Abstract
We present a review of some ad hoc subdifferentials which have been devised for the needs of generalized convexity such as the quasi-subdifferentials of Greenberg-Pierskalla, the tangential of Crouzeix, the lower subdifferential of Plastria, the infradifferential of Gutiérrez, the subdifferentials of Martínez-Legaz-Sach, Penot-Volle, Thach. We complete this list by some new proposals. We compare these specific subdifferentials to some all-purpose subdifferentials used in nonsmooth analysis. We give some hints about their uses. We also point out links with duality theories.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
K.J. Arrow and A. Enthoven, Quasi-concave programming, Econometrica 29 (4) (1961), 779–800.
M. Atteia, Analyse convexe projective, C.R. Acad. Set. Paris série A 276 (1973), 795–798, ibidem 855–858.
M. Atteia, A. Elqortobi, Quasi-convex duality, in “Optimization and optimal control, Proc. Conference Oberwolfach March 1980”, A. Auslender et al. eds. Lecture notes in Control and Inform. Sci. 30, Springer-Verlag, Berlin, 1981, 3–8.
J.-P. Aubin and H. Frankowska, Set-valued analysis, Birkhäuser, Basel, 1990.
D. Aussel, Theoreme de la valeur moyenne et convexité généralisée en analyse non régulière, thesis, Univ. B. Pascal, Clermont, Nov. 1994.
D. Aussel, Subdifferential properties of quasiconvex and pseudoconvex functions: a unified approach, preprint, Univ. B. Pascal, Clermont-Ferrand, April 1995.
D. Aussel, J.-N. Corvellec and M. Lassonde, Mean value property, and subdifferential criteria for lower semicontinuous functions, Trans. Amer. Math. Soc. 347 (1995), 4147–4161.
D. Aussel, J.-N. Corvellec and M. Lassonde, Subdifferential characterization of quasiconvexity and convexity, J. Convex Anal. 1 (2) (1994) 195–202.
D. Aussel, J.-N. Corvellec and M. Lassonde, Nonsmooth constrained optimization and multidirectional mean value inequalities, preprint, Univ. Antilles-Guyane, Pointe-à -Pitre, Sept. 1996.
M. Avriel, Nonlinear programming. Analysis and methods, Prentice Hall, Englewood Cliffs, New Jersey, 1976.
M. Avriel and S. Schaible, Second order characterizations of pseudoconvex functions, Math. Prog. 14 (1978), 170–145.
D. Azé and M. Voile, A stability result in quasi-convex programming, J. Optim. Th. Appl. 67 (1) (1990), 175–184.
M. Avriel, W.E. Diewert, S. Schaible and I. Zang, Generalized Concavity, Plenum Press, New York and London 1988.
M. Avriel and S. Schaible, Second order characterization of pseudoconvex functions, Math. Program. 14 (1978), 170–185.
E.J. Balder, An extension of duality-stability relations to non-convex optimization problems, SIAM J. Control Opt. 15 (1977), 329–343.
E.N. Barron, R. Jensen and W. Liu, u t + H(u,Du) = 0. J. Differ. Eq. 126 (1996), 48–61.
H.P. Benson, Concave minimization theory. Applications and algorithms, in Handbook of Global Optimization, R. Horst and P.M. Pardalos, eds. Kluwer, Dordrecht, Netherlands (1995), 43–148.
D. Bhatia and P. Jain, Nondifferentiable pseudo-convex functions and duality for minimax programming problems, Optimization 35 (3) (1995), 207–214.
D. Bhatia and P. Kumar, Duality for variational problems with B-vex functions, Optimization 36 (4) (1996), 347–360.
M. Bianchi, Generalized quasimonotonicity and strong pseudomonotonicity of bifunctions, Optimization 36 (1) (1996), 1–10.
C.R. Bector and C. Singh, B-vex functions, J. Optim. Th. Appl 71 (2) (1991), 237–254.
A. Ben Israel and B. Mond, What is invexity?, J. Aust. Math. Soc. Ser. B 28 (1986), 1–9.
J. Borde and J.-P. Crouzeix, Continuity properties of the normal cone to the level sets of a quasiconvex function, J. Opt. Th. Appl. 66 (1990), 415–429.
J.M. Borwein, S.P. Fitzpatrick and J.R. Giles, The differentiability of real functions on normed linear spaces using generalized subgradients, J. Math. Anal. Appl. 128 (1987), 512 - 534.
J.M. Borwein and Q.J. Zhu, Viscosity solutions and viscosity subderivatives in smooth Banach spaces with application to metric regularity, SIAM J. Control and Opt. 1996.
