• Corpus ID: 14336572

# Gradient methods for minimizing composite objective function

@article{Nesterov2007GradientMF,
title={Gradient methods for minimizing composite objective function},
author={Yurii Nesterov},
journal={Research Papers in Economics},
year={2007}
}
• Y. Nesterov
• Published 1 September 2007
• Computer Science, Mathematics
• Research Papers in Economics
In this paper we analyze several new methods for solving optimization problems with the objective function formed as a sum of two convex terms: one is smooth and given by a black-box oracle, and another is general but simple and its structure is known. Despite to the bad properties of the sum, such problems, both in convex and nonconvex cases, can be solved with eciency typical for the good part of the objective. For convex problems of the above structure, we consider primal and dual variants…
1,264 Citations
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## References

SHOWING 1-10 OF 29 REFERENCES
Accelerating the cubic regularization of Newton’s method on convex problems
An accelerated version of the cubic regularization of Newton’s method that converges for the same problem class with order, keeping the complexity of each iteration unchanged and arguing that for the second-order schemes, the class of non-degenerate problems is different from the standard class.
Rounding of convex sets and efficient gradient methods for linear programming problems
• Y. Nesterov
• Mathematics, Computer Science
Optim. Methods Softw.
• 2008
It is proved that the upper complexity bound for both schemes is O((√(n ln m)/δ)ln n) iterations of a gradient-type method, where n and m are the sizes of the corresponding linear programming problems.
Introductory Lectures on Convex Optimization - A Basic Course
It was in the middle of the 1980s, when the seminal paper by Kar markar opened a new epoch in nonlinear optimization, and it became more and more common that the new methods were provided with a complexity analysis, which was considered a better justification of their efficiency than computational experiments.
Gradient Projection for Sparse Reconstruction: Application to Compressed Sensing and Other Inverse Problems
• Computer Science
IEEE Journal of Selected Topics in Signal Processing
• 2007
This paper proposes gradient projection algorithms for the bound-constrained quadratic programming (BCQP) formulation of these problems and test variants of this approach that select the line search parameters in different ways, including techniques based on the Barzilai-Borwein method.
Just relax: convex programming methods for identifying sparse signals in noise
• J. Tropp
• Computer Science
IEEE Transactions on Information Theory
• 2006
A method called convex relaxation, which attempts to recover the ideal sparse signal by solving a convex program, which can be completed in polynomial time with standard scientific software.
Smooth minimization of non-smooth functions
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.
Iterative solution of nonlinear equations in several variables
• Mathematics
Computer science and applied mathematics
• 1970
Convergence of Minimization Methods An Annotated List of Basic Reference Books Bibliography Author Index Subject Index.
Regression Shrinkage and Selection via the Lasso
A new method for estimation in linear models called the lasso, which minimizes the residual sum of squares subject to the sum of the absolute value of the coefficients being less than a constant, is proposed.
Atomic Decomposition by Basis Pursuit
• Computer Science
SIAM J. Sci. Comput.
• 1998
Basis Pursuit (BP) is a principle for decomposing a signal into an "optimal" superposition of dictionary elements, where optimal means having the smallest l1 norm of coefficients among all such decompositions.
Deconvolution with the l 1 norm
• Geology
• 1979
Given a wavelet w and a noisy trace t = s * w + n, an approximation s of the spike train s can be obtained using the l 1 norm. This extraction has the advantage of preserving isolated spikes in s. On