A constructive approach to strengthen algebraic descriptions of function and operator classes

Abstract

It is well known that functions (resp. operators) satisfying a property~$p$ on a subset $Q\subset {\mathbb{R}}^d$ cannot necessarily be extended to a function (resp. operator) satisfying~$p$ on the whole of~${\mathbb{R}}^d$. Given $Q\subseteq {\mathbb{R}}^d$, this work considers the problem of obtaining necessary and ideally sufficient conditions to be satisfied by a function (resp. operator) on $Q$, ensuring the existence of an extension of this function (resp. operator) satisfying $p$ on ${\mathbb{R}}^d$. More precisely, given some property $p$, we present a refinement procedure to obtain stronger necessary conditions to be imposed on $Q$. This procedure can be applied iteratively until the stronger conditions are also sufficient. We illustrate the procedure on a few examples, including the strengthening of existing descriptions for the classes of smooth functions satisfying a Łojasiewicz condition, convex blockwise smooth functions, Lipschitz monotone operators, strongly monotone cocoercive operators, and uniformly convex functions. In most cases, these strengthened descriptions can be represented, or relaxed, to semi-definite constraints, which can be used to formulate tractable optimization problems on functions (resp. operators) within those classes.