A. Cambini, E. Castagnoli, L. Martein, P. Mazzoleni, S. Schaible (eds), Generalized Convexity and fractional programming with economic applications, Proc. Pisa, 1988, Lecture Notes in Economics and Math. Systems 345, Springer Verlag, Berlin, 1990.
A. Cambini and L. Martein, Generalized concavity and optimality conditions in vector and scalar optimization,, in“ Generalized convexity” S. Komlósi, T. Racsák, S. Schaible, eds., Springer Verlag, Berlin, 1994, 337–357.
R. Cambini, Some new classes of generalized concave vector-valued functions, Optimization 36 (1) (1996), 11–24.
F.H. Clarke, Optimization and Nonsmooth Analysis, Wiley-Interscience, New-York, 1983.
F.H. Clarke and Yu S. Ledyaev, New finite increment formulas, Russian Acad. Dokl. Math. 48 (1) (1994), 75–79.
F.H. Clarke, R.J. Stern and P.R. Wolenski, Subgradient criteria for monotonicity, the Lipschitz condition and monotonicity, Canadian J. Math. 45 (1993), 1167–1183.
R. Correa, A. Jofre and L. Thibault, Characterization of lower semicontinuous convex functions, Proc. Am. Math. Soc. 116 (1992), 67–72.
R. Correa, A. Jofre and L. Thibault, Subdifferential monotonicity as a characterization of convex functions, Numer. Funct. Anal. Opt. 15 (1994), 531–535.
B.D. Craven, Invex functions and constrained local minima, Bull. Aust. Math. Soc. 24 (1981), 357–366.
B.D. Craven and B.M. Glover, Invex functions and duality, J. Aust. Math. Soc. Ser. A 39 (1985), 1–20.
B.D. Craven, D. Ralph and B.M. Glover, Small convex-valued subdifferentials in mathematical programming, Optimization 32 (1) (1995), 1–22.
J.-P. Crouzeix, Polaires quasi-convexes et dualite, C.R. Acad. Sci. Paris série A 279 (1974), 955–958.
J.-P. Crouzeix, Contribution à l’étude des fonctions quasi-convexes, Thése d’Etat, Univ. de Clermont II, 1977.
J.-P. Crouzeix, Some differentiability properties of quasiconvex functions on IR n, in“ Optimization and optimal control, Proceedings Conference Oberwolfach 1980”, A. Auslender, W. Oettli and J. Stoer, eds. Lecture Notes in Control and Information Sciences 30, Springer-Verlag (1981), 89–104.
J.-P. Crouzeix, Continuity and differentiability properties of quasiconvex functions on IRn , in“ Generalized concavity in optimization and economics”, S. Schaible and W.T. Ziemba, eds. Academic Press, New York, (1981), 109–130.
J.-P. Crouzeix, About differentiability of order one of quasiconvex functions on IRn, J. Optim. Th. Appl. 36 (1982), 367–385.
J.-P. Crouzeix, Duality between direct and indirect utility functions, J. Math. Econ. 12 (1983), 149–165.
J.-P. Crouzeix and J.A. Ferland, Criteria for quasiconvexity and pseudo- convexity: relationships and comparisons, Math. Programming 23 (1982), 193–205.
J.-P. Crouzeix and J.A. Ferland, Criteria for differentiate generalized monotone maps, Math. Programming 75 (1996), 399–406.
J.-P. Crouzeix, J.A. Ferland and S. Schaible, Generalized convexity on affine subspaces with an application to potential functions, Math. Programming 56 (1992) 223–232.
J.-P. Crouzeix, J.A. Ferland and C. Zalinescu, α-convex sets and strong quasiconvexity, preprint, Univ. B. Pascal, Clermont, 1996.
R.A. Danao, Some properties of explicitely quasiconcave functions, J. Op-tim. Th. Appl. 74 (3) (1992) 457–468.
R. Deville, G. Godefroy and V. Zizler, Smoothness and renormings in Banach spaces, Pitman Monographs in Math. 64, Longman, 1993.
W.E. Diewert,“ Alternative characterizations of six kinds of quasiconcavity in the nondifferentiable case with applications to nonsmooth programming”, in: S. Schaible and W.T. Ziemba (eds.) Generalized Concavity in Optimization and Economics, Academic Press, New-York, 1981.
W.E. Diewert, Duality approaches to microeconomics theory, in: Handbook of Mathematical Economics, vol. 2, K.J. Arrow and M.D. Intriligator, eds. North Holland, Amsterdam, 1982, 535–599.
