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How to deal with the unbounded in optimization: Theory and algorithms

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Abstract

The aim of this survey is to show how the unbounded arises in optimization problems and how it leads to fundamental notions which are not only useful for proving theoretical results such as convergence of algorithms and the existence of optimal solutions, but also for constructing new methods.

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References

  1. P. Angleraud, Caracterisation duale du bon comportement de fonctions,C.R. Acad. Sci. Paris Ser. I 314 (1992) 583–586.

    MATH  MathSciNet  Google Scholar 

  2. A. Auslender,Optimisation: Théorie et Algorithmes (Masson, Paris, 1976).

    Google Scholar 

  3. A. Auslender, Convergence of stationary sequences for variational inequalities with maximal monotone operators,Appl. Math. Optim. 28 (1993) 161–172.

    Article  MATH  MathSciNet  Google Scholar 

  4. A. Auslender, Asymptotic properties of the Fenchel’s dual functional and their applications to decomposition problems,J. Optim. Theory Appl. 73 (1992) 427–450.

    Article  MATH  MathSciNet  Google Scholar 

  5. A. Auslender and J.-P. Crouzeix, Well behaved asymptotical convex functions, in:Analyse Non-linéaire (Gauthier-Villars, Paris, 1989) 101–122.

    Google Scholar 

  6. A. Auslender, R. Cominetti and J.-P. Crouzeix, Functions with unbounded level sets and applications to duality theory,SIAM J. Optimization 3 (4) (1993) 669–687.

    Article  MATH  MathSciNet  Google Scholar 

  7. A. Auslender, Noncoercive optimization problems,Mathematics of Operations Research 21 (4) (1996) 769–782.

    MATH  MathSciNet  Google Scholar 

  8. A. Auslender, R. Cominetti and M. Haddou, Asymptotic analysis for penalty and barrier methods in convex and linear programming,Mathematics of Operations Research (1997), to appear.

  9. C. Baiocchi, G. Buttazzo, F. Gastaldi and F. Tomarelli, General existence theorems for unilateral problems in continuum mechanies,Arch. Rational Mech. Anal. 100 (1988) 149–189.

    Article  MATH  MathSciNet  Google Scholar 

  10. A. Ben-Tal and G. Roth, A truncated log-barrier algorithm for large scale convex programming and minmax problems: implementation and computational results, to appear.

  11. A. Ben-Tal and M. Zibulevski, Penalty/Barrier multiplier methods: A new class of augmented Lagrangian algorithms for large scale convex programming problems, Research Report 4/93, Optimization Laboratory, Faculty of Industrial Engineering and Management, Technion-Israel Institute of Technology, Haifa.

  12. E.G. Belousov,Introduction to Convex Analysis and Integer Optimization (University Publisher, Moscow, 1977), in Russian.

    Google Scholar 

  13. G. Butazzo and F. Tomarelli, Nonlinear Neuman problems,Adv. Math. 89 (1991) 126–142.

    Article  Google Scholar 

  14. L. Caklovic, S. Li and M. Willem, A note on Palais Smale condition and coercivity,Diff. Int. Equa. 3 (1990) 799–800.

    MATH  MathSciNet  Google Scholar 

  15. C.W. Carrol, The created response surface technique for optimizing nonlinear restrained systems,Operations Research 9 (2) (1961).

  16. Y. Censor and A. Lent, Optimization of log (x) entropy over linear equality constraints,SIAM J. Control Optim. 25 (1987) 921–933.

    Article  MATH  MathSciNet  Google Scholar 

  17. C. Chen and O.L. Mangasarian, A class of smoothing functions for nonlinear and mixed complementarity problems,Mathematical Programming 71 (1) (1995) 51–70.

    Article  MathSciNet  Google Scholar 

  18. C.C. Chou, K.F. Ng and J.S. Pang, Minimizing and stationary sequences for optimization problems, Manuscript, Department of Mathematical Sciences, The Johns Hopkins University, Baltimore, MD, 1995.

