Journal ArticleDOI
Smooth minimization of non-smooth functions
TLDR
A new approach for constructing efficient schemes for non-smooth convex optimization is proposed, based on a special smoothing technique, which can be applied to functions with explicit max-structure, and can be considered as an alternative to black-box minimization.Abstract:
In this paper we propose a new approach for constructing efficient schemes for non-smooth convex optimization. It is based on a special smoothing technique, which can be applied to functions with explicit max-structure. Our approach can be considered as an alternative to black-box minimization. From the viewpoint of efficiency estimates, we manage to improve the traditional bounds on the number of iterations of the gradient schemes from ** keeping basically the complexity of each iteration unchanged.read more
Citations
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Proceedings Article
Adaptive Subgradient Methods for Online Learning and Stochastic Optimization.
TL;DR: Adaptive subgradient methods as discussed by the authors dynamically incorporate knowledge of the geometry of the data observed in earlier iterations to perform more informative gradient-based learning, which allows us to find needles in haystacks in the form of very predictive but rarely seen features.
Journal Article
Adaptive Subgradient Methods for Online Learning and Stochastic Optimization
TL;DR: This work describes and analyze an apparatus for adaptively modifying the proximal function, which significantly simplifies setting a learning rate and results in regret guarantees that are provably as good as the best proximal functions that can be chosen in hindsight.
Journal ArticleDOI
Robust principal component analysis
TL;DR: In this paper, the authors prove that under some suitable assumptions, it is possible to recover both the low-rank and the sparse components exactly by solving a very convenient convex program called Principal Component Pursuit; among all feasible decompositions, simply minimize a weighted combination of the nuclear norm and of the e1 norm.
Journal ArticleDOI
A First-Order Primal-Dual Algorithm for Convex Problems with Applications to Imaging
Antonin Chambolle,Thomas Pock +1 more
TL;DR: A first-order primal-dual algorithm for non-smooth convex optimization problems with known saddle-point structure can achieve O(1/N2) convergence on problems, where the primal or the dual objective is uniformly convex, and it can show linear convergence, i.e. O(ωN) for some ω∈(0,1), on smooth problems.
Book
Proximal Algorithms
Neal Parikh,Stephen Boyd +1 more
TL;DR: The many different interpretations of proximal operators and algorithms are discussed, their connections to many other topics in optimization and applied mathematics are described, some popular algorithms are surveyed, and a large number of examples of proxiesimal operators that commonly arise in practice are provided.
References
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Book
Introductory Lectures on Convex Optimization: A Basic Course
TL;DR: A polynomial-time interior-point method for linear optimization was proposed in this paper, where the complexity bound was not only in its complexity, but also in the theoretical pre- diction of its high efficiency was supported by excellent computational results.
Book
An introduction to optimization
TL;DR: This review discusses mathematics, linear programming, and set--Constrained and Unconstrained Optimization, as well as methods of Proof and Some Notation, and problems with Equality Constraints.
Book
Convex analysis and minimization algorithms
TL;DR: In this article, the cutting plane algorithm is used to construct approximate subdifferentials of convex functions, and the inner construction of the subdifferential is performed by a dual form of Bundle Methods.
MonographDOI
Lectures on modern convex optimization: analysis, algorithms, and engineering applications
TL;DR: The authors present the basic theory of state-of-the-art polynomial time interior point methods for linear, conic quadratic, and semidefinite programming as well as their numerous applications in engineering.