Invex function information

In vector calculus, an invex function is a differentiable function $f$ from $\mathbb {R} ^{n}$ to $\mathbb {R}$ for which there exists a vector valued function $\eta$ such that

f(x)-f(u)\geq \eta (x,u)\cdot \nabla f(u),\,

for all x and u.

Invex functions were introduced by Hanson as a generalization of convex functions.^[1] Ben-Israel and Mond provided a simple proof that a function is invex if and only if every stationary point is a global minimum, a theorem first stated by Craven and Glover.^[2]^[3]

Hanson also showed that if the objective and the constraints of an optimization problem are invex with respect to the same function $\eta (x,u)$ , then the Karush–Kuhn–Tucker conditions are sufficient for a global minimum.

^ Hanson, Morgan A. (1981). "On sufficiency of the Kuhn-Tucker conditions". Journal of Mathematical Analysis and Applications. 80 (2): 545–550. doi:10.1016/0022-247X(81)90123-2. hdl:10338.dmlcz/141569. ISSN 0022-247X.
^ Ben-Israel, A.; Mond, B. (1986). "What is invexity?". The ANZIAM Journal. 28 (1): 1–9. doi:10.1017/S0334270000005142. ISSN 1839-4078.
^ Craven, B. D.; Glover, B. M. (1985). "Invex functions and duality". Journal of the Australian Mathematical Society. 39 (1): 1–20. doi:10.1017/S1446788700022126. ISSN 0263-6115.

[1] Hanson, Morgan A. (1981). "On sufficiency of the Kuhn-Tucker conditions". Journal of Mathematical Analysis and Applications. 80 (2): 545–550. doi:10.1016/0022-247X(81)90123-2. hdl:10338.dmlcz/141569. ISSN 0022-247X.

[2] Ben-Israel, A.; Mond, B. (1986). "What is invexity?". The ANZIAM Journal. 28 (1): 1–9. doi:10.1017/S0334270000005142. ISSN 1839-4078.

[3] Craven, B. D.; Glover, B. M. (1985). "Invex functions and duality". Journal of the Australian Mathematical Society. 39 (1): 1–20. doi:10.1017/S1446788700022126. ISSN 0263-6115.

Invex function information

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