Papers › A constructive approach to strengthen algebraic descriptions of function and operator classes
A constructive approach to strengthen algebraic descriptions of function and operator classes
Anne Rubbens, Julien M. Hendrickx, Adrien Taylor
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It is well known that functions (resp. operators) satisfying a property~p on a subset Q⊂ℝᵈ cannot necessarily be extended to a function (resp. operator) satisfying~p on the whole of~ℝᵈ. Given Q ⊆ℝᵈ, 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 ℝᵈ. 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 \L{}ojasiewicz 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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