Geodesic
On Carlier's Inequality
Journal of convex analysis, Tome 30 (2023) no. 2, pp. 499-514
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The Fenchel-Young inequality is fundamental in Convex Analysis and Optimization. It states that the difference between certain function values of two vectors and their inner product is nonnegative. Recently, Carlier introduced a very nice sharpening of this inequality, providing a lower bound that depends on a positive parameter. In this note, we expand on Carlier's inequality in three ways. First, a duality statement is provided. Secondly, we discuss asymptotic behaviour as the underlying parameter approaches zero or infinity. Thirdly, relying on cyclic monotonicity and associated Fitzpatrick functions, we present a lower bound that features an infinite series of squares of norms. Several examples illustrate our results.
Classification : 26B25, 47H05, 26D07, 90C25
Mots-clés : Carlier's inequality, cyclic monotonicity, Fenchel conjugate, Fenchel-Young inequality, Fitzpatrick function, maximally monotone operator, proximal mapping, resolvent 
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     author = {H. H. Bauschke and S. Singh and X. Wang},
     title = {On {Carlier's} {Inequality}},
     journal = {Journal of convex analysis},
     pages = {499--514},
     year = {2023},
     volume = {30},
     number = {2},
     url = {http://geodesic.mathdoc.fr/item/JCA_2023_30_2_JCA_2023_30_2_a6/}
}
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H. H. Bauschke; S. Singh; X. Wang. On Carlier's Inequality. Journal of convex analysis, Tome 30 (2023) no. 2, pp. 499-514. http://geodesic.mathdoc.fr/item/JCA_2023_30_2_JCA_2023_30_2_a6/