Convex Analysis and Monotone Operator Theory in Hilbert Spaces

By Heinz H. Bauschke, Patrick L. Combettes. 2011

6,369 citations recorded by the source corpus.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Geometry and Topology, Mathematical Optimization

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Computational Theory and Mathematics

In Geometry and Topology

In Mathematical Optimization

Citations in the corpus

Stochastic Gradient Descent-Ascent and Consensus Optimization for Smooth Games: Convergence Analysis under Expected Co-coercivity (2021, methodology): “L -Lipschitz gradients) [Bauschke et al., 2011].”
Frank-Wolfe Splitting via Augmented Lagrangian Method (2018, background): “In this proof, for x ∈ X we note N X c ( x ) the normal cone to X at x deﬁned as the reader can refers to (Bauschke and Combettes, 2011) for more properties on the normal cone.”
Distributed Fixed Point Methods with Compressed Iterates (2019, methodology): “It is a generalization of Gradient Descent, Proximal Gradient Descent, and Douglas Rachford algorithms [5,8] and it takes the form of fixed point iterations x = TDY(x).”
Graph Oracle Models, Lower Bounds, and Gaps for Parallel Stochastic Optimization (2018, methodology): “We use the η-Moreau envelope [5] of this function in order to prove our lower bound:”
On the Convergence of Decentralized Gradient Descent (2013, background): “From the classic analysis of gradient descent iteration in [1] and [21], [x(i)(k)], and hence x(i)(k), will converge to a certain point when α ≤ (1 + λn(W ))/Lh.”
A Universal Catalyst for First-Order Optimization (2015, background): “The catalyst acceleration can be interpreted as a variant of the proximal point algorithm [3, 9], which is a central concept in convex optimizati on, underlying augmented Lagrangian approaches, and composite minimization schemes [5, 20].”
Stochastic Variance Reduction Methods for Saddle-Point Problems (2016, positive): “Following ∈ In this situation, prox σM ( x ′ , y ′ ) = (prox σf ( x ′ ) , prox σg ( y ′ )) the usual set-up of composite optimization [10], no smoothness assumption is made on M and hence on f or g .”
Computational Optimal Transport (2018, background): “We refer, for instance, to the monograph [Bauschke and Combettes, 2011] for a detailed account on these solvers and their use for large-scale applications.”
Finite-Sample Analysis of Proximal Gradient TD Algorithms (2015, methodology): “In this paper, we take a major step forward in resolving this problem by showing a principled way of designing true stochastic gradient TD algorithms by using a primal-dual saddle point objective function, derived from the original objective functions, coupled with the principled use of operator spl”
Asynchronous distributed optimization using a randomized alternating direction method of multipliers (2013, background): “In order to solve the optimization problem (1) on the graph G, we first provide an equivalent formulation of (1) that will be revealed useful.”
Calculus of the Exponent of Kurdyka–Łojasiewicz Inequality and Its Applications to Linear Convergence of First-Order Methods (2016, methodology): “To this end, recall from [4, Proposition 12.29] that, for all x ∈ R n , and that ∇ F λ is Lipschitz continuous with a Lipschitz constant of 1 λ .”
Proximal Splitting Algorithms: A Tour of Recent Advances, with New Twists. (2020, methodology): “…problem: minimize From Fermat’s rule [25, Theorem 16.3], the problem (34) is equivalent to (11) with = 𝜕𝑓 , which is maximally monotone, and 𝐶 = ∇ ℎ , which is 𝜉 -cocoercive, with 𝜉 = 1 𝛽 [25, Corollary 18.17] (and as mentioned in the introduction, a zero of + ∇ ℎ is supposed to exist).”