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Invex functions and constrained local minima

  • B.D. Craven (a1)

Abstract

If a certain weakening of convexity holds for the objective and all constraint functions in a nonconvex constrained minimization problem, Hanson showed that the Kuhn-Tucker necessary conditions are sufficient for a minimum. This property is now generalized to a property, called K-invex, of a vector function in relation to a convex cone K. Necessary conditions and sufficient conditions are obtained for a function f to be K-invex. This leads to a new second order sufficient condition for a constrained minimum.

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References

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[1]Craven, B.D., Mathematical programming and control theory (Chapman and Hall, London; John Wiley & Sons, New York; 1978).
[2]Craven, B.D., “Duality for generalized convex fractional programs”, Generalized concavity in optimization and economics (Academic Press, New York, London, to appear).
[3]Craven, B.D. and Mond, B., “Sufficient Fritz John optimality conditions for nondifferentiable convex programming”, J. Austral. Math. Soc. Ser. B 19 (1975/1976), 462468.
[4]Fiacco, Anthony V., McCormick, Garth P., Nonlinear programming: sequential unconstrained minimization techniques (John Wiley and Sons, New York, London, Sydney, 1968).
[5]Hanson, Morgan A., “On sufficiency of the Kuhn-Tucker conditions”, J. Math. Anal. Appl. 80 (1981), 545550.
[6]Hanson, M.A. and Mond, B., “Further generalizations of convexity in mathematical programming” (Pure Mathematics Research Paper No. 80–6, Department of Mathematics, La Trobe University, Melbourne, 1980). See also: J. Inform. Optim. Sci. (to appear).
[7]Mond, B. and Hanson, M.A., “On duality with generalized convexity” (Pure Mathematical Research Paper No. 80–4, Department of Mathematics, La Trobe University, Melbourne, 1980).
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