Newton-like dynamics associated to nonconvex optimization problems

الأصل · EN

We consider the dynamical system equation*{ arrayll v(t)∈∂ϕ(x(t)) λ x(t) + v(t) + v(t) + ∇ ψ(x(t))=0, array.equation* where ϕ:ⁿ→∪{+∞} is a proper, convex and lower semicontinuous function, ψ:ⁿ→ is a (possibly nonconvex) smooth function and λ>0 is a parameter which controls the velocity. We show that the set of limit points of the trajectory x is contained in the set of critical points of the objective function ϕ+ψ, which is here seen as the set of the zeros of its limiting subdifferential. If the objective function satisfies the Kurdyka-Łojasiewicz property, then we can prove convergence of the whole trajectory x to a critical point. Furthermore, convergence rates for the orbits are obtained in terms of the Łojasiewicz exponent of the objective function, provided the latter satisfies the Łojasiewicz property.