A new nonlinear Lagrangian method for nonconvex semidefinite programming

• Autorzy: Li Y. , Zhang L.
• Języki publikacji: EN

Abstrakty

EN

We study convergence properties of a new nonlinear Lagrangian method for nonconvex semidefinite programming. The convergence analysis shows that this method converges locally when the penalty parameter is less than a threshold and the error bound of solution is proportional to the penalty parameter under the constraint nondegeneracy condition, the strict complementarity condition and the strong" second order sufficient conditions. The major tools used in the analysis include the second implicit function theorem and differentials of Lowner operators.

Słowa kluczowe

EN

semidefinite programming; nonlinear Lagrangian method; convergence analysis

PL

półokreślone programowanie; nieliniowa metoda Lagrangiana; analiza zbieżności

• Wydawca: De Gruyter
• Czasopismo: Journal of Applied Analysis
• Rocznik: 2009
• Tom: Vol. 15, nr 2
• Strony: 149--172
• Opis fizyczny: Bibliogr. 22 poz.

Twórcy

autor

Li Y.

autor

Zhang L.

• School of Science Dalian Nationalities University Dalian 116600, China,

Bibliografia

• [1] Apkarian, P., Noll, D., Tuan, H. D., Fixed-order H? control design via a partially augmented Lagrangian method, Internal. J. Robust Nonlinear Control 13 (2003). 1137-1148.
• [2] Ben-Tal, A., Zibulevsky, M., Penalty/barrier multiplier methods for convex programming problems, SIAM J. Optim. 7 (1997), 347-366.
• [3] Bertsekas, D. P., Constrained Optimization and Lagrange Multiplier Methods, Academic Press, New York, 1982.
• [4] Bonnans, J. F., Shapiro, A., Perturbation Analysis of Optimization Problems, Springer-Verlag, New York, 2000.
• [5] Debreu. G., Definite and semidefinite quadratic forms, Econometrica 20 (1952), 295-300.
• [6] Fares, B., Apkarian, P., Noll, D., An augmented Lagrangian method for a class of LMI-constrained problems in robust control theory, Internal. J. Control 74 (2001), 348-360.
• [7] Fares, B., Noll, D., Apkarian, P., Robust control via. sequential semidefinite programming, SIAM J. Control Optim. 40 (2002), 1791-1820.
• [8] Horn. R. A., Johnson, C. R., Topics in Matrix Analysis. Vol 2, Cambridge University Press, Cambridge. 1991.
• [9] Kocvara, M., Stingl, M., PENNON: a generalized, augmented Lagrangian method, for semidefinite programming. Optim. Methods Softw. 18 (2003), 317-333.
• [10] Löwner, K., Über monotone matrixfunktionen, Math. Z. 38 (1934), 177 216.
• [11] Mosheyev, L., Zibulevsky, M., Penalty/Barrier multiplier algorithm for semidefinite programming. Optim. Methods Softw. 13 (2000). 235-261.
• [12] Noll, D., Local convergence of an augmented Lagrangian method for matrix inequality constrained programming, Optim. Methods Softw. 22 (2007), 777-802.
• [13] Noll, D., Torki, M., Apkarian, P., Partially augmented, Lagrangian method, for matrix inequality constraints, SIAM J. Optim. 15 (2004), 161-184.
• [14] Penrose, R., A generalized inverse for matrices, Proc. Cambridge Philos. Soc. 51 (1955), 406-413.
• [15] Polyak, R. A., Log-Sigmoid multipliers method, in constrained optimization. Ann. Oper. Res. 101 (2001), 427-460.
• [16] Polyak, R. A.. Teboulle, M., Nonlinear reseating and proximal-like methods in convex optimization, Math. Program. 76 (1997), 265-284.
• [17] Powell. M. ,J. D., A method for nonlinear constraints in minimization problems, in "Optimization", R. Fletcher, ed., Academic Press, New York, 1969, 283-298.
• [18] Rockafellar, R. T., A dual approach to solving nonlinear programming problems by unconstrained optimization, Math. Program. 5 (1973), 354-373.
• [19] Shapiro, A., First and, second order analysis of nonlinear semidefinite programs. Math. Program. 77 (1977), 301-320.
• [20] Stingl, M., On the Solution of Nonlinear Semidefinite Programs by Augmented, Lagrangian Methods. Dissertation, Shaker Verlag, Aachen, 2006.
• [21] Sun, D. F., The strong second order sufficient condition and, constraint nondegeneracy in nonlinear semidefinite programming and their implications, Math. Oper. Res. 31 (2006), 761-776.
• [22] Sun, D. F., Sun, J., Zhang L., The. rate of convergence of the augmented Lagrangian method for nonlinear semidefinite programming, Math. Program. 114 (2008). 349-391.