Monotone Operators and the Proximal Point Algorithm

By R. Tyrrell Rockafellar. 1976

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In Optimization & Training

Citations in the corpus

A Tight and Unified Analysis of Extragradient for a Whole Spectrum of Differentiable Games (2019, methodology): “The gradient method is known not to converge in such game while the proximal point and extragradient methods converge Rockafellar (1976); Tseng (1995).”
Train faster, generalize better: Stability of stochastic gradient descent (2015, background): “mal point operator associated with the ℓ1-norm. For more information, see the surveys by Combettes and Wajs [6] or Parikh and Boyd [32]. An elementary proof of the following Lemma, due to Rockafellar [35], can be found in the appendix. Lemma 4.6. If f is convex, the proximal update (4.1) is 1-expans”
Stochastic Variance Reduction Methods for Saddle-Point Problems (2016, neutral): “,m} end for Output: Approximate solution(x, y) linearly [27] with rate(1 + τ−1)−1 = (1 − 1 1+τ ).”
A unified view of entropy-regularized Markov decision processes (2017, methodology): “A direct application of the Mirror Descent algorithm (Nemirovski and Yudin, 1983; Beck and Teboulle, 2003; Martinet, 1978; Rockafellar, 1976) to our case is deﬁned as where D R is the Bregman divergence associated with the convex regularization function R .”
Distributed optimization of deeply nested systems (2012, background): “Below we show this reduces the ill-conditioning caused by the nesting, and partially decouples many variables, affording an efficient and distributed optimization.”
Asynchronous distributed optimization using a randomized alternating direction method of multipliers (2013, methodology): “The firm non expansiveness of JρT plays a central role in the proof of the following result: Lemma 2 (Proximal point algorithm [16]):If T is a maximal monotone operator and ρ > 0, then the iterates ζ = JρT(ζ ) starting at any point ofY converge to a point of fix(JρT) whenever this set is non-empty.”
Templates for convex cone problems with applications to sparse signal recovery (2010, methodology): “When Yj is updated in this manner, the algorithm is known as the proximal point method, which has been studied since at least [57].”
Robust Online Multi-Task Learning with Correlative and Personalized Structures (2017, background): “In the following lemma, we show that the problem (9) can be turned into a linearized version of the proximal algorithm [45].”
A Component-Wise EM Algorithm for Mixtures (2001, unknown): “The proximal point algorithm was rst studied in [26].”
Accelerated schemes for a class of variational inequalities (2014, methodology): “The accelerated mirror-prox (AMP) method for solving a weak solution of V I ( Z ; G, H, J ) Observe that the AMP method diﬀers from the mirror-prox method in that we introduced two new sequences, i.e., { w mdt } and { w } (here “md” stands for “middle”, and “ag” stands for “aggregated”).”
The DC (Difference of Convex Functions) Programming and DCA Revisited with DC Models of Real World Nonconvex Optimization Problems (2005, background): “It seemed to take rise in the works of Pham Dinh Tao on the computation of bound-norms of matrices (i.e. maximizing a seminorm over the unit ball of a norm) in 1974 (Tao, 1975, 1976, 1981, 1984, 1986; and references therein).”
Linear convergence of the Douglas–Rachford method for two closed sets (2014, methodology): “n element of PAx that is a solution of (2). Thus, DR can be used to solve (2). The Douglas–Rachford method was originally deﬁned for (maximally monotone) operators and has been extensively studied in [7, 17, 21, 28, 31]. It is well-known that for convex subsets of a Hilbert spaces, the DR sequence c”
A projection method for solving nonlinear problems in reflexive Banach spaces (2011, background): “For more information regarding these problems see, for example, [2], [29], [7] and [14], respectively.”