S. Dolecki and S. Kurcyusz, On Φ-convexity in extremal problems, SIAM J. Control Optim. 16 (1978), 277–300.
A. Eberhard and M. Nyblom, Generalized convexity, proximal normality and differences of functions, preprint, Royal Melbourne Institute of Technology, Melbourne, Dec. 1995
A. Eberhard, M. Nyblom, D. Ralph, Applying generalized convexity notions to jets, preprint, Royal Melbourne Institute of Technology and Univ. Melbourne, Sept. 1996.
R. Ellaia and H. Hassouni, Characterization of nonsmooth functions through their generalized gradients, Optimization 22 (1991), 401–416.
A. Elqortobi, Inf-convolution quasi-convexe des fonctionnelles positives, Rech. Oper. 26 (1992), 301–311.
K.-H. Elster and J. Thierfelder: Abstract cone approximations and generalized differentiability of in nonsmooth optimization, Optimization 19 (1988), 315–341.
K. H. Elster and A. Wolf, Recent results on generalized conjugate functions,
M. Fabian, Subdifferentials, local [-supports and Asplund spaces, J. London Math. Soc. (2) 34 (1986), 568–576.
M. Fabian, On classes of subdifferentiability spaces of Ioffe, Nonlinear Anal., Th. Meth. Appl. 12 (1) (1988), 63–74.
M. Fabian, Subdifferentiability and trustworthiness in the light of a new variational principle of Borwein and Preiss, Acta Univ. Carolinae 30 (1989), 51–56.
F. Flores-Bazan, On a notion of subdifferentiability for non-convex functions, Optimization 33, (1995), 1–8.
P. Georgiev, Submonotone mappings in Banach spaces and applications, Set- Valued Anal, to appear.
G. Giorgi and S. Komlósi, Dini derivatives in optimization, Part I, Revista di Mat. per le sc. econ. e sociali 15 (1), 1993, 3-30, Part II, idem 15 (2)(1993), 3–24, Part III, idem 18 (1) (1996), 47–63
G. Giorgi and S. Mitutelu, Convexités généralisées et propriétés, Revue Roumaine Math. Pures Appl. 38 (2) (1993), 125–142.
B.M. Glover, Generalized convexity in nondifferentiable programming, Bull. Australian Math. Soc. 30 (1984) 193–218.
B.M. Glover, Optimality and duality results in nonsmooth programming, preprint, Ballarat Univ. College.
H.P. Greenberg and W.P. Pierskalla, Quasiconjugate function and surrogate duality, Cahiers du Centre d’Etude de Recherche Oper. 15 (1973), 437–448.
J.M. Gutiérrez, Infragradientes y direcciones de decrecimiento, Rev. Real Acad. C. Ex., Fis. y Nat. Madrid 78 (1984), 523–532.
J.M. Gutiérrez, A generalization of the quasiconvex optimization problem, to appear in J. Convex Anal. 4 (2) (1997).
J. Gwinner, Bibliography on non-differentiable optimization and non- smooth analysis, J. Comp. Appl. Math. 7 (1981), 277–285.
S. Hackman and U. Passy, Projectively-convex sets and functions, J. Math. Econ. 17 (1988) 55–68.
M.A. Hanson, On sufficiency of the Kuhn-Tucker conditions, J. Math. Anal. Appl. 80 (1981), 545–550.
H. Hartwig, On generalized convex functions, Optimization 14 (1983), 49–60.
H. Hartwig, Local boundedness and continuity of generalized convex functions, Optimization 26 (1992), 1–13.
A. Hassouni, Sous-différentiel des fonctions quasi-convexes, thèse de troisieme cycle, Univ. P. Sabatier, Toulouse, 1983.
K. Hinderer and M. Stiegglitz, Minimization of quasi-convex symmetric and of discretely quasi-convex symmetric functions, Optimization 36 (4) (1996), 321–332.
R. Horst and P.M. Pardalos (eds.), Handbook of global optimization, Kluwer, Dordrecht, 1995.
R. Horst, P.M. Pardalos and N.V. Thoai, Introduction to global optimization, Kluwer, Dordrecht, 1995.
R. Horst and H. Tuy, Global optimization, deterministic approaches, Springer Verlag, Berlin, 1990.
A.D. Ioffe, On subdifferentiability spaces, Ann. N. Y. Acad. Sci. 410 (1983), 107–119.
A.D. Ioffe, Subdifferentiability spaces and nonsmooth analysis, Bull. Amer. Math. Soc. 10 (1984), 87–90.
A.D. Ioffe, Approximate subdifferentials and applications I. The finite dimensional theory, Trans. Amer. Math. Soc. 281 (1984), 289–316.
A.D. Ioffe, On the theory of subdifferential, Fermat Days 85: Mathematics for Optimization, J.B. Hiriart-Urruty, ed., Math. Studies series, North Holland, Amsterdam, (1986), 183–200.
