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Continuous programming containing arbitrary norms

Published online by Cambridge University Press:  09 April 2009

S. Chandra ,
B. D. Craven  and
I. Husain
S. Chandra
Affiliation:
Mathematics DepartmentIndian Institute of TechnologyHauz Khas, New Delhi 110016, India
B. D. Craven
Affiliation:
Mathematics DepartmentRegional Engineering CollegeSrinagar, Kashmir, India
I. Husain
Affiliation:
Mathematics DepartmentUniversity of MelbourneParkville, Victoria 3052, Australia
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Abstract

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Optimality conditions and duality results are obtained for a class of cone constrained continuous programming problems having terms with arbitrary norms in the objective and constraint functions. The proofs are based on a Fritz John theorem for constrained optimization in abstract spaces. Duality results for a fractional analogue of such continuous programming problems are indicated and a nondifferentiable mathematical programming duality result, not explicitly reported in the literature, is deduced as a special case.

MSC classification

Secondary: 90C48: Programming in abstract spaces 90C30: Nonlinear programming
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 39 , Issue 1 , August 1985 , pp. 28 - 38
Copyright
Copyright © Australian Mathematical Society 1985

References

[1]Abrham, J. and Buie, R. N., ‘Kuhn-Tucker conditions and duality in continuous programming’, Utilitas Math. 16 (1979), 15–37.Google Scholar
[2]Abrham, J. and Buie, R. N., ‘Duality in continuous fractional programming’, Utilitas Math. 17 (1980), 35–44.Google Scholar
[3]Abrham, J. and Buie, R. N., ‘Duality in continuous programming and optimal control’, Utilitas Math. 17 (1980), 103–109.Google Scholar
[4]Bector, C. R., Chandra, S. and Gulati, T. R., ‘Duality for fractional control problems’, Proc. Fifth Manitoba Conference on Numerical Math., (1975), 231–241.Google Scholar
[5]Bector, C. R., Chandra, S. and Husian, I., ‘Generalized concavity and duality in continuous programming’, communicated to Utilitos Math.Google Scholar
[6]Chandra, S., Craven, B. D. and Husian, I., ‘A class of nondifferentiable continuous programming problems’, J. Math. Anal. Appl., to appear.Google Scholar
[7]Clarke, F. H., ‘A new approach to Lagrange multipliers’, Math. Oper. Res. 1 (1976), 165–174.CrossRefGoogle Scholar
[8]Clarke, F. H., ‘The maximum principle under minimal hypothesis’, SIAM. J. Control Optim. 14 (1976), 1078–1091.CrossRefGoogle Scholar
[9]Clarke, F. H., ‘Inequality constraints in the calculus of variations’, Canadian J. Math. 29 (1977), 528–540.CrossRefGoogle Scholar
[10]Clarke, F. H., ‘Generalized gradients of Lipschitz functionals’, Advances in Math. 40 (1981), 52–67.CrossRefGoogle Scholar
[11]Courant, R., Differential and integral calculus, vol. 2, Blackie, London and Edinburgh, 1936/1948.Google Scholar
[12]Courant, R. and Hubert, D., Methods of mathematical physics, vol. 1, Wiley Interscience, New York, 1948.Google Scholar
[13]Craven, B. D., Mathematical programming and control theory, Chapman & Hall, London, 1978.CrossRefGoogle Scholar
[14]Craven, B. D. and Mond, B., ‘Sufficient Fritz John conditions for nondifferentiable convex programming’, J. Agstral. Math. Soc., Ser. B, 19 (1976), 462–468.CrossRefGoogle Scholar
[15]Craven, B. D. and Mond, B., ‘Lagrangian conditions for quasi-differentiable optimization’, Proc. of 9th Int. Conf. on Mathematical Programming, Budapest 1976, Prékopa, A. (ed.), vol. 1, 177–192, in Survey of mathematical programming, Akadémiai Kiado, Budapest and North-Holland, Amsterdam, 1979.Google Scholar
[16]Craven, B. D. and Mond, B., ‘Continuous nonlinear fractional programming’, unpublished.Google Scholar
[17]Dinkelbach, W., ‘On nonlinear fractional programming’, Management Sci. 13 (1967), 492–498.CrossRefGoogle Scholar
[18]Fletcher, R. and Watson, G. A., ‘First and second order conditions for a class of nondifferentiable optimization problems’, Math. Programming 18 (1980), 291–307.CrossRefGoogle Scholar
[19]Jagannathan, R., ‘Duality for nonlinear fractional programs’, Z. Oper. Res. 17 (1973), 1–3.Google Scholar
[20]Mond, B., ‘A class of nondifferentiable fractional programming problems’, ZAMM 58 (1978), 337–341.CrossRefGoogle Scholar
[21]Mond, B., ‘A class of nondifferentiable mathematical programming problems’, J. Math. Anal. Appl. 46 (1974), 169–174.CrossRefGoogle Scholar
[22]Mond, B. and Hanson, M. A., ‘Duality for variational problems’, J. Math. Anal. Appl. 18 (1967), 355–364.CrossRefGoogle Scholar
[23]Mond, B. and Schechter, M., ‘A programming problem with an Lp norm in the objective function’, J. Austral. Math. Soc. (Ser. B) 19 (1976), 333–342.CrossRefGoogle Scholar
[24]Mond, B. and Weir, T., ‘Generalized concavity and duality’, Generalized concavity in optimization and economics, Schaible, S. and Ziemba, W. T. (ed), Academic Press, New York, 1981.Google Scholar
[25]Reiland, T. W., ‘Optimality conditions and duality in continuous programs’, J. Math. Anal. Appl. 77 (1980), 329–343.CrossRefGoogle Scholar
[26]Reiland, T. W. and Hanson, M. A., ‘Generalized Kuhn-Tucker conditions and duality for continuous nonlinear programming problems’, J. Math. Anal. Appl. 74 (1980), 578–598.CrossRefGoogle Scholar
[27]Robinson, S. M., ‘Stability theory for systems of inequalities, part II: Differentiable nonlinear systems’, SIAM J. Numer. Anal. 13 (1976), 476–513.CrossRefGoogle Scholar
[28]Rockafellar, R. T., ‘Conjugate convex functions in optimal control and the calculus of variations’, J. Math. Anal. Appl. 32 (1970), 174–222.CrossRefGoogle Scholar
[29]Rockafellar, R. T., ‘Existence and duality theorems for convex problems of Bolza’, Trans. Amer. Math. Soc. 159 (1971), 1–40.CrossRefGoogle Scholar
[30]Rockafellar, R. T., ‘Dual problems of optimal control’, in Techniques of optimization, Balakrishnan, A. V. (ed.), Academic Press, New York, 1972.Google Scholar
[31]Schechter, M., ‘More on subgradient duality’, J. Math. Anal. Appl. 71 (1979), 251–262.CrossRefGoogle Scholar
[32]Watson, G. A., ‘A class of programming problems whose objective function contains a norm’, J. Math. Anal. Appl. 23 (1978), 401–411.Google Scholar
[33]Weir, T., Hanson, M. A. and Mond, B., ‘Generalized concavity and duality in continuous programming’, Technical Report M649, Florida State University, Talahassee, Florida, 11. 1982.Google Scholar