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- Scott Saobing Chen, David L. Donoho, Michael A. Saunders
- SIAM Review
- 1998

The time-frequency and time-scale communities have recently developed a large number of overcomplete waveform dictionaries — stationary wavelets, wavelet packets, cosine packets, chirplets, and warplets, to name a few. Decomposition into overcomplete systems is not unique, and several methods for decomposition have been proposed, including the method of… (More)

- Christopher C. Paige, Michael A. Saunders
- ACM Trans. Math. Softw.
- 1982

An iterative method is given for solving Ax ~ffi b and minU Ax b 112, where the matrix A is large and sparse. The method is based on the bidiagonalization procedure of Golub and Kahan. It is analytically equivalent to the standard method of conjugate gradients, but possesses more favorable numerical properties. Reliable stopping criteria are derived, along… (More)

- Philip E. Gill, Walter Murray, Michael A. Saunders
- SIAM Review
- 2002

Sequential quadratic programming (SQP) methods have proved highly effective for solving constrained optimization problems with smooth nonlinear functions in the objective and constraints. Here we consider problems with general inequality constraints (linear and nonlinear). We assume that first derivatives are available and that the constraint gradients are… (More)

- Christopher C. Paige, Michael A. Saunders
- ACM Trans. Math. Softw.
- 1982

Received 4 June 1980; revised 23 September 1981, accepted 28 February 1982 This work was supported by Natural Sciences and Engineering Research Council of Canada Grant A8652, by the New Zealand Department of Scientific and Industrial Research; and by U S. National Science Foundation Grants MCS-7926009 and ECS-8012974, the Department of Energy under Contract… (More)

- Bruce A. Murtagh, Michael A. Saunders
- Math. Program.
- 1978

This paper descr ibes our efforts to develop a non l inea r p rog ramming a lgor i thm for p rob lems charac te r ized by a large sparse set of l inear cons t ra in t s and a signif icant degree of non l inea r i ty in the ob jec t ive func t ion . It has been our expe r i ence that m a n y l inear p rog ramming prob lems are inord ina te ly large because… (More)

- David Chin-Lung Fong, Michael A. Saunders
- SIAM J. Scientific Computing
- 2011

An iterative method LSMR is presented for solving linear systems Ax = b and leastsquares problems min ‖Ax−b‖2, with A being sparse or a fast linear operator. LSMR is based on the Golub-Kahan bidiagonalization process. It is analytically equivalent to the MINRES method applied to the normal equation ATAx = ATb, so that the quantities ‖Ark‖ are monotonically… (More)

- Jason D. Lee, Yuekai Sun, Michael A. Saunders
- SIAM Journal on Optimization
- 2014

We generalize Newton-type methods for minimizing smooth functions to handle a sum of two convex functions: a smooth function and a nonsmooth function with a simple proximal mapping. We show that the resulting proximal Newton-type methods inherit the desirable convergence behavior of Newton-type methods for minimizing smooth functions, even when search… (More)

- PHILIP E . GILLt, Walter Murray, DULCE B . PONCELEN, Michael A. Saunders
- 1992

PHILIP E. GILLt, WALTER MURRAY$, DULCE B. PONCELEN, AND MICHAEL A. SAUNDERS$ Dedicated to Gene Golub on the occasion of his 60th birthday Abstract. Methods are discussed for the solution of sparse linear equations Ky z, where K is symmetric and indefinite. Since exact solutions are not always required, direct and iterative methods are both of interest. An… (More)

- Jason D. Lee, Yuekai Sun, Michael A. Saunders
- NIPS
- 2012

We seek to solve convex optimization problems in composite form: minimize x∈Rn f(x) := g(x) + h(x), where g is convex and continuously differentiable and h : R → R is a convex but not necessarily differentiable function whose proximal mapping can be evaluated efficiently. We derive a generalization of Newton-type methods to handle such convex but nonsmooth… (More)

We study the use of black-box LDL factorizations for solving the augmented systems (KKT systems) associated with least-squares problems and barrier methods for linear programming (LP). With judicious regularization parameters, stability can be achieved for arbitrary data and arbitrary permutations of the KKT matrix. This offers improved efficiency compared… (More)