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Rotundity, smoothness and duality

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EN
Abstrakty
EN
The duality between smoothness and rotundity of functions is studied in a nonlinear abstract framework. Here smoothness is enlarged to subdifferentiability properties and rotundity is formulated by means of approximation properties.
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Strony
711--733
Opis fizyczny
Bibliogr. 109 poz.
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autor
  • Laboratoire de Math́ematiques appliquees CNRS FRE 2570 Av. de l’Universite, 64000 Pau, France
Bibliografia
  • Agadi, A. and Penot, J.-P. (1996) A comparative study of various notions of approximation of sets. Preprint, Univ. of Pau.
  • Asplund, E. (1968) Fr ́echet differentiability of convex functions. Acta Math 121, 31–47.
  • Asplund, E. and Rockafellar, R.T. (1969)Gradients of convex functions. Trans. Amer. Math. Soc. 139, 443–467.
  • Atteia, M. and Elqortobi, A. (1981) Quasi-convex duality. In: A. Auslen- der et al., eds., Optimization and Optimal Control, Proc. Conference Oberwolfach March 1980. Lecture Notes in Control and Inform. Sci. 30, Springer-Verlag, Berlin, 3–8.
  • Aubin, J.-P. (1999) Mutational and Morphological Analysis. Birkh ̈auser, Basel.
  • Aze, D. (1997) El ́ements d’analyse convexe et variationnelle. Ellipses, Paris.
  • Aze, D. (1999) On the remainder of the first order development of convex functions. Ann. Math. Qu ́ebec 23, 1–14.
  • Aze, D., Corvellec, J.-N. and Lucchetti, R. (2002) Variational pairs and applications to stability in nonsmooth analysis. Nonlinear Anal. 49, 643-670.
  • Aze, D. and Penot, J.-P. (1995) Uniformly convex and uniformly smooth convex functions. Ann. Fac. Sci. Toulouse 4, 705–730.
  • Aze, D. and Rahmouni, A. (1994) Lipschitz behavior of the Legendre-Fenchel transform. Set-Valued Anal. 2 (1-2), 35–48.
  • Aze, D. and Rahmouni, A. (1995) Intrinsic bounds for Kuhn-Tucker points of perturbed convex programs. In: R. Durier and C. Michelot, eds., Recent developments in optimization, Seventh French-German Conference on Optimization. Lecture Notes in Economics and Math. 429, Springer Verlag, Berlin, 17–35.
  • Aze, D. and Rahmouni, A. (1996) On primal dual stability in convex opti- mization. J. Convex Anal. 3, 309–329.
  • Balder, E.J. (1977) An extension of duality-stability relations to non-convex optimization problems. SIAM J. Control Opt. 15, 329–343.
  • Beauzamy, B. (1992) Introduction to Banach Spaces and their Geometry. Math. Studies 68, North-Holland, Amsterdam.
  • Bronstedt, A. (1964) Conjugate convex functions on topological vector spaces. Mat-Fys. Medel Danska Vod Selsk. 2.
  • Cominetti, R. (1994) Some remarks on convex duality in normed spaces with and without compactness. Control and Cybern. 23 (1-2), 123–138.
  • Crouzeix, J.-P. (1977) Contribution `a l’ ́etude des fonctions quasi-convexes. These d’Etat, Univ. de Clermont II.
  • Dellacherie C. and Meyer, P.A. (1975) Probabilit ́es et Potentiel. Hermann, Paris.
  • Deville, R., Godefroy, G. and Zizler, D. (1993) Smoothness and renorming in Banach spaces. Longman, Harlow.
  • Diestel, J. (1975) Geometry of Banach Spaces. Selected Topics. Lecture Notes in Mathematics 485, Springer Verlag.
  • Diewert, W.E. (1982) Duality approaches to microeconomics theory. In: K.J. Arrow and M.D. Intriligator, eds., Handbook of Mathematical Economics, vol. 2, North Holland, Amsterdam, 535–599.
