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Constraint qualifications in input optimisation

Published online by Cambridge University Press:  17 February 2009

M. Van Rooyen ,
M. Sears  and
S. Zlobec
M. Van Rooyen
Affiliation:
Department of Computational and Applied Mathematics, University of the Witwatersrand, Johannesburg, South Africa2001.
M. Sears
Affiliation:
Department of Computational and Applied Mathematics, University of the Witwatersrand, Johannesburg, South Africa2001.
S. Zlobec
Affiliation:
Department of Mathematics and Statistics, McGill University, Burnside Hall, 805 Sherbrooke Street West, Montreal, Quebec, CanadaH3A 2K6.
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Abstract

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We introduce assumptions in input optimisation that simplify the necessary conditions for an optimal input. These assumptions, in the context of nonlinear programming, give rise to conceptually new kinds of constraint qualifications.

Type
Research Article
Information
The ANZIAM Journal , Volume 30 , Issue 3 , January 1989 , pp. 326 - 342
Copyright
Copyright © Australian Mathematical Society 1989

References

[1]Bank, B., Guddat, J., Klatte, D., Kummer, B. and Tammer, K., Nonlinear Parametric Optimization, (Academie-Verlag, Berlin, 1982).CrossRefGoogle Scholar
[2]Ben-Israel, A., and Mond, B., “First-order optimality conditions for generalized convex functions: A feasible directions approach”, Utilitas Math. 25 (1984) 249–262.Google Scholar
[3]Ben-Israel, A., Ben-Tal, A., and Zlobec, S., Optimality in Nonlinear Programming: A Feasible Directions Approach, (Wiley-Interscience, New York, 1981).Google Scholar
[4]Berge, C., Espace Topologiques, Fonctions Multivogues, (Dunod, Paris, 1959).Google Scholar
[5]Charnes, A., and Zlobec, S., “Stability of efficiency evaluations in data envelopment analysis”, Research Report CCS 560, Center for Cybernetic Studies, The University of Texas, Austin03 (1987).Google Scholar
[6]Craven, B., “Alternative theorems”, ZAMM 62 (12) (1982) 699–705.CrossRefGoogle Scholar
[7]Craven, B., “Nondifferentiable optimization by smooth approximations”, Optimization 17 (1986) 3–17.CrossRefGoogle Scholar
[8]Craven, B., and Jeyakumar, V., “Alternative and duality theorems with weakened convexity”, Utilitas Math (to appear).Google Scholar
[9]Guignard, M., “Generalized Kuhn-Tucker conditions for mathematical programming problems in a Banach space”, SIAM J. Control Optim. 7 (1969) 232–241.Google Scholar
[10]Hogan, W. W., “Point-to-set maps in mathematical programming”, SIAM Rev. 15 (1973) 591–603.CrossRefGoogle Scholar
[11]Karush, W., “Minima of Functions of Several Variables with Inequalities as Side Constraints”, M. Sc. Thesis, University of Chicago, 1939.Google Scholar
[12]Klatte, D., “On the lower semicontinuity of optimal sets in convex parametric optimization”, Math. Programming Stud. 10 (1979) 104–109.CrossRefGoogle Scholar
[13]Kuhn, H. W., and Tucker, A. W., “Nonlinear programming”, in (ed. Neymann, J.) Proc. 2nd Berkeley Symp. Math. Stat. Prob. (1951) 481–492.Google Scholar
[14]Mangasarian, O., Nonlinear Programming (McGraw-Hill, New York, 1969).Google Scholar
[15]Penot, J. P., “A new constraint qualification conditions”, J. Optim. Theory Appl. 48 (1986) 459–468.CrossRefGoogle Scholar
[16]Peterson, D. W, “A review of constraint qualifications in mathematical programming,” SIAM Rev. 15 (1973) 639–654.CrossRefGoogle Scholar
[17]Semple, J., and Zlobec, S., “Continuity of a Lagrange multiplier function in input optimization”, Math. Programming 34 (1986) 362–369.CrossRefGoogle Scholar
[18]Semple, J., and Zlobec, S., “On a necessary condition for stability in perturbed linear convex programming.’ ZOR, Series A: Theorie 31 (1987) 161–172.Google Scholar
[19]van Rooyen, M., and Zlobec, S., “A complete characterization of optimal economic systems with respect to stable perturbations”, TWISK Rep. CSIR. (07 1987).Google Scholar
[20]Wei, Q.-L., “Stability in mathematical programming in the sense of lower semicontinuity and continuity”. Journal of the Qufu Teachers College 1 (1981). (In Chinese.)Google Scholar
[21]Zlobec, S., “Regions of stability for ill-posed convex programs”, Aplikace Maternatiky 27 (1982) 176–191. Addendum,Google Scholar
Zlobec, S., “Regions of stability for ill-posed convex programs”, Aplikace Maternatiky 31 (1986) 109–117.Google Scholar
[22]Zlobec, S., “Characterizing an optimal input in perturbed convex programming,” Math. Programming 25 (1983) 109–121;CrossRefGoogle Scholar
Corrigendum, Zlobec, S., “Characterizing an optimal input in perturbed convex programming,” Math. Programming 35 (1986) 368–371.CrossRefGoogle Scholar
[23]Zlobec, S., “Input optimization: I. Optimal realizations of mathematical models”, Math. Programming 31 (1985) 245–268.CrossRefGoogle Scholar
[24]Zlobec, S., “Input optimization: III. Optimal realizations of multi-objective models”, optimization 17 (1986) 429–445.Google Scholar
[25]Zlobec, S., “Survey of input optimization”, Optimization 18 (1987) 309–348.CrossRefGoogle Scholar
[26]Zlobec, S., “Optimality conditions for a class of nonlinear programs”, Utilitas Math. 32 (1987) 217–230.Google Scholar
[27]Zlobec, S., “An index condition in input optimization”, Utilitas Math. 33 (1988) 183–192.Google Scholar
[28]Zlobec, S., and Ben-Israel, A., “Perturbed convex programs: Continuity of optimal solutions and optimal values”, Methods of Operations Research, 31 (1979) 739–749.Google Scholar