A convex function satisfying the Lojasiewicz inequality but failing the gradient conjecture both at zero and infinity

2021/02/10 by Daniilidis, Aris, Haddou, Mounir, Ley, Olivier · 1 citation
#Dynamical Systems (math.DS) #FOS: Mathematics #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.2102.05342

Abstract

We construct an example of a smooth convex function on the plane with a strict minimum at zero, which is real analytic except at zero, for which Thom's gradient conjecture fails both at zero and infinity. More precisely, the gradient orbits of the function spiral around zero and at infinity. Besides, the function satisfies the Lojasiewicz gradient inequality at zero.

Cited by

• SAD Neural Networks: Divergent Gradient Flows and Asymptotic Optimality via o-minimal Structures

Related

• Proof of the gradient conjecture of R. Thom
• Lojasiewicz inequalities and applications
• Convex functions with unbounded gradient
• Characterizations of Lojasiewicz inequalities and applications
• Some resonances of Lojasiewicz inequalities