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A programming problem with an Lp norm in the objective function

Published online by Cambridge University Press:  17 February 2009

Bertram Mond  and
Murray Schechter
Bertram Mond
Affiliation:
Department of Pure Mathematics, Latrobe University, Melbourne, Australia Department of Applied Mathematics, Technion, Israel Institute of Technology, Haifa, Israel.
Murray Schechter
Affiliation:
Department of Mathematics, Lehigh University, Bethlehem, Penna, U.S.A. Department of Applied Mathematics, Technion, Israel Institute of Technology, Haifa, Israel.
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Abstract

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We consider a programming problem in which the objective function is the sum of a differentiable function and the p norm of Sx, where S is a matrix and p > 1. The constraints are inequality constraints defined by differentiable functions. With the aid of a recent transposition theorem of Schechter we get a duality theorem and also a converse duality theorem for this problem. This result generalizes a result of Mond in which the objective function contains the square root of a positive semi-definite quadratic function.

Type
Research Article
Information
The ANZIAM Journal , Volume 19 , Issue 3 , June 1976 , pp. 333 - 342
Copyright
Copyright © Australian Mathematical Society 1976

References

[1]Eisenberg, E., ‘Supports of a convex function’, Bull. Am. Math. Soc. 68, (1962), 192–195.CrossRefGoogle Scholar
[2]Eisenberg, E., ‘A gradient inequality for a class of non-differentiable functions’, Oper. Res. 14 (1966), 157–163.CrossRefGoogle Scholar
[3]Kantorovich, L. V. and Akilov, G. P., Functional Analysis in Normed Spaces, Macmillan Company, New York, (1971).Google Scholar
[4]Kuhn, H. W. and Tucker, A. W., ‘Nonlincar programming’ in Proceedings 2nd Berkeley Symposium on Mathematical Statistics and Probability, pp. 481–492, University of California Press, (1951).Google Scholar
[5]Mangasarian, O. L. and Fromovitz, S., ‘The Fritz John necessary optimality conditions in the presence of equality and inequality constraints’, J. Math. Anal. Appl. 17 (1967), 37–47.Google Scholar
[6]Mond, B., ‘Aclass of non-differentiable mathematical programming problems’, J. Math. Anal. Appl. 46 (1974), 169–174.CrossRefGoogle Scholar
[7]Mond, B. and Schechter, M., ‘On a constraint qualification in a non-differentiable programming problem’, Nav. Res. Log. Quart. 23 (1976), 611–613.CrossRefGoogle Scholar
[8]Rockafeller, R. T., Convex Analysis, Princeton University Press, Princeton, N.J., (1969).Google Scholar
[9]Schechter, M., ‘A solvability theorem for homogeneous functions’, SIAM J. Math. Anal., 7 (1976), 696–701.Google Scholar