First- and Second-Order Asymptotic Analysis with Applications in Quasiconvex Optimization

A Quasiconvex Asymptotic Function with Applications in Optimization

“…The next proposition is straightforward since f and f r has the same infimum. A characterization result for boundedness from below for convex functions using first and second order asymptotic functions can be found in [6,Section 3.3].…”

“…Now, we will recall [6,Theorem 3.1]. To that end, we first recall the following class of functions which includes those functions that are convex or coercive.…”

“…(2.5) The problem to find an adequate definition of an asymptotic function has been studied in the last years, since the usual asymptotic function is not well suited for the description of the behavior of a nonconvex function at infinity. Several attempts to deal with the quasiconvex case has been made in [1,5,6,10] while applications to optimization can be found in [6,9].…”

Optimality Conditions for Vector Equilibrium Problems with Applications

“…As was noted in [12][13][14], the usual asymptotic function is not good enough to describe the behavior of a nonconvex function at the infinity. Alternatives for quasiconvex functions were given in [13,14], while the authors in [15] concluded that the notion of q-asymptotic function (defined below) is the "best" suited for quasiconvexity. Several connections with other issues on generalized convexity were developed in [16,17], supporting the conclusions of [15].…”

“…In this paper, and based on previous results provided in [14][15][16][17][18], we introduce asymptotic functions for dealing with pseudomonotone operators and quasiconvex functions, simultaneously. We also use asymptotic functions for dealing with vector optimization problems in the quasiconvex case.…”