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- Coralia Cartis, Nicholas I. M. Gould, Philippe L. Toint
- Math. Program.
- 2011

An Adaptive Regularisation algorithm using Cubics (ARC) is proposed for unconstrained optimization, generalizing at the same time an unpublished method due to Griewank (Technical Report NA/12, 1981, DAMTP, Univ. of Cambridge), an algorithm by Nesterov & Polyak (Math. Programming 108(1), 2006, pp 177-205) and a proposal by Weiser, Deuflhard & Erdmann (Optim.… (More)

- Coralia Cartis, Nicholas I. M. Gould, Philippe L. Toint
- Math. Program.
- 2011

An Adaptive Regularisation framework using Cubics (ARC) was proposed for unconstrained optimization and analysed in Cartis, Gould & Toint (Part I, 2007). In this companion paper, we further the analysis by providing worst-case global iteration complexity bounds for ARC and a second-order variant to achieve approximate first-order, and for the latter even… (More)

An Adaptive Cubic Overestimation (ACO) algorithm for unconstrained optimization, generalizing a method due to Nesterov & Polyak (Math. Programming 108, 2006, pp 177-205), is proposed. At each iteration of Nesterov & Polyak’s approach, the global minimizer of a local cubic overestimator of the objective function is determined, and this ensures a significant… (More)

- Coralia Cartis, Nicholas I. M. Gould, Philippe L. Toint
- SIAM Journal on Optimization
- 2010

It is shown that the steepest descent and Newton’s method for unconstrained nonconvex optimization under standard assumptions may be both require a number of iterations and function evaluations arbitrarily close to O(ǫ) to drive the norm of the gradient below ǫ. This shows that the upper bound of O(ǫ) evaluations known for the steepest descent is tight, and… (More)

- Jeffrey D. Blanchard, Coralia Cartis, Jared Tanner
- SIAM Review
- 2011

Compressed Sensing (CS) seeks to recover an unknown vector with N entries by making far fewer than N measurements; it posits that the number of compressed sensing measurements should be comparable to the information content of the vector, not simply N . CS combines the important task of compression directly with the measurement task. Since its introduction… (More)

- Coralia Cartis, Nicholas I. M. Gould, Philippe L. Toint
- SIAM Journal on Optimization
- 2011

We estimate the worst-case complexity of minimizing an unconstrained, nonconvex composite objective with a structured nonsmooth term by means of some first-order methods. We find that it is unaffected by the nonsmoothness of the objective in that a first-order trust-region or quadratic regularization method applied to it takes at most O(ǫ−2)… (More)

Currently there is no framework for the transparent comparison of sparse approximation recoverability results derived using different methods of analysis. We cast some of the most recent recoverability results for `1-regularization in terms of the phase transition framework advocated by Donoho. To allow for quantitative comparisons across different methods… (More)

- Coralia Cartis, Nicholas I. M. Gould, L. Toint
- 2011

The adaptive cubic regularization algorithm described in Cartis, Gould and Toint (2009, 2010) is adapted to the problem of minimizing a nonlinear, possibly nonconvex, smooth objective function over a convex domain. Convergence to first-order critical points is shown under standard assumptions, without any Lipschitz continuity requirement on the objective’s… (More)

- Coralia Cartis, Nicholas I. M. Gould, Philippe L. Toint
- J. Complexity
- 2012

This paper examines worst-case evaluation bounds for finding weak minimizers in unconstrained optimization. For the cubic regularization algorithm, Nesterov and Polyak (2006) [15] and Cartis et al. (2010) [3] show that at most O(ε−3) iterations may have to be performed for finding an iterate which is within ε of satisfying second-order optimality… (More)

Consider a measurement matrix A of size n×N , with n < N , y a signal in RN , and b = Ay the observed measurement of the vector y. From knowledge of (b, A), compressed sensing seeks to recover the k-sparse x, k < n, which minimizes ‖b − Ax‖. Using various methods of analysis — convex polytopes, geometric functional analysis, and the restricted isometry… (More)