  • Dolecki, S. and Kurcyusz, S. (1978) On Φ-convexity in extremal problems. SIAM J. Control Optim. 16, 277–300
  • Dontchev, A. and Hager, A. (1994) Implicit functions, Lipschitz maps and stability in optimization. Math. Oper. Res. 19, 753–768.
  • Eberhard, A. and Nyblom, M. (1998) Jets, generalised convexity, proximal normality and differences of functions. Nonlinear Anal. 34, 319–360.
  • Elqortobi, A. (1992) Inf-convolution quasi-convexe des fonctionnelles positives. Rech. Oper. 26, 301–311.
  • Elqortobi, A. (1993) Conjugaison quasi-convexe des fonctionnelles positives Annales Sci. Math. Qu ́ebec 17 (2), 155–167.
  • Elster, K.-H. and Nehse, R. (1974) Zur theorie der polarfunktionale. Math. Oper. Stat. 5, 3–21.
  • Elworthy, K.D. (1975) Measures on infinite dimensional manifolds. In: Functional Integration and Applications. Clarendon Press, Oxford, 60–68.
  • Evers, J.J.M. and van Maaren, H. (1981) Duality principles in mathematics and their relations to conjugate functions. Preprint, Univ. of Technology of Twente.
  • Fan, K. (1963) On the Krein-Milman theorem. In: Convexity, Proc. Pure Math. 7, Amer. Math. Soc. Providence, 211–220.
  • Flachs, J. and Pollatschek, A.M. (1979) Duality theorems for certain programs involving minimum or maximum operations. Math. Prog. 16, 348–370.
  • Flores-Bazan, F. (1995) On a notion of subdifferentiability for non-convex functions. Optimization 33, 1–8.
  • Flores-Bazan, F. and Martinez-Legaz, J.-E. (1998) Simplified global optimality conditions in generalized conjugacy theory. In: J.-P. Crouzeix, J.-E. Martiınez-Legaz and M. Volle, eds., Generalized Convexity, Generalized Monotonicity: Recent Results. Kluwer, Dordrecht, 305–329.
  • Fougeres, A. (1977) Coercivite, convexite, relaxation: une extension naturelle du theoreme d’inf- ́equicontinuit ́e de J.-J. Moreau. C. R. Acad. Sci., Paris, S ́er. A 285, 711–713.
  • Fougeres, A. (1977) Proprietes ǵeometriques et minoration des integrandes convexes normales coercives. C. R. Acad. Sci., Paris, S ́er. A 284, 873–876.
  • Fujishige, S. (1984) Theory of submodular programs: A Fenchel-type min-max theorem and subgradients of submodular functions. Math. Programming 29, 142–155.
  • Gromov, M. (1999) Metric Structures for Riemannian and Non-Riemannian Spaces. Birkh ̈auser, Boston.
  • Horst, R. and Tuy, H. (1990) Global 0ptimization. Springer Verlag, Berlin.
  • Ioffe, A.D. (2001) Abstract convexity and non-smooth analysis. Adv. Math. Econ. 3, 45–61.
  • Ioffe, A.D. (2001) Towards metric theory of metric regularity. In: M. Lassonde, ed., Approximation, Optimization and Mathematical Economics. Physica-Verlag, Heidelberg, 165–176.
  • Janin, R. (1973) Sur la dualit ́e en programmation dynamique. C.R. Acad. Sci., Paris, S ́er. A 277, 1195–1197.
  • John, R. (2001) A note on Minty variational inequalities and generalized monotonicity. In: N. Hadjisavvas, et al., eds., Generalized convexity and generalized monotonicity. Proceedings of the 6th international symposium, Samos, Greece, September 1999. Lect. Notes Econ. Math. Syst. 502, Springer, Berlin, 240–246.
  • Lafontaine, J. and Pansu, P. (1981) Structures M ́etriques pour les Varietes Riemanniennes. Cedic/Fernand Nathan, Paris.
  • Lau, L.J. (1970) Duality and the structure of utility functions. J. Econ. Theory, 1, 374–396.
