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Generalized convexity in nondifferentiable programming

Published online by Cambridge University Press:  17 April 2009

Bevil M. Glover
Affiliation:
43 Wirbill Street, Cobram, Victoria, 3644, Australia.
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Abstract

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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
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, 2150.Google Scholar
[2]Bector, C.R., “Programming problems with convex fractional functions,” Operations Research 16 (1968), 383389.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), 417437.Google Scholar
[4]Borwein, J.M., “Fractional programming without differentiability”, Math. Prog. 11 (1976), 283290.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), 77100.CrossRefGoogle Scholar
[6]Clarke, F.H., “A new approach to Lagrange multipliers”, Math. Oper. Res. 1, (1976), 165174.CrossRefGoogle Scholar
[7]Clarke, F.H., “Generalized gradients of Lipschitz functionals”, Advances in Math. 40 (1981), 5267.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, 661680.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), 363378.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), 278279.Google Scholar
[13]Craven, B.D. and Zlobec, S., “Complete characterization of optimality for convex programming in Banach spaces”, Applicable Analysis 11 (1980), 6178.CrossRefGoogle Scholar
[14]Cobzas, S., “On the Lipschitz properties of continuous convex functions”, Mathematica (Cluj) 21(44) (1979), 123125.Google Scholar
[15]Chandra, S., “Strong pseudo-convex programming”, Indian J. of Pure and Applied Math. 3 (1972), 178182.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, 109130.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), 5194.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), 125142.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), 7386.CrossRefGoogle Scholar
[24]Hiriart-Urruty, J.B., “New concepts in nondifferentiable programming”, Bull. Soc. Math. France, Mémoire 60 (1979), 5785.CrossRefGoogle Scholar
[25]Karamardian, S., “Strictly quasiconvex (concave) functions and duality in mathematical programmingJ. Math. Analysis and Applns. 20 (1967), 344358.CrossRefGoogle Scholar
[26]Lebourg, G., “Generic differentiability of lipschitzian functions”, Trans. Amer. Math. Soc. 256 (1979), 125144.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), 959972.CrossRefGoogle Scholar
[29]Mond, B. and Zlobec, S., “Duality for nondifferentiable programming without a constraint qualification”, Utilitas Mathematica 15 (1979), 291302.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, 103309).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), 531542.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), 203220.CrossRefGoogle Scholar
[36]Zlobec, S. and Jacobson, D.H., “Minimizing an arbitrary function subject to convex constraints”, Utilitas Mathematica 17 (1980), 239257.Google Scholar
[37]Craven, B.D. and Glover, B.M., “Invex functions and duality”, J. Austral. Math. Soc., (to appear).Google Scholar