A general asymptotic function with applications in nonconvex optimization

We introduce a new concept of asymptotic functions which allows us to simultaneously study convex and concave functions as well as quasiconvex and quasiconcave functions. We provide some calculus rules and most relevant properties of the new asymptotic functions for application purpose. We also compare them with the classical asymptotic functions of convex analysis. By using the new concept of asymptotic functions we establish sufficient conditions for the nonemptiness and for the boundedness of the solution set of quasiconvex minimization problems and quasiconcave maximization problems. Applications are given for quadratic and fractional quadratic problems.