A.D. Ioffe, Approximate subdifferentials and applications II. The metric theory, Mathematika 36 (1989), 1–38.
A.D. Ioffe, Proximal analysis and approximate subdifferentials, J. London Math. Soc. 41 (1990), 261–268.
A.D. Ioffe, Codirectional compactness, metric regularity and subdifferential calculus, preprint, Technion, Haifa, 1996.
A.D. Ioffe and J.-P. Penot, Subdifferential of performance functions and calculus of coderivatives of set-valued mappings, Serdica Math. J. 22 (1996), 359–384.
E.H. Ivanov and R. Nehse, Relations between generalized concepts of convexity and conjugacy, Math. Oper. Stat. Optimization 13 (1982), 9–18.
R. Janin, Sur une classe de fonctions sous-linearisables, C.R. Acad. Sci. Paris A 277 (1973), 265–267.
R. Janin, Sur la dualité en programmation dynamique, C.R. Acad. Set. Paris A 277 (1973), 1195–1197.
V. Jeyakumar, Nondifferentiable programming and duality with modified convexity, Bull. Australian Math. Soc. 35 (1987) 309–313.
V. Jeyakumar, W. Oettli and M. Natividad, A solvability theorem for a class of quasiconvex mappings with applications to optimization, J. Math. Anal. Appl. 179 (1993), 537–546.
V. Jeyakumar and V.F. Demyanov, A mean value theorem and a characterization of convexity using convexificators, preprint, Univ. New South Wales, Sydney, 1996.
R. John, Demand-supply systems, variational inequlities and (generalized) monotone functions, preprint Univ. Bonn, August 1996.
C. Jouron, On some structural design problems, in: Analyse non convexe, Pau, 1979, Bulletin Soc. Math. France, Mémoire 60, 1979, 87–93.
S. Karamardian, Complementarity over cones with monotone and pseudomonotone maps, J. Optim. Theory Appl. 18 (1976), 445–454.
S. Karamardian and S. Schaible, Seven kinds of monotone maps, J. Optim. Theory Appl. 66 (1990), 37–46.
S. Karamardian, S. Schaible and J.-P. Crouzeix, Characterizations of Generalized Monotone Maps, J. Opt. Th. Appl. 76 (3) (1993), 399–413.
S. Komlósi, On a possible generalization of Pshenichnyi’s quasidifferentiability, Optimization 21 (1990), 3–11.
S. Komlósi, Some properties of nondifferentiable pseudoconvex functions, Math. Programming 26 (1983), 232–237.
S. Komlósi, On generalized upper quasidifferentiability, in: F. Giannessi (ed.)“ Nonsmooth Optimization: Methods and Applications”, Gordon and Breach, London, 1992, 189–200.
S. Komlósi, Quasiconvex first order approximations and Kuhn-Tucker type optimality conditions, European J. Opt. Res. 65 (1993), 327–335.
S. Komlósi, Generalized monotonicity in nonsmooth analysis, in Generalized convexity, S. Komlósi,, T. Rapcsáck, S. Schaible, eds. Lecture Notes in Economics and Math. Systems 405, Springer Verlag, Berlin, (1994), 263–275.
S. Komlosi, Generalized monotonicity and generalized convexity, J. Opt. Theory Appl. 84 (1995), 361–376.
S. Komlósi, Monotonicity and quasimonotonicity in nonsmooth analysis, in:“ Recent Advances in Nonsmooth Optimization”, D.Z. Du, L. Qi, R.S. Womersley (eds.) World Scientific Publishers, Singapore, 1995, 193–214.
A. Ya. Kruger, Properties of generalized differentials, Siberian Math. J. 26 (1985), 822–832.
G. Lebourg, Valeur moyenne pour un gradient généralisé, C.R. Acad. Sci. Paris, 281 (1975), 795–797.
G. Lebourg, Generic differentiability of Lipschitzian functions, Trans. Amer. Math. Soc. 256 (1979), 125–144.
P.O. Lindberg, A generalization of Fenchel conjugation giving generalized lagrangians and symmetric nonconvex duality, Survey of Mathematical Programming, (Proc. 9th Intern. Progr. Symposium) Akad. Kiado and North Holland, 1 (1979), 249–268.
J.C. Liu, Optimization and duality for multiobjective fractional programming involving nonsmooth (F,p)-convex functions, Optimization 36 (4) (1996), 333–346.
J.C. Liu, Optimization and duality for multiobjective fractional programming involving nonsmooth pseudoconvex functions, Optimization 37 (1) (1996), 27–40.
D.T. Luc, Characterizations of quasiconvex functions, Bull. Austral. Math. Soc. 48 (1993), 393–405.
