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Invexity and generalized convexity. (English) Zbl 0731.26009

A differentiable function f: n is said to be invex if there exists a function η(y,x) n such that, for all y,x n

f(y)-f(x)η(y,x) t f(x)·

Various extensions of such functions including pseudo- and quasi-invex have been defined and their relationship to each other and other generalizations of convexity have been studied. For the non- differentiable case, f is said to be pre-invex if

f(x+tη(y,x))tf(y)+(1-t)f(x),0t1·

This comprehensive paper brings together many of these scattered results which are then studied, compared, and extended. Some definitions that are introduced include η-invex subsets, pseudo and quasi-pre-invex functions. One very minor correction. Reference 2 should be to the Journal, not the Bulletin, of the Australian Mathematical Society.


MSC:
26B25Convexity and generalizations (several real variables)
90C30Nonlinear programming