The Łojasiewicz Inequality for Nonsmooth Subanalytic Functions with Applications to Subgradient Dynamical Systems

Characterizations of Łojasiewicz inequalities: Subgradient flows, talweg, convexity

“…Let f : X −→ R ∪ {+∞} be a lower semicontinuous function defined on a complete metric space X and ϕ ∈ K(0, r 0 ) (see (8) (i ) The multivalued mapping…”

“…This was one of the motivations for the nonsmooth K L-inequalities developed in [8,9]. Due to its considerable impact on several fields of applied mathematics: minimization and algorithms [1,5,8,39], asymptotic theory of differential inclusions [48], neural networks [28], complexity theory [47] (see [47,Definition 3], where functions satisfying a K L-type inequality are called gradient dominated functions), and partial differential equations [51,35,30,31], we hereby tackle the problem of characterizing such inequalities in a nonsmooth infinite-dimensional setting and provide further clarifications for several application aspects. Our framework is rather broad (infinite dimensions, nonsmooth functions); nevertheless, to the best of our knowledge, most of the present results are also new in a smooth finite-dimensional framework: readers who feel unfamiliar with notions of nonsmooth and variational analysis may, as a first stage, consider that all functions involved are differentiable, replace subdifferentials by usual derivatives and subgradient systems by smooth ones.…”

“…This result is named after S. Lojasiewicz (see [42], [43]), who was the first to establish its validity for the class of real-analytic functions. We refer to [38] for an extension of this result to the class of C 1 subanalytic functions and to [8] for a further extension to (nonsmooth) lower semicontinuous subanalytic functions. At the same time it is known that the Lojasiewicz inequality fails for C ∞ functions in general (see the classical example of the function x −→ exp(−1/x 2 ) if x = 0, and 0 if x = 0 around the pointx = 0).…”

“…In the framework of a C 1 function f defined on a real Hilbert space [H, ·, · ], and assuming for simplicity thatf = 0 is a critical value, this generalized inequality (that we hereby call the Kurdyka-Lojasiewicz inequality, or in short, the K L-inequality) states that (1) ||∇(ϕ • f )(x)|| ≥ 1, for some continuous function ϕ : [0, r) → R, C 1 on (0, r) with ϕ > 0 and all x in [0 < f < r] := {y ∈ H : 0 < f(y) < r}. The class of such functions ϕ will be further denoted by K(0,r); see (8). Note that the Lojasiewicz inequality corresponds to the case ϕ(t) = t 1−ρ .…”

“…where ϕ : (0, r) → R is a C 1 function bounded from below such that ϕ > 0 (see (8)). A byproduct of this result (though not an equivalent statement, as we show in Section 4.3; see Remark 37 (c)) is the fact that bounded subgradient curves have finite lengths and hence converge to a generalized critical point.…”