Ph. Loewen, A Mean Value Theorem for Fréchet subgradients, Nonlinear Anal Th. Methods, Appl. 23 (1994), 1365–1381.
D.T. Luc, On generalised convex nonsmooth functions, Bull. Aust. Math. Soc. 49 (1994), 139–149.
D.T. Luc, Characterizations of quasiconvex functions, Bull. Austral. Math. Soc. 48 (1993), 393–405.
D.T. Luc and S. Schaible, Generalized monotone nonsmooth maps, J. Convex Anal. 3 (2) (1996), 195–206.
D.T. Luc and S. Swaminathan, A characterization of convex functions, Nonlinear Analysis, Theory, Methods & Appl., 30 (1993), 697 - 701.
O.L. Mangasarian, Pseudoconvex functions, SIAM J. Control 3 (1965), 281–290.
O.L. Mangasarian, Nonlinear Programming, Mc Graw-Hill, New-York, 1969.
D.H. Martin, The essence of invexity, J. Opt Th. Appl. 47 (1985), 65–76.
J.-E. Martínez-Legaz, Level sets and the minimal time function of linear control processes, Numer. Fund. Anal. Optim. 9 (1-2) (1987), 105–129.
J.-E. Martínez-Legaz, Quasiconvex duality theory by generalized conjugation methods, Optimization, 19 (1988) 603–652.
J.-E. Martínez-Legaz, On lower subdifferentiable functions, Trends in Mathematical Optimization, K.H. Hoffmann et al. eds, Int. Series Numer. Math. 84 Birkhauser, Basel, 1988, 197–232.
J.-E. Martínez-Legaz, Generalized conjugation and related topics, in Generalized convexity and fractional programming with economic applications, Proceedings Symp. Pisa, A. Cambini et al. eds, Lecture Notes in Economics and Math. Systems 345, Springer-Verlag, Berlin, 1990, pp. 168–197.
J.-E. Martínez-Legaz, Weak lower subdifferentials and applications, Optimization 21 (1990), 321–341.
J.-E. Martínez-Legaz, Duality between direct and indirect utility functions under minimal hypothesis, J. Math. Econ. 20 (1991) 199–209.
J.-E. Martínez-Legaz, On convex and quasiconvex spectral functions, in Proceedings of the second Catalan days on Applied Mathematics, M. Sofonea and J.-N. Corvellec, eds., Presses Univ. Perpignan (1995), 199–208.
J.-E. Martínez-Legaz, Dual representation of cooperative games based on Fenchel-Moreau conjugation, Optimization 36 (4) (1996), 291–320.
J.-E. Juan-Enrique Martínez-Legaz and Romano-Rodriguez, ower sub-differentiability of quadratic functions, Math. Prog. 60 (1993), 93 - 113.
J.-E. Martfnez-Legaz and S. Romano-Rodriguez, α—lower subdifferentiable functions, Siam J. Optim. 3 (4) (1993), 800–825.
J.-E. Martínez-Legaz and P.H. Sach, A new subdifferential in quasiconvex analysis, preprint 95/9, Hanoi Institute of Math, 1995.
J.-E. Martínez-Legaz and M.S. Santos, Duality between direct and indirect preferences, Econ. Theory 3 (1993), 335–351.
B. Martos, Nonlinear programming, theory and methods, North Holland, Amsterdam, 1975.
P. Mazzoleni, Generalized concavity for economic applications, Proc. Workshop Pisa 1992, Univ. Verona.
Ph. Michel and J.-P. Penot, A generalized derivative for calm and stable functions, Differential and Integral Equations, 5 (2) (1992), 433–454.
B.S. Mordukhovich, Nonsmooth Analysis with Nonconvex Generalized Differentials and Adjoint Mappings, Dokl. Akad. Nauk Bielorussia SSR, 28 (1984) 976–979.
B.S. Mordukhovich, Approximation Methods in Problems of Optimization and Control, Nauka, Moscow, Russia, 1988.
B.S. Mordukhovich, and Y. Shao, Nonsmooth Sequential Analysis in Asplund Spaces, Trans. Amer. Math. Soc. 348 (4) (1996) 1235–1280.
J.-J. Moreau, Inf-convolution, sous-additivité, convexité des fonctions numériques, J. Math. Pures et Appl. 49 (1970), 109–154.
D. Pallaschke and S. Rolewicz, Foundations of mathematical optimization, book to appear.
U. Passy and E.Z. Prisman, A convexlike duality scheme for quasiconvex programs, Math. Programming 32 (1985), 278–300.
B.N. Pchenitchny and Y. Daniline, Méthodes numériques dans les problemes d’extrémum, Mir, French transl. Moscow, (1975).
