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- Farid Alizadeh, Donald Goldfarb
- Math. Program.
- 2003

Second-order cone programming (SOCP) problems are convex optimization problems in which a linear function is minimized over the intersection of an affine linear manifold with the Cartesian product of second-order (Lorentz) cones. Linear programs, convex quadratic programs and quadratically constrained convex quadratic programs can all be formulated as SOCP… (More)

- Shiqian Ma, Donald Goldfarb, Lifeng Chen
- Math. Program.
- 2011

The linearly constrained matrix rank minimization problem is widely applicable in many fields such as control, signal processing and system identification. The tightest convex relaxation of this problem is the linearly constrained nuclear norm minimization. Although the latter can be cast as a semidefinite programming problem, such an approach is… (More)

- Stanley Osher, Martin Burger, Donald Goldfarb, Jinjun Xu, Wotao Yin
- Multiscale Modeling & Simulation
- 2005

We introduce a new iterative regularization procedure for inverse problems based on the use of Bregman distances, with particular focus on problems arising in image processing. We are motivated by the problem of restoring noisy and blurry images via variational methods, by using total variation regularization. We obtain rigorous convergence results, and… (More)

- Donald Goldfarb, Garud Iyengar
- Math. Oper. Res.
- 2003

In this paper we show how to formulate and solve robust portfolio selection problems. The objective of these robust formulations is to systematically combat the sensitivity of the optimal portfolio to statistical and modeling errors in the estimates of the relevant market parameters. We introduce “uncertainty structures” for the market parameters and show… (More)

- Donald Goldfarb, Ashok U. Idnani
- Math. Program.
- 1983

An efficient and numerically stable dual algorithm for positive definite quadratic programming is described which takes advantage of the fact lhat the unconstrained minimum of the objective function can be used as a starling point. Its implementation utilizes the Cholesky and QR factorizations and procedures for updating them. The performance of the dual… (More)

- Wotao Yin, Stanley Osher, Donald Goldfarb, Jérôme Darbon
- SIAM J. Imaging Sciences
- 2008

We propose simple and extremely efficient methods for solving the basis pursuit problem min{‖u‖1 : Au = f, u ∈ R}, which is used in compressed sensing. Our methods are based on Bregman iterative regularization, and they give a very accurate solution after solving only a very small number of instances of the unconstrained problem minu∈Rn μ‖u‖1+ 12‖Au−f ‖2… (More)

- Zaiwen Wen, Donald Goldfarb, Wotao Yin
- Math. Program. Comput.
- 2010

We present an alternating direction method based on an augmented Lagrangian framework for solving semidefinite programming (SDP) problems in standard form. At each iteration, the algorithm, also known as a two-splitting scheme, minimizes the dual augmented Lagrangian function sequentially with respect to the Lagrange multipliers corresponding to the linear… (More)

- Zaiwen Wen, Wotao Yin, Donald Goldfarb, Yin Zhang
- SIAM J. Scientific Computing
- 2010

for recovering sparse solutions to an undetermined system of linear equations Ax = b. The algorithm is divided into two stages that are performed repeatedly. In the first stage a first-order iterative method called “shrinkage” yields an estimate of the subset of components of x likely to be nonzero in an optimal solution. Restricting the decision variables… (More)

- Donald Goldfarb, Zhiwei Qin
- SIAM J. Matrix Analysis Applications
- 2014

- Cun Mu, Bo Huang, John Wright, Donald Goldfarb
- ICML
- 2014

Recovering a low-rank tensor from incomplete information is a recurring problem in signal processing and machine learning. The most popular convex relaxation of this problem minimizes the sum of the nuclear norms of the unfoldings of the tensor. We show that this approach can be substantially suboptimal: reliably recovering a K-way tensor of length n and… (More)