Convergence of the augmented Lagrangian method for nonlinear optimization problems over second-order cones

Journal of Optimization Theory and Applications

Volumn 139, Issue 3, 2008, Pages 557-575

  Liu, Y J ^a   Zhang, L W ^b  

^a SHENYANG AEROSPACE UNIVERSITY   (China)

^b DALIAN UNIVERSITY OF TECHNOLOGY   (China)

Author keywords

Augmented Lagrangian methods; Nonlinear second order cone constrained optimization problems; Rates of local convergence; Semismooth functions

Indexed keywords

CONSTRAINED OPTIMIZATION; NONLINEAR PROGRAMMING;

AUGMENTED LAGRANGIAN METHODS; CONDITION; CONSTRAINED OPTIMI-ZATION PROBLEMS; LOCAL CONVERGENCE; NONLINEAR OPTIMIZATION PROBLEMS; NONLINEAR SECOND-ORDER CONE CONSTRAINED OPTIMIZATION PROBLEM; PAPER ANALYSIS; RATE OF LOCAL CONVERGENCE; SECOND ORDER CONE; SEMISMOOTH FUNCTIONS;

LAGRANGE MULTIPLIERS;

EID: 55649083953     PISSN: 00223239     EISSN: 15732878     Source Type: Journal    
DOI: 10.1007/s10957-008-9390-6     Document Type: Article

References (28)

1. Arrow, K.J., Solow, R.M.: Gradient methods for constrained maxima with weakened assumptions. In: Arrow, K.J., Hurwicz, L., Uzawa, H. (eds.) Studies in Linear and Nonlinear Programming, pp. 165-176. Stanford University Press, Stanford (1958)
2. Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, New York (1998)
3. A. Shapiro J. Sun 2004 Some properties of the augmented Lagrangian in cone constrained optimization Math. Oper. Res. 29 3 479 491
4. M.R. Hestences 1969 Multiplier and gradient methods J. Optim. Theory Appl. 4 303 320
5. Powell, M.J.D.: A Method for Nonlinear Constraints in Minimization Problems. In: Fletcher, R. (ed.) Optimization. Academic Press, New York (1969)
6. R.T. Rockafellar 1973 A dual approach to solving nonlinear programming problems by unconstrained optimization Math. Program. 5 354 373
7. R.T. Rockafellar 1973 The multiplier method of Hestenes and Powell applied to convex programming J. Optim. Theory Appl. 12 555 562
8. N.V. Tretyakov 1973 A method of penalty estimates for convex programming problems Èkonomika i Matematicheskie Metody 9 525 540
9. Buys, J.D.: Dual Algorithms for Constrained Optimization Problems. PhD Thesis, University of Leiden (1972)
10. D.P. Bertsekas 1976 On penalty and multiplier methods for constrained minimization SIAM J. Control Optim. 14 216 235
11. A.R. Conn N.I.M. Gould Ph.L. Toint 1991 A gloablly convergent augmented Lagrangian algorithm for optimization with general constraints and simple bounds SIAM J. Numer. Analysis 28 545 572
12. L. Contesse-Becker 1993 Extended convergence results for the method of multipliers for non-strictly binding inequality constraints J. Optim. Theory Appl. 79 273 310
13. K. Ito K. Kunisch 1990 The augmented Lagrangian method for equality and inequality constraints in Hilbert spaces Math. Program. 46 341 360
14. Sun, J., Zhang, L.-W., Wu, Y.: Some properties of the augmented Lagrangian in nonlinear semidefinite optimization. J. Optim. Theory Appl. (to appear)
15. Sun, D., Sun, J., Zhang, L.W.: The rate of convergence of the augmented Lagrangian method for nonlinear semidefinite programming. Mathematical Programming online
16. S. Wang X.Q. Yang K.L. Teo 2006 Power penalty method for a linear complementarity problem arising from American option valuation J. Optim. Theory Appl. 129 2 227 254
17. J.F. Bonnans C.H. Ramírez 2005 Perturbation analysis of second-order cone programming problems Math. Program. B 104 205 227
18. R. Mifflin 1977 Semismooth and semiconvex functions in constrained optimization SIAM J. Control Optim. 15 957 972
19. L. Qi J. Sun 1993 A nonsmooth version of Newton's method Math. Program. 58 353 367
20. Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)
21. D. Sun 2001 A further result on an implicit function theorem for locally Lipschitz functions Oper. Res. Lett. 28 193 198
22. F. Alizadeh D. Goldfarb 2003 Second-order cone programming Math. Program. 95 3 51
23. X.D. Chen D. Sun J. Sun 2003 Complementarity functions and numerical experiments on some smoothing Newton methods for second-order-cone complementarity problems Comput. Optim. Appl. 25 39 56
24. J.-S. Pang D. Sun J. Sun 2003 Semismooth homeomorphisms and strong stability of semidefinite and Lorenz complementarity problems Math. Oper. Res. 28 39 63
25. Zarantonello, E.H.: Projections on convex sets in Hilbert space and spectral theory I and II. In: Zarantonello, E.H. (ed.) Contributions to Nonlinear Functional Analysis, pp. 237-424. Academic Press, San Diego (1971)
26. Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2000)
27. G. Debreu 1952 Definite and semidefinite quadratic forms Econometrica 20 295 300
28. Y.-J. Liu L.-W. Zhang 2007 Convergence analysis of the augmented Lagrangian method for nonlinear second-order cone optimization problems Nonlinear Analysis: Theory, Methods Appl. 67 1359 1373