J.-P. Penot, Modified and augmented Lagrangian theory revisited and augmented, unpublished lecture, Fermat Days 85, Toulouse (1985).
J.-P. Penot, On the Mean Value Theorem, Optimization, 19 (1988) 147–156.
J.-P. Penot, Optimality conditions for composite functions, preprint 90–15, Univ. of Pau, partially published in“ Optimality conditions in mathematical programming and composite optimization”, Math. Programming 67 (1994), 225–245.
J.-P. Penot, Miscellaneous incidences of convergence theories in optimization, Part II: applications to nonsmooth analysis, in“Recent advances in nonsmooth optimization”, D.-Z. Du et al. eds, World Scientific, Singapore, (1995), pp. 289–321.
J.-P. Penot, A mean value theorem with small subdifferentials, J. Optim. Th. Appl. 94 (1) (1997), 209–221.
J.-P. Penot, Generalized Convexity in the Light of Nonsmooth Analysis, Recent Developments in Optimization, Edited by R. Durier and C. Michelot., Lecture Notes in Economics and Mathematical Systems Springer Verlag, Berlin, Germany,, Vol. 429, pp. 269–290, 1995.
J.-P. Penot, Views on nonsmoooth analysis, unpublished manuscript for the Conference on Nonsmooth Analysis, Pau, June 1995.
J.-P. Penot, Subdifferential calculus and subdifferential compactness, Proceedings of the 2nd Catalan Days on Applied Mathematics, Presses Universitaires Perpignan, (1995), 209–226.
J.-P. Penot, Favorable classes of mappings and multimappings in nonlinear analysis and optimization, J. Convex Anal. 3 (1) (1996), 97–116.
J.-P. Penot, Duality theories for anticonvex problems, preprint Univ. of Pau, 1996.
J.-P. Penot, Conjugacies for radiant and shady problems, preprint, Univ. of Pau, 1996.
J.-P. Penot, Nonsmooth analysis, from subdifferential calculus to codifferential calculus, in preparation.
J.-P. Penot and P.H. Quang, On generalized convex functions and generalized monotonicity of set-valued maps, preprint Univ. Pau, Nov. 1992, to appear in J. Opt. Th. Appl. 92 (2) (1997), 343–356.
J.-P. Penot and P.H. Quang, On the cutting plane algorithm, preprint Univ. of Pau.
J.-P. Penot and P.H. Sach, Generalized monotonicity of subdifferentials and generalized convexity, J. Optim. Theory and Appl. 64 (1) (1997), 251–262.
J.-P. Penot and P. Terpolilli, Cones tangents et singularités, C.R. Acad. Sc. Paris série I, 296 (1983), 721–724.
J.-P. Penot and M. Voile, Dualité de Fenchel et quasi-convexité, C.R. Acad. Sciences Paris série I, 304 (13) (1987), 269–272.
J.-P. Penot and M. Voile, On quasi-convex duality, Math. Operat. Research 15 (4) (1990), 597–625.
J.-P. Penot and M. Volle, Another duality scheme for quasiconvex problems, Trends in Mathematical Optimization, K.H. Hoffmann et al. eds, Int. Series Numer. Math. 84 Birkhauser, Basel, 1988, 259–275.
J.-P. Penot and M. Volle, On strongly convex and paraconvex dualities, in Generalized convexity and fractional programming with economic applications, Proceedings Symp. Pisa, A. Cambini et al. eds, Lecture Notes in Economics and Math. Systems 345, Springer-Verlag, Berlin, 1990, pp. 198–218.
J.-P. Penot and M. Voile, Surrogate duality in quasiconvex programming, preprint 1997.
R. Phelps, Convex Functions, Monotone Operators and Differentiability, Lecture Notes in Mathematics, Springer Verlag, Berlin, Germany, Vol. 1364, 1989.
H.X. Phu, Six kinds of roughly convex functions, J. Opt. Th. Appl. 92 (2), 357-376.
H.X. Phu, Some properties of globally -convex functions, Optimization 35 (1995), 23–41.
H.X. Phu, -subdifferential and γ-convexity of functions on the real line, Applied Math. Optim. 27 (1993), 145–160.
H.X. Phu, -subdifferential and -convexity of functions on a normed vector space, J. Optim. Th. Appl. 85 (1995), 649–676.
H.X. Phu and P.T. An, Stable generalization of convex functions, Optimization 38 (4) (1996), 309–318.
R. Pini and C. Singh, A survey of recent advances in generalized convexity with applications to duality theory and optimality conditions (1985-1995), Optimization, 39 (4) (1997), 311–360.
F. Plastria, Lower subdifferentiable functions and their minimization by cutting plane, J. Opt. Th. Appl 46 (1) (1985), 37–54.
