Zero-scale asymptotic functions and quasiconvex optimization

We introduce the notion of a zero-scale asymptotic function. In contrast to the usual asymptotic function, which is related to the slopes of a function at infinity along a given direction, the new function is related to the jumps of the function along that direction. Applications are given to the unconstrained and the constrained optimization of quasiconvex functions. Also, the problem of quasiconvex maximization is discussed. Further, a class of quasiconvex problems is introduced, that is shown to have zero duality gap. Finally, new results on quasiconvex quadratic programming are obtained.