Abstract. In real algebraic geometry, Lojasiewicz’s theorem asserts that any integral curve of the gradient flow of an analytic function that has an accumulation point has a unique limit. …

Polyak inequality A special case of the Łojasiewicz inequality, due to Polyak [ru], is commonly used to prove linear convergence of gradient descent algorithms. This section is based on …

2014年2月20日 · In real algebraic geometry, Lojasiewicz's theorem asserts that any integral curve of the gradient flow of an analytic function that has an accumulation point has a unique limit. …

2009年12月22日 · The classical Lojasiewicz inequality and its extensions for partial differential equation problems (Simon) and to o-minimal structures (Kurdyka) have a considerable impact …

As can be seen from a counterexample due to Palis and De Melo [19, p. 14], the set of cluster points of a bounded trajectory generated by the gradient of a C∞ function is, in general, far …

The Lojasiewicz exponent of f at the origin, denoted by ρf, is the infimum of the exponents ρ satisfying the Lojasiewicz’s gradient inequality. Bochnak and Risler (cf. [Bo-Ri]) proved that ρf …

In a metric context, we show that a generalized form of theLojasiewicz inequality (hereby ca lled the Kurdyka-Lojasiewicz inequali-ty) relates to metric regularity and to the Lipschitz continuity …

2021年1月14日 · The Lojasiewicz inequality has found rather striking applications in the theory of ordinary and partial differential equations, in particular to gradient flows.

One of these inequalities is a growth measure of the energy functional with respect to the Monge-Kantorovich distance to stationary points, while the other provides a relationship between the …

2021年4月12日 · R. Chill, The Lojasiewicz-Simon gradient inequality on Hilbert spaces, in Proceedings of the 5th European-Maghrebian Workshop on Semigroup Theory, Evolution …