Quasikonvexers

Quasikonvexers are a theoretical class in convex analysis and related fields that encompasses objects—functions, operators, and collections of sets—that preserve or induce quasiconvexity. The term is used to describe both individual operators and families of structures that interact with quasiconvex objects in a controlled way. The concept generalizes the idea of quasiconvexity, which requires that all sublevel sets {x | f(x) ≤ α} are convex, by focusing on how such properties are maintained under transformations and combinations.

Operationally, a practical formulation identifies a quasikonxer as an operator T on a function space such that for any quasiconvex function f, the composition f ∘ T remains quasiconvex. Alternatively, a quasikonxer can be a family of sets or a class of functions closed under a quasiconvexity-preserving operation, such as pointwise maxima, convex combinations of inputs, or certain monotone transformations.

Properties and examples often emphasize closure behaviors. The composition of two quasikonxers is typically a quasikonxer, and the identity operator is a trivial quasikonxer. Examples include translations and certain linear maps that preserve convex sublevel sets, as well as the operation of taking the maximum of two quasiconvex functions, which yields a quasiconvex function and suggests a corresponding quasikonxer in the function class.

Applications of quasikonvexers appear in optimization, where they aid in constructing surrogate objectives that retain tractability, in economic theory for modeling quasi-preferential structures, and in machine learning for designing robust objective functions. They relate to, but are distinct from, quasiconvex functions, convex operators, and monotone transformations.