Weak Convergence of a Relaxed and Inertial Hybrid Projection-Proximal Point Algorithm for Maximal Monotone Operators in Hilbert Space

By Felipe Alvarez. SIAM Journal on Optimization, 2003

330 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/a4657eadf8e266c509d3fb5444f73ac0cb6680d4

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Citations in the corpus

iPiano: Inertial Proximal Algorithm for Nonconvex Optimization (2014, methodology): “In [2, 1], the Heavy-ball method has been extended to maximal monotone operators, e.”
An Inertial Forward-Backward Algorithm for Monotone Inclusions (2014, background): “Indeed, it was shown in [1] that one can even use overrelaxation and inertial forces simultaneously.”
Convergence theorems for inertial KM-type algorithms (2008, methodology): “13) hence, for wn ∈ [0, 2] we get |xn+1 − q|2 − |xn − q|2 n(|xn − q|2 − |xn−1 − q|2)+ 2 n|xn−1 − xn|.”
A Hybrid Proximal-Extragradient Algorithm with Inertial Effects (2014, unknown): “Lemma 3 (see [1–3]) Let (φk)k∈N, (δk)k∈N and (αk)k∈N be sequences in [0,+∞) such that φk+1 ≤ φk + αk(φk − φk−1) + δk for all k ≥ 1, ∑ k∈N δk < +∞ and there exists a real number α with 0 ≤ αk ≤ α < 1 for all k ∈ N.”
A note on the inertial proximal point method (2015, background): “Subsequently, inexact and hybrid type iPPMs were studied in [26, 3, 23, 22].”
ASYMPTOTIC BEHAVIOR OF SECOND-ORDER DISSIPATIVE EVOLUTION EQUATIONS COMBINING POTENTIAL WITH NON-POTENTIAL EFFECTS ∗ (2009, background): “35) naturally leads to a second order relaxed proximal algorithm, whose trajectories (sequences) converge to the set of zeroes A−1(0), for A a general maximal monotone operator, see [3] and references herein.”
An introduction to continuous optimization for imaging (2016, background): “F. Alvarez (2003), ‘Weak convergence of a relaxed and inertial hybrid projectionproximal point algorithm for maximal monotone operators in Hilbert space’, SIAM J. Optim.”
Inertial Douglas-Rachford splitting for monotone inclusion problems (2014, unknown): “Remark 10 In case Bx = 0 for all x ∈ H, the iterative scheme in Theorem 8 becomes xn+1 = λnJγA ( xn + αn(xn − xn−1) ) + (1− λn)(xn + αn(xn − xn−1)) ∀n ≥ 1, which was already considered in [2] as a proximal-point algorithm (see [34]) in the context of solving the monotone inclusion problem 0 ∈ Ax.”