algorithm with super-linear convergence. J. Optim. Theor. Appl. 83(2), November 1994
Cominetti, R., Martin., J.S.: Asymptotic behavior of the exponential penalty trajectory in linear programming. Math. Program. 67, 169–187 (1994)
Dussault., J.-P.: Numerical stability and efficiency of penalty algorithms. SIAM J. Numer. Anal. 32(1), 296–317, February 1995
Dussault., J.-P.: Augmented penalty algorithms. IMA J. Numer. Anal. 18, 355–372 (1998)
Dussault, J.-P.: Improved convergence order for augmented penalty algorithms. Submitted, 2003
Gould., N.I.M.: On the convergence of a sequential penalty function method for constrained minimization. SIAM J. Numer. Anal. 26, 107–108 (1989)
Hock, W., Schittkowski, K.: Test examples for nonlinear programming codes. Number 187 in Lecture notes in economics and mathematical systems. Springer Verlag, Berlin, 1981
Kort, B.W., Bertsekas, D.P.: A new penalty function method for constrained minimization. In: Proceedings of the 1972 I.E.E.E. conference on decision and control, New Orleans, LA, December 1972, pp. 162–166
Nash, S., Polyak, R., Sofer, A.: A numerical comparison of barrier and modified barrier methods for large-scale constrained optimization. Technical Report 93-02, Department of Operations Research, George Mason University, Fairfax, Virginia 22030, 1993
Nguyen, V.H., Strodiot, J.-J.: Convergence rate results for a penalty function method. In: Optimization techniques, proceedings of the 8th IFIP conference on optimization techniques, Würzburg, 1977, pp. 101–106
Nguyen, V.H., Strodiot, J.-J.: On the convergence rate for a penalty function method of exponential type. J. Optim. Theor. Appl. 27 495–508, 1979
Polyak., R.: Modified barrier functions: theory and methods. Math. Progam. 54, 177–222 (1992)
Scilab group. Πlab. Institut National de Recherche en Informatique et Automatique – Domaine de Voluceau – Rocquencourt – B.P. 105 – 78153 – LE CHESNAY Cedex – FRANCE, email : Scilab@inria.fr, 1996
Wächter, A., Biegler, L.T.: Failure of global convergence for a class of interior point methods for nonlinear programming. CAPD Technical Report B99-07, Department of Chemical Engineering, Carnegie Mellon University, Pittsburg, PA 15213, May 2000
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This research was partially supported by NSERC grant OGP0005491
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Dussault, JP. Augmented non-quadratic penalty algorithms. Math. Program., Ser. A 99, 467–486 (2004). https://doi.org/10.1007/s10107-003-0459-6
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DOI: https://doi.org/10.1007/s10107-003-0459-6