General existence of an invex-to-convex homotopy preserving the minimizer

Determine whether, in general, there exists a homotopy from a coercive invex function on R^n to a convex function that preserves the same global minimizer throughout the homotopy.

Background

Example 3.2 exhibits an explicit homotopy on R that transforms a specific coercive invex function to a convex one while preserving the global minimizer.

The authors note that, beyond special cases and the situations covered by their semiflow-based result (which ensures a continuous Lyapunov-function homotopy for GAS equilibria), the general existence of such invex-to-convex homotopies remains unsettled.