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Encyclopedia of Optimization pp 1556–1561Cite as

Ill-posed Variational Problems

IVP

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References

  1. Alart P, Lemaire B (1991) Penalization in non‐classicalconvex programming via variational convergence. Math Program 51:307–331

    Article  MathSciNet  MATH  Google Scholar 

  2. Antipin AS (1975) Regularization methods for convex programming problems. Ekonomika i Mat Metody 11:336–342 (in Russian).

    Google Scholar 

  3. Antipin AS (1976) On a method for convex programs using a symmetrical modification of the Lagrange function. Ekonomika i Mat Metody 12:1164–1173 (in Russian).

    MATH  Google Scholar 

  4. Auslender A (1987) Numerical methods for nondifferentiable convex optimization. Math Program Stud 30:102–126

    MathSciNet  MATH  Google Scholar 

  5. Auslender A, Crouzeix JP, Fedit P (1987) Penalty proximal methods in convex programming. J Optim Th Appl 55:1–21

    Article  MathSciNet  MATH  Google Scholar 

  6. Auslender A, Haddou M (1995) An interior-proximal method for convex linearly constrained problems and its extension to variational inequalities. Math Program 71:77–100

    MathSciNet  Google Scholar 

  7. Bakushinski AB, Goncharski AV (1989) Ill-posed problems – Numerical methods and applications. Moscow Univ. Press, Moscow

    Google Scholar 

  8. Bakushinski AB, Polyak BT (1974) About the solution of variational inequalities. Soviet Math Dokl 15:1705–1710

    Google Scholar 

  9. Beer G (1993) Topologies on closed and convex closed sets. Math and its Appl, vol 268. Kluwer, Dordrecht

    MATH  Google Scholar 

  10. Bonnans J, Gilbert JCh, Lemaréchal C, Sagastizabal C (1995) A family of variable metric proximal methods. Math Program 68:15–47

    Google Scholar 

  11. Chen G, Teboulle M (1992) A proximal-based decomposition method for convex minimization problems. Math Program 66:293–318

    Google Scholar 

  12. Dontchev AL, Zolezzi T (1993) Well-posed optimization problems. Lecture Notes Math, vol 1543. Springer, Berlin

    MATH  Google Scholar 

  13. Eckstein J (1993) Nonlinear proximal point algorithms using Bregman functions, with application to convex programming. Math Oper Res 18:202–226

    Article  MathSciNet  MATH  Google Scholar 

  14. Eckstein J, Bertsekas D (1992) On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators. Math Program 55:293–318

    Article  MathSciNet  MATH  Google Scholar 

  15. Furi M, Vignoli A (1970) About well-posed minimization problems for functionals in metric spaces. J Optim Th Appl 5:225–290

    Article  MATH  Google Scholar 

  16. Hettich R,Kortanek KO (1993) Semi-infinite programming: Theory, methods and applications. SIAM Rev 35:380–429

    Article  MathSciNet  MATH  Google Scholar 

  17. Ibaraki S, Fukushima M, Ibaraki T (1992) Primal-dual proximal point algorithm for linearly constrained convex programming problems. Comput Optim Appl 1:207–226

    Article  MathSciNet  MATH  Google Scholar 

  18. Iusem AN, Svaiter BF (1995) A proximal regularization of the steepest descent method. RAIRO Rech Opérat 29:123–130

    MathSciNet  MATH  Google Scholar 

  19. Kaplan A (1982) Algorithm for convex programming using a smoothing for exact penalty functions. Sibirsk Mat Zh 23:53–64

    MathSciNet  MATH  Google Scholar 

  20. Kaplan A, Tichatschke R (1994) Stable methods for ill-posed variational problems: Prox-regularization of elliptical variational inequalities and semi-infinite optimization problems. Akad. Verlag, Berlin

    Google Scholar 

  21. Kaplan A, Tichatschke R (1996) Path-following proximal approach for solving ill-posed convex semi-infinite programming problems. J Optim Th Appl 90:113–137

