Hostname: page-component-5c6d5d7d68-lvtdw Total loading time: 0 Render date: 2024-08-16T14:30:49.285Z Has data issue: false hasContentIssue false
  • English
  • Français

Generalized convexity in nondifferentiable programming

Published online by Cambridge University Press:  17 April 2009

Bevil M. Glover
Bevil M. Glover
Affiliation:
43 Wirbill Street, Cobram, Victoria, 3644, Australia.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

For an abstract mathematical programming problem involving quasidifferentiable cone-constraints we obtain necessary (and sufficient) optimality conditions of the Kuhn-Tucker type without recourse to a constraint qualification. This extends the known results to the non-differentiable setting. To obtain these results we derive several simple conditions connecting various concepts in generalized convexity not requiring differentiability of the functions involved.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 30 , Issue 2 , October 1984 , pp. 193 - 218
Copyright
Copyright © Australian Mathematical Society 1984

References

[1]Avriel, M., Diewert, W.E., Schaible, S. and Ziemba, W.T., “Introduction to concave and generalized concave functions”, in Generalized Concavity in Optimization and Economics, Schaible, S. and Ziemba, W.T. (eds.), Academic Press, New York, 1981, 21–50.Google Scholar
[2]Bector, C.R., “Programming problems with convex fractional functions,” Operations Research 16 (1968), 383–389.CrossRefGoogle Scholar
[3]Tal, A. Ben, Ben-Israel, A. and Zlobec, S., “Characterization of optimality in convex programming without a constraint qualification”, J.O.T.A. 20 (1976), 417–437.Google Scholar
[4]Borwein, J.M., “Fractional programming without differentiability”, Math. Prog. 11 (1976), 283–290.CrossRefGoogle Scholar
[5]Borwein, J.M. and Wolkowicz, H., “Characterization of optimality without constraint qualification for the abstract convex program”, Math. Programming Study 19 (1982), 77–100.CrossRefGoogle Scholar
[6]Clarke, F.H., “A new approach to Lagrange multipliers”, Math. Oper. Res. 1, (1976), 165–174.CrossRefGoogle Scholar
[7]Clarke, F.H., “Generalized gradients of Lipschitz functionals”, Advances in Math. 40 (1981), 52–67.CrossRefGoogle Scholar
[8]Craven, B.D., Mathematical Programming and Control Theory, (Chapman and Hall, London, 1978).CrossRefGoogle Scholar
[9]Craven, B.D., “Vector-valued optimization”, in Generalized Concavity in Optimization and Economics, Schaible, S. and Ziemba, W.T. (eds.) Academic Press, New York, 1981, 661–680.Google Scholar
[10]Craven, B.D., Glover, B.M. and Zlobec, S., “On minimization subject to cone constraints”, Numer. Func. Analysis Optimiz., 6(4), (1983), 363–378.CrossRefGoogle Scholar
[11]Craven, B.D. and Mond, B., “Lagrangean conditions for quasidifferentiable optimization” in Survey of Mathematical Programming, Prékopa, A. (ed.), Akadémiai Kiadó, Budapest, 1, 1976.Google Scholar
[12]Craven, B.D. and Mond, B., “On duality for fractional programming”, Z.A.M.M. 58 (1959), 278–279.Google Scholar
[13]Craven, B.D. and Zlobec, S., “Complete characterization of optimality for convex programming in Banach spaces”, Applicable Analysis 11 (1980), 61–78.CrossRefGoogle Scholar
[14]Cobzas, S., “On the Lipschitz properties of continuous convex functions”, Mathematica (Cluj) 21(44) (1979), 123–125.Google Scholar
[15]Chandra, S., “Strong pseudo-convex programming”, Indian J. of Pure and Applied Math. 3 (1972), 178–182.Google Scholar
