Abstract
In this paper we present penalty and barrier methods for solving general convex semidefinite programming problems. More precisely, the constraint set is described by a convex operator that takes its values in the cone of negative semidefinite symmetric matrices. This class of methods is an extension of penalty and barrier methods for convex optimization to this setting. We provide implementable stopping rules and prove the convergence of the primal and dual paths obtained by these methods under minimal assumptions. The two parameters approach for penalty methods is also extended. As for usual convex programming, we prove that after a finite number of steps all iterates will be feasible.
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Auslender A (1999) Penalty and barrier methods: a unified framework. SIAM J Optim 10:211–230
Auslender A (2003) Variational inequalities over the cone of semidefinite positive matrices and over the Lorentz cone. Optimization methods and software, pp 1–18
Auslender A, Teboulle M (2003) Asymptotic cones and functions in optimization and variational inequalities. Springer monographs in mathematics, Springer, Berlin Heidelberg New York
Auslender A, Cominetti R, Haddou M (1997) Asymptotic analysis of penalty and barrier methods in convex and linear programming. Math Opers Res 22:43–62
Bhatia R (1997) Matrix analysis. Springer graduate texts in mathematics, Springer, Berlin Heidelberg New York
Bonnans JF, Shapiro A (2000) Perturbation analysis of optimization problems. Springer series in operations research, Springer, Berlin Heidelberg New York
Ben-Tal A, Nemirowskii A (2002) Lectures on modern convex optimization, analysis, algorithms, and engineering applications. MPS-SIAM series on optimization, SIAM, Philadelphia
Ben-Tal A, Zibulevsky M (1997) Penalty-barrier methods for convex programming problems. SIAM J Optim 7:347–366
Chen C, Mangasarian OL (1996) A class of smoothing functions for nonlinear and mixed complementary problems. Compt Optim Appl 5:97–138
Cominetti R, Dussault JP (1994) A stable exponential penalty method with superlinear convergence. JOTA 83:285–309
Den Hertzog D, Roos C, Terlaky T (1991) Inverse barrier method for linear programming. Report 91–27, Faculty of Technical Mathematics and Informatics, Delft University of Technology, Netherlands
Frisch KR (1995) The logarithmic potential method of convex programming. Memorandum of May 13, 1995, University Institute of Economics, Oslo, Norway
Golub GH, Van Loan CF (1996) Matrix computations, 3rd edn. The Johns Hopkins University Press, Baltimore
Gonzaga C, Castillo RA (2003) A nonlinear programming algorithm based on non-coercive penalty functions. Math Program Ser A 96:87–101
Graña Drummond LM, Peterzil Y (2002) The central path in smooth convex semidefinite programming. Optimization 51:207–233
Kato T (1970) Perturbation theory for linear operators. Springer, Berlin Heidelberg New York
Lewis AS (1996) Convex analysis on the Hermitian matrices. SIAM J Optim 6:164–177
Nesterov YN, Nemirovski AS (1994) Interior point polynomial algorithms in convex programing. SIAM, Philapdelphia
Polyak RA (1992) Modified barrier functions (theory and methods). Math Prog 54:177–222
Rockafellar RT (1970) Convex analysis. Princeton University Press, Princeton
Seeger A (1997) Convex analysis of spectrally defined matrix functions. SIAM J Optim 7:679–696
Vandenbergue L, Boyd S (1995) Semidefinite programming. SIAM Rev 38:49–95
Xavier AE (1992) Hyperbolic penalization, PhD thesis, COPPE - Federal University of Rio de Janeiro, Rio de Janeiro, Brazil (in Portuguese)
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Auslender, A., C., H.R. Penalty and Barrier Methods for Convex Semidefinite Programming. Math Meth Oper Res 63, 195–219 (2006). https://doi.org/10.1007/s00186-005-0054-0
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DOI: https://doi.org/10.1007/s00186-005-0054-0