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Penalty/barrier path-following in linearly constrained optimization

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EN
Abstrakty
EN
In the present paper rather general penalty/barrier path-following methods (e.g. with p-th power penalties, logarithmic barriers, SUMT, exponential penalties) applied to linearly constrained convex optimization problems are studied. In particular, unlike in previous studies [1,11], here simultaneously different types of penalty/barrier embeddings are included. Together with the assumed 2nd order sufficient optimality conditions this required a significant change in proving the local existence of some continuously differentiable primal and dual path related to these methods. In contrast to standard penalty/barrier investigations in the considered path-following algorithms only one Newton step is applied to the generated auxiliary problems. As a foundation of convergence analysis the radius of convergence of Newton's method depending on the penalty/barrier parameter is estimated. There are established parameter selection rules which guarantee the overall convergence of the considered path-following penalty/barrier techniques.
Twórcy
  • Institute of Numerical Mathematics, Dresden University of Technology, D-01062 Dresden, Germany
Bibliografia
  • [1] D. Al-Mutairi, C. Grossmann and K.T. Vu, Path-following barrier and penalty methods for linearly constrained problems, Preprint MATH-NM-10-1998, TU Dresden 1998 (to appear in Optimization).
  • [2] A. Auslender, R. Cominetti and M. Haddou, Asymptotic analysis for penalty and barrier methods in convex and linear programming, Math. Operations Res. 22 (1997), 43-62.
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  • [10] R.M. Freund and S. Mizuno, Interior point methods: current status and future directions, OPTIMA, Math. Programming Soc. Newsletter 51 (1996).
  • [11] C. Grossmann, Asymptotic analysis of a path-following barrier method for linearly constrained convex problems, Optimization 45 (1999), 69-88.
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  • [13] M. Halická and M. Hamala, Duality transformation functions in the interior point methods, Acta Math. Univ. Comenianae LXV (1996), 229-245.
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  • [17] M.H. Wright, Ill-conditioning and computational error in interior methods for nonlinear programming, SIAM J. Opt. 9 (1998), 84-111.
  • [18] S.J. Wright, On the convergence of the Newton/Log-barrier method, Preprint ANL/MCS-P681-0897, MCS Division, Argonne National Lab., August 1997 (revised version 1999).
  • [19] H. Yamashita and H. Yabe, An interior point method with a primal-dual l₂ barrier penalty function for nonlinear optimization, Working paper, March 1999, University of Tokyo.
Typ dokumentu
Bibliografia
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bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1001
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