    Google Scholar 

  19. D.G. Costa and E.A. de B. e Silva, The Palais Smale condition versus coercivity,Nonlin. Anal. Th. Math. Appli. 16 (1991) 371–381.

    Article  MATH  Google Scholar 

  20. A. Fiacco and G. McCormick,Nonlinear Programming: Sequential Unconstrained Minimization Techniques (Wiley, New York, 1968).

    MATH  Google Scholar 

  21. M. Frank and P. Wolfe, An algorithm for quadratic programming,Naval Research Logistics Quarterly 3 (1956) 95–110.

    MathSciNet  Google Scholar 

  22. K.R. Frisch, The logarithmic potential method of convex programming, Memorandum of May 13, 1955, University Institute of Economics, Oslo.

  23. M. Fukushima and J.S. Pang, Minimizing and stationary sequences of Merit Functions for Complementarity Problems and Variational Inequalities, Technical Report, 1995, Nara Institute.

  24. M. Fukushima, Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems,Mathematical Programming 53 (1992) 99–110.

    Article  MATH  MathSciNet  Google Scholar 

  25. M. Fukushima, M. Haddou, V.H. Nguyen, J.J. Strodiot, T. Sugimato and E. Yamkawa, A parallel descent algorithm for convex programming,Computational Optimization and Applications 5 (1996) 5–37.

    Article  MATH  MathSciNet  Google Scholar 

  26. M. Goberna and A. Lopez, Optimal value function in semi-infinite programming,J. Optim. Theory Appl. 59 (1988) 261–278.

    MATH  MathSciNet  Google Scholar 

  27. S.P. Han and G. Lou, A parallel algorithm for a class of convex programs,SIAM J. Control Optim. 26 (1988) 345–355.

    Article  MATH  MathSciNet  Google Scholar 

  28. B. Lemaire, Bonne Position, conditionnement et bon comportement asymptotique, Seminaire d’Analyse Convexe, Montpellier (1992), exp. no. 5.

    Google Scholar 

  29. R. Polyak, Modified barrier functions: theory and methods,Mathematical Programming 54 (1992) 177–222.

    Article  MATH  MathSciNet  Google Scholar 

  30. R. Polyak and M. Teboulle, Nonlinear rescaling and proximal-like methods in convex optimization,Mathematical Programming, submitted.

  31. M.J.D. Powell, Some global convergence properties of a variable metric algorithm without exact line searches,SIAM-AMS Proceedings, Vol. 9, Nonlinear Programming 1976 (American Mathematical Society, Providence, RI).

    Google Scholar 

  32. R.T. Rockafellar,Convex Analysis (Princeton University Press, Princeton, NJ, 1970).

    MATH  Google Scholar 

  33. R.T. Rockafellar and R. Wets,Variational Analysis, to appear.

  34. E. Steinitz, Bedingt konvergente Reien und konvexe Systeme I,Journal fur Reine und Angewandte Mathematik 143 (1913) 128–175.

    Article  MATH  Google Scholar 

  35. M.J. Todd, On convergence properties of algorithms for unconstrained minimization,IMA Journal of Numerical Analysis 9 (1989) 435–441.

    Article  MATH  MathSciNet  Google Scholar 

  36. P. Tseng and D. Bertsekas, Relaxation methods for problems with strictly convex costs and linear constraints,Mathematical Programming 38 (1987) 303–321.

    Article  MATH  MathSciNet  Google Scholar 

  37. J. Wang and J.S. Pang, Global error bounds for quadratic inequality systems,Optimisation 31 (1994) 1–12.

    MATH  MathSciNet  Google Scholar 

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Auslender, A. How to deal with the unbounded in optimization: Theory and algorithms. Mathematical Programming 79, 3–18 (1997). https://doi.org/10.1007/BF02614308

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