2022/06/29 by Bauschke, Heinz H., Singh, Shambhavi, Wang, Xianfu · 2 citations
#26B25 #47H05 (Primary) 26D07 #90C25 (Secondary) #FOS: Mathematics #Optimization and Control (math.OC)
paper · doi:10.48550/arxiv.2206.14872
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.