    Article  MathSciNet  MATH  Google Scholar 

  22. Kaplan A, Tichatschke R (1997) Prox-regularization and solution of ill-posed elliptic variational inequalities. Appl Math 42:111–145

    Article  MathSciNet  MATH  Google Scholar 

  23. Karmanov VG (1980) Mathematical programming. Fizmatgiz, Moscow

    MATH  Google Scholar 

  24. Kiwiel K (1995) Proximal level bundle methods for convex nondifferentiable optimization, saddle point problems and variational inequalities. Math Program 69B:89–109

    MathSciNet  MATH  Google Scholar 

  25. Kiwiel K (1996) Restricted step and Levenberg–Marquardt techniques in proximal bundle methods for non-convex nondifferentiable optimization. SIAM J Optim 6:227–249

    Article  MathSciNet  MATH  Google Scholar 

  26. Lemaire B (1988) Coupling optimization methods and variational convergence. ISNM 84:163–179

    MathSciNet  Google Scholar 

  27. Levitin ES, Polyak BT (1966) Minimization methods under constraints. J Vycisl Mat i Mat Fiz 6:787–823

    MATH  Google Scholar 

  28. Liskovets OA (1987) Regularization of problems with monotone operators when the spaces and operators are discretly approximated. USSR J Comput Math Math Phys 27:1–8

    Article  MathSciNet  MATH  Google Scholar 

  29. Louis AK (1989) Inverse and schlecht gestellte Probleme. Teubner, Leipzig Studienbücher Math

    MATH  Google Scholar 

  30. Martinet B (1970) Régularisation d'inéquations variationelles par approximations successives. RIRO 4:154–159

    MathSciNet  Google Scholar 

  31. Mifflin R (1996) A quasi-second-order proximal bundle algorithm. Math Program 73:51–72

    MathSciNet  Google Scholar 

  32. Mosco U (1969) Convergence of convex sets and of solutions of variational inequalities. Adv Math 3:510–585

    Article  MathSciNet  MATH  Google Scholar 

  33. Polyak BT (1987) Introduction to optimization. Optim. Software, New York

    Google Scholar 

  34. Qi L, Chen X (1997) A preconditioning proximal Newton method for non-differentiable convex optimization. Math Program 76B:411–429

    MathSciNet  MATH  Google Scholar 

  35. Revalski JP (1995) Various aspects of well-posedness of optimization problems. In: Lucchetti R, Revalski J (eds) Recent Developments in Well-Posed Variational Problems. Kluwer, Dordrecht

    Google Scholar 

  36. Rockafellar RT (1976) Augmented Lagrange multiplier functions and applications of the proximal point algorithm in convex programming. Math Oper Res 1:97–116

    MathSciNet  MATH  Google Scholar 

  37. Rockafellar RT (1976) Monotone operators and the proximal point algorithm. SIAM J Control Optim 14:877–898

    Article  MathSciNet  MATH  Google Scholar 

  38. Tikhonov AN (1966) On the stability of the functional optimization problems. USSR J Comput Math Math Phys 6:631–634

    Google Scholar 

  39. Tikhonov AN, Arsenin VJ (1977) Methods for solving ill-posed problems. Wiley, New York

    Google Scholar 

  40. Vasil'ev FP (1981) Methods for solving extremal problems. Nauka, Moscow

    MATH  Google Scholar 

  41. Wei Z, Qi L (1996) Convergence analysis of a proximal Newton method. Numer Funct Anal Optim 17:463–472

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Universität Trier, Trier, Germany

    Alexander Kaplan & Rainer Tichatschke

Authors
  1. Alexander Kaplan
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  2. Rainer Tichatschke
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Editor information

Editors and Affiliations

  1. Department of Chemical Engineering, Princeton University, Princeton, NJ, 08544-5263, USA

    Christodoulos A. Floudas

  2. Center for Applied Optimization, Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL, 32611-6595, USA

    Panos M. Pardalos

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© 2008 Springer-Verlag

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Kaplan, A., Tichatschke, R. (2008). Ill-posed Variational Problems . In: Floudas, C., Pardalos, P. (eds) Encyclopedia of Optimization. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-74759-0_273

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