Convex and quasiconvex truncations of nonconvex functions

TL;DR

This paper studies nonconvex functions with truncations that are convex or quasiconvex, focusing on smooth functions with positive definite Hessian regions, and proves injectivity of their restricted gradients in these regions.

Contribution

It introduces a class of nonconvex functions with convex or quasiconvex truncations and establishes gradient injectivity properties within positive definite Hessian regions.

Findings

Injectivity of restricted gradients in positive definite Hessian regions.

Characterization of smooth functions with level sets in positive definite regions.

Examples of functions with convex/quasiconvex truncations starting from certain levels.

Abstract

We consider nonconvex real valued functions whose truncations are either quasiconvex or even convex starting with a certain level. Among them, the $C^2$-smooth functions whose level sets are all completely contained in the positive definite region of their Hessian matrices, starting with a certain level, are good examples of such functions. For such a function we show the injectivity of its restricted gradient to a large subset of the positive definite region of its Hessian matrices.