  • Lemaire, B. (1992) Bonne position, conditionnement, et bon comportement asymptotique. S ́eminaire d’Analyse Convexe, Univ. of Montpellier 22, 5.1–5.12.
  • Lemaire, B. (1995) Duality in reverse convex optimization. In: M. Sofonea and J.-N. Corvellec, eds., Proceedings of the Second Catalan Days on Applied Mathematics. Presses Universitaires de Perpignan, Perpignan, 173–182.
  • Lemaire, B. and Volle, M. (1998) Duality in d.c. programming. In: J.-P. Crouzeix, J.-E. Mart ́ınez-Legaz and M. Volle, eds., Generalized Convexity, Generalized Monotonicity: Recent Results. Kluwer, Dordrecht, 331–342.
  • Martinez-Legaz, J.-E. (1988) On lower subdifferentiable functions. In: K.H. Hoffmann et al. eds., Trends in Mathematical Optimization. Series Numer. Math. 84, Birkhauser, Basel, 197–232.
  • Martinez-Legaz, J.-E. (1988) Quasiconvex duality theory by generalized conjugacy methods. Optimization 19 (5), 603–652.
  • Martinez-Legaz, J.-E. (1990) Generalized conjugacy and related topics. In: A. Cambini, E. Castagnoli, L. Martein, P. Mazzoleni, S. Schaible, eds., Generalized convexity and fractional programming with economic applications. Lecture Notes in Econ. and Math. Systems 345, Springer Verlag, Berlin, 168–197.
  • Martinez-Legaz, J.-E. (1991) Duality between direct and indirect utility functions under minimal hypothesis. J. Math. Econ. 20, 199–209.
  • Martinez-Legaz, J.-E. (1995) Fenchel duality and related properties in generalized conjugacy theory. Southeast Asian Bull. Math. 19 (2), 99–106.
  • Mignot, F. (1976) Controle dans les in ́egalit ́es variationnelles elliptiques. J. Funct. Anal. 22, 130–185.
  • Moreau, J.-J. (1970) Inf-convolution, sous-additivit ́e, convexit ́e des fonctions num ́eriques. J. Math. Pures et Appl. 49, 109–154.
  • Oettli, W. and Schl ̈ager, D. (1998) Conjugate functions for convex and nonconvex duality. J. Glob. Optim. 13 (4), 337–347.
  • Pallaschke, D. and Rolewicz, S. (1997) Foundations of Mathematical Optimization: Convex Analysis without Linearity. Mathematics and its Aplications 388, Kluwer, Dordrecht.
  • Rotundity, smoothness and duality 731 Pang, J.-S. (1990) Newton’s method for B-differentiable equations. Math. Oper. Res. 15, 311–341.
  • Pang, J.-S. (1995) Necessary and sufficient conditions for solution stability of parametric nonsmooth equations. In: D.Z. Du et al., eds., Recent Advances in Nonsmooth Optimization. World Scientific, Singapore, 261–288.
  • Passy, U. and Prisman, E.Z. (1985) A convexlike duality scheme for quasi-onvex programs. Math. Programming 32, 278–300.
  • Penot, J.-P. (1982) Regularity conditions in mathematical programming. Math. Prog. Study 19, 167–199.
  • Penot, J.-P. (1985) Modified and augmented Lagrangian theory revisited and augmented. Unpublished lecture. Fermat Days, Toulouse.
  • Penot, J.-P. (1995) Conditioning convex and nonconvex problems. J. Optim. Th. Appl. 90 (3), 539–558.
  • Penot, J.-P. (1997) Duality for radiant and shady problems. Acta Math. Vietnamica 22 (2), 541–566.
  • Penot, J.-P. (1998) Are generalized derivatives useful for generalized convex functions? In: J.-P. Crouzeix, J.-E. Martinez-Legaz and M. Volle, eds., Generalized Convexity, Generalized Monotonicity: Recent Results. Kluwer, Dordrecht, 3–59.
