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Original Articles

Convergence of a local regularization approach for mathematical programmes with complementarity or vanishing constraints

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Pages 483-512 | Received 22 Feb 2010, Accepted 22 Oct 2010, Published online: 26 Nov 2010

References

  • Achtziger , W. and Kanzow , C. 2008 . Mathematical programs with vanishing constraints: Optimality conditions and constraint qualifications . Math. Program. , 114 : 69 – 99 .
  • Achtziger , W. , Hoheisel , T. and Kanzow , C. November 2008 . “ A smoothing-regularization approach to mathematical programs with vanishing constraints ” . November , Institute of Mathematics, University of Würzburg . Technical Report
  • Achtziger , W. , Hoheisel , T. and Kanzow , C. “ On a relaxation method for mathematical programs with vanishing constraints and its application in topology optimization ” . Institute of Mathematics, University of Wüurzburg . Technical Report, forthcoming
  • Andreani , R. , Martínez , J. M. and Schuverdt , M. L. 2005 . The CPLD condition of Qi and Wei implies the quasinormality constraint qualification . J. Optim. Theory Appl. , 125 : 473 – 485 .
  • Anitescu , M. 2005 . On using the elastic mode in nonlinear programming approaches to mathematical programs with complementarity constraints . SIAM J. Optim. , 15 : 1203 – 1236 .
  • Anitescu , M. 2005 . Global convergence of an elastic mode approach for a class of mathematical programs with complementarity constraints . SIAM J. Optim. , 16 : 120 – 145 .
  • Bertsekas , D. P. and Ozdaglar , A. E. 2002 . Pseudonormality and a Lagrange multiplier theory for constrained optimization . J. Optim. Theory Appl. , 114 : 187 – 343 .
  • Clarke , F. H. 1983 . “ Optimization and Nonsmooth Analysis ” . New York : John Wiley . (reprinted by SIAM, Philadelphia, PA, 1990)
  • Cheng , G. D. and Guo , X. 1997 . ϵ -Relaxed approach in structural topology optimization . Struct. Optim. , 13 : 258 – 266 .
  • DeMiguel , V. , Friedlander , M. P. , Nogales , F. J. and Scholtes , S. 2005 . A twosided relaxation scheme for mathematical programs with equilibrium constraints . SIAM J. Optim. , 16 : 587 – 609 .
  • Dorsch , D. , Shikhman , V. and Stein , O. June 2010 . “ MPVC: Critical point theory ” . June , Department of Mathematics – C, RWTH Aachen University . Preprint No. 138
  • Facchinei , F. , Jiang , H. and Qi , L. 1999 . A smoothing method for mathematical programs with equilibrium constraints . Math. Program. , 85 : 107 – 134 .
  • Flegel , M. L. and Kanzow , C. 2005 . Abadie-type constraint qualification for mathematical programs with equilibrium constraints . J. Optim. Theory Appl. , 124 : 595 – 614 .
  • Flegel , M. L. and Kanzow , C. 2005 . On M-stationary points for mathematical programs with equilibrium constraints . J. Math. Anal. Appl. , 310 : 286 – 302 .
  • Flegel , M. L. and Kanzow , C. 2005 . On the Guignard constraint qualification for mathematical programs with equilibrium constraints . Optimization , 54 : 517 – 534 .
  • Flegel , M. L. , Kanzow , C. and Outrata , J. V. 2007 . Optimality conditions for disjunctive programs with applications to mathematical programs with equilibrium constraints . Set-Valued Anal. , 15 : 139 – 162 .
  • Fletcher , R. and Leyffer , S. 2004 . Solving mathematical programs with complementarity constraints as nonlinear programs . Optim. Methods Softw. , 19 : 15 – 40 .
  • Hoheisel , T. and Kanzow , C. 2007 . First- and second-order optimality conditions for mathematical programs with vanishing constraints . Appl. Math. , 52 : 495 – 514 . (special issue dedicated to J.V. Outrata's 60. birthday)
  • Hoheisel , T. and Kanzow , C. 2008 . Stationary conditions for mathematical programs with vanishing constraints using weak constraint qualifications . J. Math. Anal. Appl. , 337 : 292 – 310 .
  • Hoheisel , T. and Kanzow , C. 2009 . On the Abadie and Guignard constraint qualification for mathematical programs with vanishing constraints . Optimization , 58 : 431 – 448 .
  • Hu , X. M. and Ralph , D. 2004 . Convergence of a penalty method for mathematical programs with equilibrium constraints . J. Optim. Theory Appl. , 123 : 365 – 390 .
