351
Views
22
CrossRef citations to date
0
Altmetric
Original Articles

Convergence properties of augmented Lagrangian methods for constrained global optimization

, &
Pages 763-778 | Received 21 Sep 2007, Published online: 17 Sep 2008

References

  • Andreani , R. , Birgin , E. G. , Martínez , J. M. and Schuverdt , M. L. 2007 . On augmented Lagrangian methods with general lower-level constraints . SIAM J. Optim. , 18 : 1286 – 1309 .
  • Andreani , R. , Birgin , E. G. , Martínez , J. M. and Schuverdt , M. L. 2008 . Augmented Lagrangian methods under the constant positive linear dependence constraint qualification . Math. Program , 111 : 5 – 32 .
  • Bartholomew-Biggs , M. C. 1987 . Recursive quadratic programming methods based on the augmented Lagrangian function . Math. Program. Study. , 31 : 21 – 41 .
  • Bertsekas , D. P. 1982 . Constrained Optimization and Lagrangian Multiplier Methods , New York : Academic Press .
  • Birgin , E. G. , Castillo , R. A. and Martínez , J. M. 2005 . Numerical comparison of augmented Lagrangian algorithms for nonconvex problems . Comput. Optim. Appl. , 31 : 31 – 55 .
  • Birgin , E. G. , Floudas , C. A. and Martínez , J. M. 2006 . “ Global minimization using an augmented Lagrangian method with variable lower-level constraints ” . In Tech. Rep , Brazil : Department of Computer Science IME-USP, University of São Paulo . Available in Optimization Online
  • Burachik , R. S. and Rubinov , A. M. 2005 . On the absence of duality gap for Lagrange-type functions . J. Industrial Manage. Optim. , 1 : 33 – 38 .
  • Burachik , R. S. and Rubinov , A. M. 2007 . Abstract convexity and augmented Lagrangians . SIAM J. Optim. , 18 : 413 – 436 .
  • Coleman , T. F. and Li , Y. 1994 . On the convergence of reflective Newton methods for large-scale nonlinear minimization subject to bounds . Math. Program , 67 : 189 – 224 .
  • Coleman , T. F. and Li , Y. 1996 . An interior trust region approach for nonlinear minimization subject to bounds . SIAM J. Optim. , 6 : 418 – 445 .
  • Di Pillo , G. and Lucidi , S. 2001 . An augmented Lagrangian function with improved exactness properties . SIAM J. Optim. , 12 : 376 – 406 .
  • Conn , A. R. , Gould , N. I.M. and Toint , P. L. 1991 . A globally convergent augmented Lagrangian algorithm for optimization with general constraints and simple bounds . SIAM J. Numer. Anal. , 28 : 545 – 572 .
  • Conn , A. R. , Gould , N. I.M. , Sartenaer , A. and Toint , P. L. 1996 . Convergence properties of an augmented Lagrangian algorithm for optimization with a combination of general equality and linear constraints . SIAM J. Optim. , 6 : 674 – 703 .
  • Contesse-Becker , L. 1993 . Extended convergence results for the method of multipliers for nonstrictly binding inequality constraints . J. Optim. Theory Appl. , 79 : 273 – 310 .
  • Floudas , C. A. , Pardalos , P. M. , Adjiman , C. S. , Esposito , W. R. , Gumus , Z. H. , Harding , S. T. , Klepeis , J. L. , Meyer , C. A. and Schweiger , C. A. 1999 . Handbook of Test Problems in Local and Global Optimization , Dordrecht : Kluwer Academic Publishers .
  • Griva , I. and Polyak , R. 2006 . A primal-dual nonlinear rescaling method with dynamic scaling parameter update . Math. Program , 106 : 237 – 259 .
  • Hager , W. W. 1987 . Dual techniques for constrained optimization . J. Optim. Theory Appl. , 55 : 37 – 71 .
  • Hartman , J. K. 1975 . Iterative determination of parameters for an exact penalty function . J. Optim. Theory Appl. , 16 : 49 – 66 .
  • Hestenes , M. R. 1969 . Multiplier and gradient methods . J. Optim. Theory Appl. , 4 : 303 – 320 .
  • Huang , X. X. and Yang , X. Q. 2003 . A unified augmented Lagrangian approach to duality and exact penalization . Math. Oper. Res. , 28 : 524 – 532 .
  • Kiwiel , K. C. 1996 . On the twice differentiable cubic augmented Lagrangian . J. Optim. Theory Appl. , 88 : 233 – 236 .
  • Kort , B. W. and Bertsekas , D. P. 1972 . A new penalty method for constrained minimization, in . Proceedings of the 1972 IEEE Conference on Decision and Control . 1972 , New Orleans. pp. 162 – 166 .