  • Penot, J.-P. (1998) Well-behavior, well-posedness and nonsmoooth analysis. Pliska Stud. Math. Bulgar. 12, 141-190.
  • Penot, J.-P. (2000) What is quasiconvex analysis? Optimization 47, 35–110.
  • Penot, J.-P. (2001) Duality for anticonvex programs. J. of Global Optim. 19, 163–182.
  • Penot, J.-P. and Volle, M. (1987) Dualit ́e de Fenchel et quasi-convexite. C. R. Acad. Sci., Paris, Serie I 304 (13), 371-374.
  • Penot, J.-P. and Volle, M. (1988) Another duality scheme for quasiconvex problems. In: K.H. Hoffmann et al., eds., Trends in Mathematical Optimization. Int. Series Numer. Math. 84, Birkh ̈auser, Basel, 259–275.
  • Penot, J.-P. and Volle, M. (1990) On quasi-convex duality. Math. Oper. Research 15, 597–625.
  • Penot, J.-P. and Volle, M. (1990) On strongly convex and paraconvex dualities. In: A. Cambini, E. Castagnoli, L. Martein, P. Mazzoleni, S. Schaible, eds., Generalized convexity and fractional programming with economic applications. Lecture notes in Econ. and Math. Systems 345, Springer Verlag, Berlin, 198–218.
  • Pichard, K. (2001) Equations diff ́erentielles dans les espaces m ́etriques. Applications `a l’ ́evolution de domaines. Thesis, Univ. of Pau.
  • Pichard, K. and Gautier, S. (2000) Equations with delays in metric spaces: the mutational approach. Numer. Funct. Anal. Optim. 21, 917–932.
  • Pini, R. and Singh, C. (1997) A survey of recent advances in generalized convexity with applications to duality theory and optimality conditions (1985-1995). Optimization 39 (4), 311–360.
  • Poliquin, R. (1992) An extension of Attouch’s Theorem and its application to second-order epi-differentiation of convexly composite functions. Trans. Amer. Math. Soc. 332, 861–874.
  • Robinson, S. (1991) Strongly regular generalized equations. Math. Oper. Research 16, 292–309.
  • Rockafellar, R.T. (1974) Augmented Lagrange multiplier functions and duality in nonconvex programming. SIAM J. Control Optim. 12, 268–285.
  • Rolewicz, S. (1993) Generalization of Asplund inequalities on Lipschitz functions. Arch. Math. 61, 484–488.
  • Rolewicz, S. (1994) On Mazur Theorem for Lipschitz functions. Arch. Math. 63, 535–540.
  • Rolewicz, S. (1994) Convex analysis without linearity. Control and Cybern. 23, 247–256.
  • Rolewicz, S. (1996) Duality and convex analysis in the absence of linear structure. Math. Japonica 44, 165–182.
  • Rubinov, A.M. (2000) Abstract Convexity and Global Optimization. Kluwer, Dordrecht.
  • Rubinov, A.M. and Andramonov, M.Yu. (1999) Minimizing increasing starshaped functions based on abstract convexity. J. of Global Optim. 15, 19–39.
  • Rubinov, A.M. and Glover, B.M. (1997) On generalized quasiconvex conjugacy. Contemporary Mathematics 204, 199–217.
  • Rubinov, A.M. and Glover, B.M. (1998) Duality for increasing positively homogeneous functions and normal sets. Rech. Op ́er. 32, 105-123.
  • Rubinov, A.M. and Glover, B.M. (1998) Quasiconvexity via two step functions. In: J.-P. Crouzeix, J.-E. Martinez-Legaz and M. Volle, eds., Generalized Convexity, Generalized Monotonicity: Recent Results. Kluwer, Dordrecht, 159–183.
  • Rubinov, A.M., Glover, B.M. and Yang, X.Q. (1999) Decreasing functions with applications to optimization. SIAM J. Optim. 10, 289–313.