  • Izmailov , A. F. and Solodov , M. V. 2009 . Mathematical programs with vanishing constraints: Optimality conditions, sensitivity and a relaxation method . J. Optim. Theory Appl. , 142 : 501 – 532 .
  • Izmailov , A. F. , Pogosyan , A. L. and Solodov , M. V. Semismooth Newton method for the lifted reformulation of mathematical programs with complementarity constraints . December . Rio de Janeiro , , Brazil Technical Report, Institute of Pure and Applied Mathematics
  • Janin , R. 1984 . Directional derivative of the marginal function in nonlinear programming . Math. Program. Study , 21 : 110 – 126 .
  • Jongen , H. Th. , Rückmann , J.-J. and Shikhman , V. 2009 . On stability of the MPCC feasible set . SIAM J. Optim. , 20 : 1171 – 1184 .
  • Kanzow , C. and Schwartz , A. 2010 . Mathematical programs with equilibrium constraints: Enhanced Fritz John-conditions, new constraint qualifications and improved exact penalty results . SIAM J. Optim. , 20 : 2730 – 2753 .
  • Kirsch , U. 1990 . On singular topologies in optimum structural design . Struct. Optim. , 2 : 133 – 142 .
  • Leyffer , S. 2000 . “ MacMPEC: AMPL collection of MPECs ” . Argonne National Laboratory . Available at www.mcs.anl.gov/leyfier/MacMPEC
  • Leyffer , S. , López-Calva , G. and Nocedal , J. 2007 . Interior methods for mathematical programs with complementarity constraints . SIAM J. Optim. , 17 : 52 – 77 .
  • Luo , Z.-Q. , Pang , J.-S. and Ralph , D. 1996 . “ Mathematical Programs with Equilibrium Constraints ” . Cambridge, New York, Melbourne : Cambridge University Press .
  • Outrata , J. V. 1999 . Optimality conditions for a class of mathematical programs with equilibrium constraints . Math. Oper. Res. , 24 : 627 – 644 .
  • Outrata , J. V. 2000 . A generalized mathematical program with equilibrium constraints . SIAM J. Control Optim. , 38 : 1623 – 1638 .
  • Outrata , J. V. , Ko_cvara , M. and Zowe , J. 1998 . “ Nonsmooth approach to optimization problems with equilibrium constraints ” . In Nonconvex Optimization and Its Applications , Dordrecht , , The Netherlands : Kluwer Academic Publishers .
  • Pang , J.-S. and Fukushima , M. 1999 . Complementarity constraint qualifications and simplified B-stationarity conditions for mathematical programs with equilibrium constraints . Comput. Optim. Appl. , 13 : 111 – 136 .
  • Qi , L. and Wei , Z. 2000 . On the constant positive linear dependence condition and its applications to SQP methods . SIAM J. Optim. , 10 : 963 – 981 .
  • Raghunathan , A. U. and Biegler , L. T. 2005 . An interior point method for mathematical programs with complementarity constraints (MPCCs) . SIAM J. Optim. , 15 : 720 – 750 .
  • Ralph , D. and Wright , S. J. 2004 . Some properties of regularization and penalization schemes for MPECs . Optim. Methods Softw. , 19 : 527 – 556 .
  • Scheel , H. and Scholtes , S. 2000 . Mathematical programs with complementarity constraints: Stationarity, optimality, and sensitivity . Math. Oper. Res. , 25 : 1 – 22 .
  • Scholtes , S. 2001 . Convergence properties of a regularization scheme for mathematical programs with complementarity constraints . SIAM J. Optim. , 11 : 918 – 936 .
  • Steffensen , S. and Ulbrich , M. 2010 . A new regularization scheme for mathematical programs with equilibrium constraints . SIAM J. Optim. , 20 : 2504 – 2539 .
  • Stein , O. Lifting mathematical programs with complementarity constraints . Math. Program , DOI: 10.1007/S10107-010-0345-y
  • Veelken , S. 2009 . “ A new relaxation scheme for mathematical programs with equilibrium constraints: Theory and numerical experience ” . Faculty of Mathematics, Technical University of Munich . Ph.D. thesis
  • Ye , J. J. 1999 . Optimality conditions for optimization problems with complementarity constraints . SIAM J. Optim. , 9 : 374 – 387 .
  • Ye , J. J. 2005 . Necessary and sufficient optimality conditions for mathematical programs with equilibrium constraints . J. Math. Anal. Appl. , 307 : 350 – 369 .
  • Ye , J. J. and Ye , X. Y. 1997 . Necessary optimality conditions for optimization problems with variational inequality constraints . Math. Oper. Res. , 22 : 977 – 997 .

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