Superlinear separation and dual properties of radiant functions

Superlinear functionals are used to separate points from a radiant set according to both a strict and a weak version. Strict separation characterizes closed radiant sets; weak separation is used to define evenly radiant sets, which are characterized by means of a property of the tangent cone to the set at points of the boundary. The separation properties can be described via a polarity relation between a normed space X and the set L of continuous superlinear functionals defined on X. Radiant functions are the ones which are increasing along rays, i.e. the ones whose lower level sets are radiant and so they extend the class of quasiconvex functions with minimum at the origin. We study two particular subclasses: the one of l.s.c. radiant functions, whose lower level sets are closed and radiant and the one of evenly radiant functions, whose lower levels are evenly radiant. We introduce a conjugate function (defined on L), in two different versions, and prove the coincidence between a function and its second conjugate when the function belongs to one of the classes mentioned above. The conjugate function is then used to give global optimality conditions for problems described by radiant objective and constraints.