A constructive approach to strengthen algebraic descriptions of function and operator classes
(2026) Journal of Nonsmooth Analysis and Optimization — Vol. Volume 6, n° Original research articles, p. 1-33 (2026)
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- Abstract
- It is well known that functions (resp. operators) satisfying a property on a subset ⊂ ℝ cannot necessarily be extended to a function (resp. operator) satisfying on the whole of ℝ. Given ⊆ ℝ , this work considers the problem of obtaining necessary and ideally sufficient conditions to be satisfied by a function (resp. operator) on , ensuring the existence of an extension of this function (resp. operator) satisfying on ℝ. More precisely, given some property , we present a refinement procedure to obtain stronger necessary conditions to be imposed on. 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.
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Rubbens, A., Hendrickx, J., & Taylor, A. (2026). A constructive approach to strengthen algebraic descriptions of function and operator classes. Journal of Nonsmooth Analysis and Optimization, Volume 6(Original research articles), 1-33. https://doi.org/10.46298/jnsao-2026-15673 (Original work published 2026)