R.A. Poliquin, Subgradient monotonicity and convex functions, Nonlinear Analysis, Theory, Meth. and Appl 14 (1990), 305–317.
J. Ponstein, Seven kinds of convexity, SIAM Review 9 (1967) 115–119.
B.N. Pshenichnyi, Necessary conditions for an extremum, Dekker, New York, 1971.
P. Rabier, Definition and properties of of a particular notion of convexity, Numer. Funct. Anal. Appl. 7 (4) (1985–1985), 279–302.
R.T. Rockafellar, Augmented Lagrangians and the proximal point algorithm in convex programming, Math. Oper. Res. 1 (1976), 97–116.
R.T. Rockafellar, The theory of subgradients and its applications to problems of optimization of convex and nonconvex functions, Presses de l’niversité de Montreal and Helderman Verlag, Berlin 1981.
R.T. Rockafellar, Favorable classes of Lipschitz continuous functions in subgradient optimization, in: Progress in nondifferentiable optimization, E. Nurminski (ed.) IIASA, Laxenburg, 1982, 125–144.
S. Rolewicz, On γ-paraconvex multifunctions, Math. Japonica 24 (3) 1979, 415–430.
A.M. Rubinov and B.M. Glover, On generalized quasiconvex conjugation, preprint, Univ. of Ballarat and Univ. Negev, Beer-Sheva, 1996.
A.M. Rubinov and B. Simsek, Conjugate quasiconvex nonnegative functions, preprint, Univ. of Ballarat, August 1994.
A.M. Rubinov and B. Simsek, Dual problems of quasiconvex maximization, Bull. Aust. Math. Soc. 51 (1995)
P.H. Sach and J.-P. Penot, Characterizations of generalized convexities via generalized directional derivatives, preprint, Univ. of Pau, January 1994.
S. Schaible, Second-order characterizations of pseudoconvex quadratic functions, J. Opt. Th. Appl. 21 (1) (1977), 15–26.
S. Schaible, Generalized monotone maps, in F. Giannessi (ed.) Nonsmooth optimization: Methods and Applications, Proc. Symp. Erice, June 1991, Gordon and Breach, Amsterdam,, 1992, 392–408.
S. Schaible, Generalized monotonicity-a survey, in Generalized convexity Proc. Pecs, Hungary 1992, Lecture Notes in Economics and Math. Systems, Springer-Verlag, Berlin, 1994, 229–249.
S. Schaible, Generalized monotonicity-concepts and uses, in“ Variational inequalities and network equilibrium problems”, Proc. 19th course, Int. School of Math. Erice, June 1994, F. Giannessi and A. Maugeri, eds. Plenum, New York, 1995, 289–299.
S. Schaible and W.T. Ziemba (eds.) Generalized Concavity in Optimization and Economics, Academic Press, New-York, 1981.
B. Simsek and A.M. Rubinov, Dual problems of quasiconvex maximization, Bull. Aust. Math. Soc. 51 (1995), 139–144.
I. Singer, Some relations between dualities, polarities, coupling functions and conjugations, J. Math. Anal. Appl. 115 (1986), 1–22.
J.E. Spingarn, Submonotone subdifferentials of Lipschitz functions, Trans. Amer. Math. Soc. 264 (1) (1981), 77–89.
C. Sutti, Quasidifferentiability of nonsmooth quasiconvex functions, Optimization 27 (4) (1993) 313–320.
C. Sutti, Quasidifferential analysis of positively homogeneous functions, Optimization 27 (1/2) (1993) 43–50.
Y. Tanaka, Note on generalized convex function, J. Optim. Th. Appl. 66 (2) (1990) 345–349.
P.D. Tao and S. El Bernoussi, Duality in D.C. (difference of convex functions). Optimization. Subgradient methods, Trends in Mathematical Optimization, K.H. Hoffmann et al. eds, Int. Series Numer. Math. 84, Birkhauser, Basel, 1988, 277 - 293.
P.D. Tao and S. El Bernoussi, Numerical methods for solving a class of global nonconvex optimization problems, New methods in optimization and their industrial uses, J.-P. Penot ed., Int. Series Numer. Math. 97 Birkhauser, Basel, 1989, 97–132.
P.D. Tao and Le Thi Hoai An, D.C. optimization algorithms for computing extreme symmetric eigenvalues, preprint INSA Rouen 1996.
P.T. Thach, Quasiconjugate of functions, duality relationships between quasiconvex minimization under a reverse convex convex constraint and quasiconvex maximization under a convex constraint and application, J. Math. Anal. Appl. 159 (1991) 299–322.
P.T. Thach, Global optimality criterion and a duality with a zero gap in nonconvex optimization, SIAM J. Math. Anal. 24 (6) (1993), 1537–1556.