  • Rubinov, A.M. and Shveidel, A.P. (2000) Separability of star-shaped sets with respect to infinity. In: X. Yang et al., eds., Progress in Optimization. Kluwer, Dordrecht, 45–63.
  • Rubinov, A.M. and Simsek, B. (1995) Conjugate quasiconvex nonnegative functions. Optimization 35, 1–22.
  • Rubinov, A.M. and Simsek, B. (1995) Dual problems of quasiconvex maximization. Bull. Aust. Math. Soc. 51, 139–144.
  • Schwartz, L. (1973) Radon measures on arbitrary topological spaces and cylindrical measures. Lecture Notes, Tata Institute 6, Bombay, Oxford University Press, London.
  • Schwartz, L. (1980) Semi-martingales sur des vari ́et ́es et martingales conformes sur des varietes reelles ou complexes. Lecture Notes in Maths. 780, Springer Verlag, Berlin.
  • Singer, I. (1986) Some relations between dualities, polarities, coupling functions and conjugations. J. Math. Anal. Appl. 115, 1–22.
  • Singer, I. (1987) Optimization by level set methods. VI: generalizations of surrogate type reverse convex duality. Optimization 18 (4), 485–499.
  • Singer, I. (1997) Abstract Convex Analysis. J. Wiley, New York.
  • Tao, P.D. and El Bernoussi, S. (1988) Duality in D.C. (difference of convex functions). Optimization. Subgradient methods. In: K.H. Hoffmann et al. eds., Trends in Mathematical Optimization. Int. Series Numer. Math. 84, Birkhauser, Basel, 277–293.
  • Tao, P.D. and El Bernoussi, S. (1889) Numerical methods for solving a class of global nonconvex optimization problems. In: J.-P. Penot, ed., New methods in optimization and their industrial uses. Int. Series Numer. Math. 97, Birkha ̈user, Basel, 97–132.
  • Thach, P.T. (1991) Quasiconjugate of functions, duality relationships between quasiconvex minimization under a reverse convex constraint and quasiconvex maximization under a convex constraint and application. J. Math. Anal. Appl. 159, 299–322.
  • Thach, P.T. (1993) Global optimality criterion and a duality with a zero gap in nonconvex optimization. SIAM J. Math. Anal. 24 (6), 1537–1556.
  • Thach, P.T. (1994) A nonconvex duality with zero gap and applications. SIAM J. Optim. 4 (1), 44–64.
  • Thach, P.T. (1995) Diewert-Crouzeix conjugation for general quasiconvex duality and applications. J. Optim. Th. Appl. 86 (1), 719–743.
  • Tuy, H. (1995) D.C. optimization: theory, methods and algorithms. In: R. Horst and P.M. Pardalos, eds., Handbook of Global Optimization. Kluwer, Dordrecht, Netherlands, 149–216.
  • Vladimirov, A.A., Nesterov, Yu. E. and Chekanov, Yu.N. (1978) Uniformly convex functions. Vestnik Moskov. Univ. Ser. 115, Vyschisl. Mat. Kibernet. 3, 12–23.
  • Volle, M. (1985) Conjugaison par tranches. Annali Mat. Pura Appl. 139, 279–312.
  • Volle, M. (1997) Quasiconvex duality for the max of two functions. In: P. Griztzman, R. Horst, E. Sachs, R. Tichatschke, eds., Recent Advances in Optimization. Lecture Notes in Econ. and Math. Systems 452, Springer Verlag, Berlin, 365–379.
  • Volle, M. (1998) Duality for the level sum of quasiconvex functions and applications. ESAIM: Control, Optimisation and the Calculus of Variations art. 1, 3, 329-343, URL: http://www.emath.fr/cocv/.
  • Wolsey, L.A. (1981) Integer programming duality: price functions and sensitivity analysis. Math. Programming 20, 173–195.
  • Zalinescu, C. (1983) On uniformly convex functions. J. Math. Anal. Appl. 95, 344–374.
  • Zalinescu, C. (2002) Convex Analysis in General Vector Spaces. World Scientific, Singapore
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Bibliografia
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