  • Kort , B. W. and Bertsekas , D. P. 1976 . Combined primal-dual and penalty methods for convex programming . SIAM J. Control Optim. , 14 : 268 – 294 .
  • Lewis , R. M. and Torczon , V. 2002 . A globally convergent augmented Lagrangian pattern search algorithm for optimization with general constraints and simple bounds . SIAM J. Optim. , 12 : 1075 – 1089 .
  • Li , D. 1995 . Zero duality gap for a class of nonconvex optimization problems . J. Optim. Theory Appl. , 85 : 309 – 324 .
  • Li , D. and Sun , X. L. 2000 . Local convexification of the Lagrangian function in nonconvex optimization . J. Optim. Theory Appl. , 104 : 109 – 120 .
  • Li , D. and Sun , X. L. 2001 . Convexification and existence of a saddle point in a pth-power reformulation for nonconvex constrained optimization . Nonlinear Anal. , 47 : 5611 – 5622 .
  • Li , D. and Sun , X. L. 2001 . Existence of a saddle point in nonconvex constrained optimization . J. Global Optim. , 21 : 39 – 50 .
  • Luo , H. Z. , Sun , X. L. and Li , D. 2007 . On the convergence of augmented Lagrangian methods for constrained global optimization . SIAM J. Optim. , 18 : 1209 – 1230 .
  • Mangasarian , O. L. 1975 . Unconstrained Lagrangians in nonlinear programming . SIAM J. Control Optim. , 13 : 772 – 791 .
  • Nguyen , V. H. and Strodiot , J. J. 1979 . On the convergence rate of a penalty function method of exponential type . J. Optim. Theory Appl. , 27 : 495 – 508 .
  • Polyak , R. 1992 . Modified barrier functions: Theory and methods . Math. Program , 54 : 177 – 222 .
  • Polyak , R. 2002 . Nonlinear rescaling vs. smoothing technique in convex optimization . Math. Program , 92 : 197 – 235 .
  • Polak , E. and Tits , A. L. 1980 . A globally convergent, implementable multiplier method with automatic penalty limitation . Appl. Math. Optim. , 6 : 335 – 360 .
  • Powell , M. J.D. 1969 . A method for nonlinear constraints in minimization problems, Optimization , Edited by: Fletcher , R. 283 – 298 . New York : Academic Press .
  • Rockafellar , R. T. 1973 . The multiplier method of Hestenes and Powell applied to convex programming . J. Optim. Theory Appl. , 12 : 555 – 562 .
  • Rockafellar , R. T. 1974 . Augmented Lagrange multiplier functions and duality in nonconvex programming . SIAM J. Control Optim. , 12 : 268 – 285 .
  • Rockafellar , R. T. 1976 . Augmented Lagrangians and applications of the proximal point algorithm in convex programming . Math. Oper. Res. , 1 : 97 – 116 .
  • Rockafellar , R. T. 1993 . Lagrange multipliers and optimality . SIAM Rev. , 35 : 183 – 238 .
  • Rockafellar , R. T. and Wets , R. J.-B. 1998 . “ Variational Analysis ” . Springer-Verlag, Berlin .
  • Rubinov , A. M. and Yang , X. Q. 2003 . “ Lagrange-type Functions in Constrained Non-Convex Optimization ” . Dordrecht : Kluwer Academic Publishers .
  • Rubinov , A. M. , Glover , B. M. and Yang , X. Q. 1999 . Decreasing functions with applications to penalization . SIAM J. Optim. , 10 : 289 – 313 .
  • Rubinov , A. M. , Glover , B. M. and Yang , X. Q. 1999 . Modified Lagrangian and penalty functions in continuous optimization . Optimization , 46 : 327 – 351 .
  • Rubinov , A. M. , Huang , X. X. and Yang , X. Q. 2002 . The zero duality gap property and lower semi- continuity of the perturbation function . Math. Oper. Res. , 27 : 775 – 791 .
  • Sun , X. L. , Li , D. and McKinnon , K. I.M. 2005 . On saddle points of augmented Lagrangians for constrained nonconvex optimization . SIAM J. Optim. , 15 : 1128 – 1146 .
  • Tseng , P. and Bertsekas , D. P. 1993 . On the convergence of the exponential multiplier method for convex programming . Math. Program , 60 : 1 – 19 .
  • Xu , Z. K. 1997 . Local saddle points and convexification for nonconvex optimization problems . J. Optim. Theory Appl. , 94 : 739 – 746 .
  • Yamashita , H. 1982 . A globally convergent constrained quasi-Newton method with an augmented Lagrangian type penalty function . Math. Program , 23 : 75 – 86 .
  • Yang , X. Q. and Huang , X. X. 2001 . A nonlinear Lagrangian approach to constrained optimization problems . SIAM J. Optim. , 11 : 1119 – 1144 .

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.