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| In [[vector calculus]], an '''invex function''' is a differentiable function ''ƒ'' from '''R'''<sup>''n''</sup> to '''R''' for which there exists a vector valued function ''g'' such that
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| : <math>f(x) - f(u) \geqq g(x, u) \cdot \nabla f(u), \, </math>
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| for all ''x'' and ''u''.
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| Invex functions were introduced by Hanson <ref>M.A. Hanson, On sufficiency of the Kuhn–Tucker conditions, J. Math. Anal. Appl. 80, pp. 545–550 (1981)</ref> as a generalization of [[convex function]]s. Ben-Israel and Mond <ref>Ben-Israel, A. and Mond, B., What is invexity?, The [[ANZIAM Journal]] 28, pp. 1–9 (1986)</ref> provided a simple proof that a function is invex if and only if every [[stationary point]] is a [[global minimum]].
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| Hanson also showed that if the objective and the constraints of an optimization problem are invex with respect to the same function ''g''(''x'', ''u''), then the [[Karush–Kuhn–Tucker conditions]] are sufficient for a global minimum.
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| A slight generalization of invex functions called '''Type 1 invex functions''' are the most general class of functions for which the [[Karush–Kuhn–Tucker conditions]] are necessary and sufficient for a global minimum.<ref>M.A. Hanson, Invexity and the Kuhn-Tucker Theorem, J. Math. Anal. Appl. vol. 236, pp. 594–604 (1999)</ref>
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| ==See also==
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| * [[Convex function]]
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| * [[Pseudoconvex function]]
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| * [[Quasiconvex function]]
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| ==References==
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| <references/>
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| ==Further reading==
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| S. K. Mishra and G. Giorgi, Invexity and optimization, Nonconvex optimization and Its Applications, Vol. 88, Springer-Verlag, Berlin, 2008.
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| [[Category:Real analysis]]
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| [[Category:Types of functions]]
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| [[Category:Convex analysis]]
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| [[Category:Generalized convexity]]
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