P.T. Thach, A nonconvex duality with zero gap and applications, SIAM J. Optim. 4 (1) (1994), 44–64.
L. Thibault and D. Zagrodny, Integration of subdifferentials of lower semi- continuous functions on Banach spaces, J. Math. Anal, and Appl. 189 (1995), 33–58.
W.A. Thompson, Jr. and D.W. Parke, Some properties of generalized concave functions, Oper. Research 21 (1) (1974), 305–313.
S. Traoré and M. Volle, On the level sum of two convex functions on Banach spaces, J. of Convex Anal. 3 (1) (1996), 141–151.
S. Traoré and M. Volle, Epiconvergence d’une suite de sommesen niveaux de fonctions convexes, Serdica Math. J. 22 (1996), 293–306.
J.S. Treiman, Shrinking generalized gradients, Nonlin. Anal. Th., Methods, Appl. 12 (1988), 1429–1450.
J.S. Treiman, An infinite class of convex tangent cones, J. Opt. Th. and Appl. 68 (3) (1991), 563–582.
J.S. Treiman, Too many convex tangent cones, preprint, Western Michigan Univ.
H. Tuy, Convex programs with an additional reverse convex constraint, J. Optim. Theory Appl. 52 (1987), 463–486.
H. Tuy, D.C. optimization: theory, methods and algorithms, in Handbook of Global Optimization, R. Horst and P.M. Pardalos, eds., Kluwer, Dordrecht, Netherlands (1995), 149–216.
H. Tuy, On nonconvex optimisation problems with separated nonconvex variables, J. Global Optim. 2 (1992), 133–144.
H. Tuy, D.C. representation, and D.C. reformulation of nonconvex global optimization problems, preprint 95/8 Institute of Math. Hanoi, 1995.
J.-P. Vial, Strong and weak convexity of sets and functions, Math. Oper. Res. 8 (2) (1983) 231–259.
M. Volle, Convergence en niveaux et en épigraphes, C.R. Acad. Sci. Paris 299 (8) (1984), pp. 295–298.
M. Volle, Conjugaison par tranches, Annali Mat. Pura Appl. 139 (1985) 279–312.
M. Volle, Conjugaison par tranches et dualité de Toland, Optimization 18 (1987) 633–642.
M. Volle, Quasiconvex duality for the max of two functions, Proc. 8 th French-German Conference on Opt., Trier, to appear.
M. Volle, Duality for the level sum of quasiconvex functions and applications, preprint, Univ. Avignon.
M. Volle, Conditions initiates quasiconvexes dans les équations de Hamilton-Jacobi, to appear C.R. Acad. Sci. Paris.
X. M. Yang, Semistrictly convex functions, Opsearch 31 (1994), 15–27.
X.Q. Yang and G.H. Chen, A class of nonconvex functions and pre- variational inequalities, J. Math. Anal, and Appl. 169 (1992) 359–373.
X.Q. Yang, Generalized convex functions and vector variational inequalities, J. Opt. Th. and Appl., 79 (1993) 563–580.
X.Q. Yang, Generalized second-order characterizations of convex functions, J. Opt. Th. and Appl., 82 (1994) 173–180.
X.Q. Yang, Continuous generalized convex functions and their characterizations, Preprint, University of Western Australia, Australia, 1997.
D. Zagrodny, Approximate mean value theorem for upper subderivatives, Nonlinear Anal. Th. Meth. Appl. 12 (1988), 1413–1428.
D. Zagrodny, A note on the equivalence between the Mean Value Theorem for the Dini derivative and the Clarke-Rockafellar derivative, Optimization, 21 (1990), 179–183.
D. Zagrodny, Some recent mean value theorems in nonsmooth analysis, in Nonsmooth Optimization. Methods and Applications, Proc. Symp. Erice 1991, F. Giannessi ed., Gordon and Breach, OPA, Amsterdam 1992, 421–428.
D. Zagrodny, General sufficient conditions for the convexity of a function, Zeitschrift fur Anal. Anwendungen 11 (1992), 277–283.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1998 Kluwer Academic Publishers
About this chapter
Cite this chapter
Penot, JP. (1998). Are Generalized Derivatives Sseful for Generalized Convex Functions?. In: Crouzeix, JP., Martinez-Legaz, JE., Volle, M. (eds) Generalized Convexity, Generalized Monotonicity: Recent Results. Nonconvex Optimization and Its Applications, vol 27. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-3341-8_1
Download citation
DOI: https://doi.org/10.1007/978-1-4613-3341-8_1
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4613-3343-2
Online ISBN: 978-1-4613-3341-8
eBook Packages: Springer Book Archive