[16]Crouziex, J.-P., “Continuity and differentiability properties of quasiconvex functions on Rn”, in Generalized Concavity in Optimization and Economics, Schaible, S. and Ziemba, W.T. (eds.), Academic Press, New York, 1981, 109–130.Google Scholar
[17]Dempster, M.A.H. and Wets, R.J.-B., “On regularity conditions in constrained optimization”, Center for Advanced Study in the Behavioral Sciences, Stanford University (1975).Google Scholar
[18]Diewert, W.E., “Alternative characterizations of six kinds of quasiconcavity in the nondifferentiable case with applications to non-smooth programming” in Generalized Concavity in Optimization and Economics, Schaible, S. and Ziemba, W.T. (eds.), Academic Press, New York (1981), 51–94.Google Scholar
[19]Flett, T.M., Differential Analysis, (Cambridge University Press, Cambridge, 1980).CrossRefGoogle Scholar
[20]Glover, B.M., “A generalized Farkas lemma with application to quasi-differentiable programming”, Z. Oper. Res. 26 (1982), 125–142.Google Scholar
[21]Glover, B.M., Nondifferentiable Programming (M.Sc. Thesis, University of Melbourne, 1982).Google Scholar
[22]Guignard, M., “Generalized Kuhn-Tucker conditions for mathematical programming problems in Banach space”, SIAM J. Control 7 (1969).CrossRefGoogle Scholar
[23]Hiriart-Urruty, J.B., “On optimality conditions in nondifferentiable programming”, Math. Prog. 14 (1978), 73–86.CrossRefGoogle Scholar
[24]Hiriart-Urruty, J.B., “New concepts in nondifferentiable programming”, Bull. Soc. Math. France, Mémoire 60 (1979), 57–85.CrossRefGoogle Scholar
[25]Karamardian, S., “Strictly quasiconvex (concave) functions and duality in mathematical programming” J. Math. Analysis and Applns. 20 (1967), 344–358.CrossRefGoogle Scholar
[26]Lebourg, G., “Generic differentiability of lipschitzian functions”, Trans. Amer. Math. Soc. 256 (1979), 125–144.CrossRefGoogle Scholar
[27]Mangasarian, O.L., Nonlinear Programming, (McGraw-Hill, New York, 1969).Google Scholar
[28]Mifflin, R., “Semismooth and semiconvex functions in constrained optimization”, SIAM J. Control and Optimiz. 15 (1977), 959–972.CrossRefGoogle Scholar
[29]Mond, B. and Zlobec, S., “Duality for nondifferentiable programming without a constraint qualification”, Utilitas Mathematica 15 (1979), 291–302.Google Scholar
[30]Nashed, M.Z., “Differentiability and related properties of nonlinear operators”, in Nonlinear Functional Analysis and Applications, Hall, L.B. (ed.), (Academic Press, New York, 1971, 103–309).CrossRefGoogle Scholar
[31]Pshenichnyi, B.N., Necessary Conditions for an Extremwn, (Marcel Dekkar, New York, 1971).Google Scholar
[32]Rockafellar, R.T., Conjugate Duality and Optimization, SIAM Regional Conference Series in Applied Mathematics 16 (1974), Philadelphia.CrossRefGoogle Scholar
[33]Yamamuro, S., Differential Calculus in Topological Linear Spaces, (Lecture Notes in Mathematics 374. Springer-Verlag, 1974).CrossRefGoogle Scholar
[34]Zalinescu, C., “On an abstract control problem”, Numer. Func. Analysis Optimiz. 2 (1980), 531–542.CrossRefGoogle Scholar
[35]Zlobec, S. and Craven, B.D., “Stabilization and determination of a set of minimal binding constraints in convex programming”, Math. Operation-forschung und Statistik, series Optimization 12 (1981), 203–220.CrossRefGoogle Scholar
[36]Zlobec, S. and Jacobson, D.H., “Minimizing an arbitrary function subject to convex constraints”, Utilitas Mathematica 17 (1980), 239–257.Google Scholar
[37]Craven, B.D. and Glover, B.M., “Invex functions and duality”, J. Austral. Math. Soc., (to appear).Google Scholar