  We propose a novel approach to obtain interpolation constraints for a wide
range of function classes, i.e. necessary and sufficient constraints that a set
of points, functions values and (sub)gradients must satisfy to ensure the
existence of a global function of the class considered, consistent with this
set. The derivation of such constraints is crucial for instance in the
performance analysis of optimization methods, since obtaining a priori tight
performance guarantees requires using a tight description of function classes
of interest. Our method allows setting aside all analytic properties of the
function class to work only at an algebraic level, and to easily obtain
counterexamples when a condition characterizing a function class cannot serve
as an interpolation constraint. As an illustration, we provide interpolation
constraints for a class of non convex non smooth functions: weakly convex
functions with bounded subgradients, and rely on these new interpolation
constraints to outperform state of the art bounds on the performance of the
subgradient method on this class.
