Set containment characterization for quasiconvex programming (Q1041441)

The authors extend well-known results from convex analysis regarding the characterization of feasible sets which are described by level sets of convex functions. In the paper, so-called evenly convex sets and \(H\)-evenly convex sets are introduced which are intersections of open half-spaces and intersections of open half-spaces containing the origin. By this, a function \(f: \mathbb{R}^n\to\mathbb{R}\) is called evenly quasiconvex or \(H\)-evenly quasiconvex if its lower level sets are evenly convex or \(H\)-evenly convex respectively. Several types of quasiconjugates and biquasiconjugates of a function are introduced which allow the dual characterization of quasiconvexity, especially the comparison of the level sets of quasiconvex functions with associated level sets of its conjugates and biconjugates. In the main theorems the authors provide dual characterizations of containments of convex sets, defined by quasiconvex constraints, in larger convex sets.