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An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping. (English) Zbl 0991.65056
An inertial proximal iteration \(x^{k+1} - x^{k} -\alpha_k(x^k-x^{k-1}) + \lambda_kA(x^{k+1})\ni 0\) is proposed for solving the problem: \[ \text{Find}\quad x \in H \quad\text{such that}\quad 0\in A(x). \] Here \(A\) is a maximal monotone operator in a real Hilbert space \(H\). This algorithm is nicely linked to a one-step discretization method of the ’heavy ball with friction’ dynamical system \(\ddot x +\gamma \dot x +\nabla f(x)=0\). Convergence of this algorithm is proven under appropriate conditions for the parameters \(\alpha_k\) and \(\lambda_k\).
Reviewer: Zhen Mei (Toronto)

65K10 Numerical optimization and variational techniques
49M25 Discrete approximations in optimal control
49J27 Existence theories for problems in abstract spaces
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