Proximité et dualité dans un espace hilbertien

By J. Moreau. 1965

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Citations in the corpus

Convex-constrained Sparse Additive Modeling and Its Extensions (2017, background): “To show how we can solve (14), it is helpful to recall the following concept: Deﬁnition 1 (Proximal Map [Mor65]) .”
Optimal Algorithms for Non-Smooth Distributed Optimization in Networks (2018, methodology): “Several other smoothing methods exist in the literature, notably the Moreau envelope [19] enjoying a dimension-free approximation guarantee.”
A Descent Lemma Beyond Lipschitz Gradient Continuity: First-Order Methods Revisited and Applications (2017, background): “where prox4 · 5 stands for the so-called Moreau’s proximal map (Moreau [26]) of a proper lsc convex function .”
Inexact Proximal Point Methods for Equilibrium Problems in Banach Spaces (2007, background): “The proximal point algorithm, whose origins can be traced back to [23, 24], attained its basic formulation in the work of Rockafellar [29], where it is presented as an algorithm for finding zeroes of a maximal monotone point-to-set operator T : H → (H ), where H is a Hilbert space.”
Equilibrium programming in Hilbert spaces (2005, unknown): “6) is Moreau’s proximity operator [21]; moreover, the Moreau envelope of index γ ∈ ]0,+∞[ of f is the function f : x 7→ miny∈H f(y) + 1 2γ ‖x− y‖ 2.”
Weak Convergence of a Relaxed and Inertial Hybrid Projection-Proximal Point Algorithm for Maximal Monotone Operators in Hilbert Space (2003, unknown): “Recall that PPA, which was proposed in [15, 16] (inspired by [18]), generates a sequence (x) ⊂ H by the successive approximation scheme x = x − λkv, v ∈ A(x), k = 0, 1, .”
Optimal Convergence Rates for Convex Distributed Optimization in Networks (2019, unknown): “Remark 3 Several other smoothing methods exist in the literature, notably the Moreau envelope (Moreau, 1965) enjoying a dimension-free approximation guarantee.”
Proximal Points are on the Fast Track (2002, methodology): “Moreover, letting F (x) := f(p(x)) + 1 2 μ‖p(x)− x‖ be the Moreau-Yosida regularization of f at x, it can be shown that g(x) := μ(x− p(x)) = ∇F (x) (27) is a Lipschitz continuous function of x, [14], [21], [18].”
An Inertial Forward-Backward Algorithm for Monotone Inclusions (2014, background): “The operator (Id +λk T )−1 is the so-called resolvent operator, that has been introduced by Moreau in [32].”
Convex Optimization in Normed Spaces: Theory, Methods and Examples (2015, background): “In some sense, it expresses the fact that the proximal iteration generalizes the concept of projection when the indicator function of a set is replaced by an arbitrary proper, lower-semicontinuous and convex function (see [80], where the author presents a survey of his previous research [77, 78, 79]”
An introduction to continuous optimization for imaging (2016, unknown): “…a ‘1/2-averaged operator’; see Appendix A. Optimization for imaging 177 Playing with this expression and (3.4), we can easily deduce Moreau’s identity (Moreau 1965) x = (I + τ∂f)−1(x) + τ ( I + 1 τ ∂f∗ )−1(x τ ) = proxτf (x) + τprox 1 τ f ∗ ( x τ ) , (3.8) which in fact holds for any maximal…”
The asymptotic behavior of the composition of two resolvents (2005, background): “Then Moreau’s decomposition [36] states that 1 2‖ · ‖2= (1 2‖ · ‖2 f )+ (1 2‖ · ‖2 f ∗).”
Plug-and-Play Methods Provably Converge with Properly Trained Denoisers (2019, background): “The notion of proximal operator and its well-deﬁnedness were ﬁrst presented in (Moreau, 1965).”
Weighted Nuclear Norm Minimization and Its Applications to Low Level Vision (2016, methodology): “By utilizing NNP as the key proximal technique (Moreau 1965), many NNM-based models have been proposed in recent years (Lin et al.”
Proximal Algorithms (2013, background): “Proximal operators took their current name and form in the 1960s in seminal work by Moreau [142, 143].”
A General Framework for a Class of First Order Primal-Dual Algorithms for Convex Optimization in Imaging Science (2010, methodology): “The main tool for drawing connections between the algorithms in this section is the Moreau decomposition [35, 15].”