Semi-Lagrangian scheme: Difference between revisions
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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 | |||
: <math>f(x) - f(u) \geqq g(x, u) \cdot \nabla f(u), \, </math> | |||
for all ''x'' and ''u''. | |||
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]]. | |||
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. | |||
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> | |||
==See also== | |||
* [[Convex function]] | |||
* [[Pseudoconvex function]] | |||
* [[Quasiconvex function]] | |||
==References== | |||
<references/> | |||
==Further reading== | |||
S. K. Mishra and G. Giorgi, Invexity and optimization, Nonconvex optimization and Its Applications, Vol. 88, Springer-Verlag, Berlin, 2008. | |||
[[Category:Real analysis]] | |||
[[Category:Types of functions]] | |||
[[Category:Convex analysis]] | |||
[[Category:Generalized convexity]] | |||
Revision as of 20:58, 15 December 2013
In vector calculus, an invex function is a differentiable function ƒ from Rn to R for which there exists a vector valued function g such that
for all x and u.
Invex functions were introduced by Hanson [1] as a generalization of convex functions. Ben-Israel and Mond [2] provided a simple proof that a function is invex if and only if every stationary point is a global minimum.
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.
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.[3]
See also
References
- ↑ M.A. Hanson, On sufficiency of the Kuhn–Tucker conditions, J. Math. Anal. Appl. 80, pp. 545–550 (1981)
- ↑ Ben-Israel, A. and Mond, B., What is invexity?, The ANZIAM Journal 28, pp. 1–9 (1986)
- ↑ M.A. Hanson, Invexity and the Kuhn-Tucker Theorem, J. Math. Anal. Appl. vol. 236, pp. 594–604 (1999)
Further reading
S. K. Mishra and G. Giorgi, Invexity and optimization, Nonconvex optimization and Its Applications, Vol. 88, Springer-Verlag, Berlin, 2008.