THE HEAVY BALL WITH FRICTION METHOD, I. THE CONTINUOUS DYNAMICAL SYSTEM: GLOBAL EXPLORATION OF THE LOCAL MINIMA OF A REAL-VALUED FUNCTION BY ASYMPTOTIC ANALYSIS OF A DISSIPATIVE DYNAMICAL SYSTEM

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GANs Trained by a Two Time-Scale Update Rule Converge to a Nash Equilibrium (2017, background): “[3] the energy, that is, a Lyapunov function, isE(t) = 1/2|θ̇(t)|(2)+f(θ(t)) and Ė(t) = −a |θ̇(t)|(2) < 0.”
Weak Convergence of a Relaxed and Inertial Hybrid Projection-Proximal Point Algorithm for Maximal Monotone Operators in Hilbert Space (2003, background): “, ∇f is monotone) and (∇f)−1({0}) = ∅, then each trajectory of HBF converges weakly in H to some x̂ ∈ H with∇f(x̂) = 0; see [4] for additional convergence results.”
APPLICATIONS OF THE ŁOJASIEWICZ–SIMON, GRADIENT INEQUALITY TO GRADIENT-LIKE EVOLUTION EQUATIONS (2009, background): “For other types of second order ordinary differential equations to which the Lojasiewicz technique applies, see [1], [2].”
GANs Trained by a Two Time-Scale Update Rule Converge to a Local Nash Equilibrium (2017, background): “Convergence has been proved for ∇ f that is L -Lipschitz and bounded from below (Theorem 3.1 in Attouch et al. [3]).”
On damped second-order gradient systems (2014, background): “This type of dynamical system has been the subject of several works in various fields and along different perspectives, one can quote for instance [4] for Nonsmooth Mechanics, [12, 11] for recent advances in Optimization and [46] for pioneer works on the topic, partial differential equations and rel”