Regular Polyak-Łojasiewicz functions are essentially strongly convex when their minimizing set is bounded
Aziz Ben Nejma – GJM, Volume 11, Issue 1 (2026), 67-76.
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- août 4, 2026
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The Polyak-Łojasiewicz (PŁ) inequality extends the favorable optimization properties of strongly convex functions to a broader class of functions. In this paper, we prove a theorem (also obtained by Criscitiello, Rebjock and Boumal in an earlier blog post) showing that the richness of the class of PŁ functions is rooted in the case of an unbounded set of minimizers since sufficient regularity and a bounded set of minimzers forces them to be essentially strongly convex. More precisely, if f is a C^2 PŁ function having a bounded set of minimizers, then it has a unique minimizer. We show that this implies a result of Asplund on properties of the squared distance function, and discuss some consequences on smoothness assumptions in results in the literature.
Milestones:
Received: December 08, 2025
Accepted: April 16, 2026
Published: July 30, 2026
Authors:
Aziz Ben NejmaCNRS, UMR 7539, Université Paris Cité and Sorbonne Université, CNRS,
Laboratoire de Probabilités, Statistique et Modélisation, F-75013 Paris, France and DMA.
Ecole normale supérieure, Université PSL, CNRS, 75005 Paris, France.