We show that the analogue of Hamilton's Ricci flow in the combinatorial setting produces solutions which converge exponentially fast to Thurston's circle packing on surfaces. As a consequence, a new proof of Thurston's existence of circle packing theorem is obtained. As another consequence, Ricci flow suggests a new algorithm to find circle packings.

We present an alternating direction dual augmented Lagrangian method for solving semidefinite programming (SDP) problems in standard form. At each iteration, our basic algorithm minimizes the augmented Lagrangian function for the dual SDP problem sequentially, first with respect to the dual variables corresponding to the linear constraints, and then with respect to the dual slack variables, while in each minimization keeping the other variables fixed, and then finally it updates the Lagrange multipliers (i.e., primal variables). Convergence is proved by using a fixed-point argument. For SDPs with inequality constraints and positivity constraints, our algorithm is extended to separately minimize the dual augmented Lagrangian function over four sets of variables. Numerical results for frequency assignment, maximum stable set and binary integer quadratic programming problems demonstrate that our

Variational models with \ell_1-norm based regularization, in particular total variation (TV) and its variants, have long been known to offer superior image restoration quality, but processing speed remained a bottleneck, preventing their widespread use in the practice of color image processing. In this paper, by extending the grayscale image deblurring algorithm proposed in [Y. Wang, J. Yang, W. Yin, and Y. Zhang, <i>SIAM J. Imaging Sci.</i>, 1 (2008), pp. 248272], we construct a simple and efficient algorithm for multichannel image deblurring and denoising, applicable to both within-channel and cross-channel blurs in the presence of additive Gaussian noise. The algorithm restores an image by minimizing an energy function consisting of an \ell_1-norm fidelity term and a regularization term that can be either TV, weighted TV, or regularization functions based on higher-order derivatives. Specifically, we use a multichannel

Recent compressive sensing results show that it is possible to accurately reconstruct certain compressible signals from relatively few linear measurements via solving nonsmooth convex optimization problems. In this paper, we propose the use of the alternating direction method - a classic approach for optimization problems with separable variables (D. Gabay and B. Mercier, A dual algorithm for the solution of nonlinear variational problems via finite-element approximations, Computer and Mathematics with Applications, vol. 2, pp. 17-40, 1976; R. Glowinski and A. Marrocco, Sur lapproximation par elements finis dordre un, et la resolution par penalisation-dualite dune classe de problemes de Dirichlet nonlineaires, Rev. Francaise dAut. Inf. Rech. Oper., vol. R-2, pp. 41-76, 1975) - for signal reconstruction from partial Fourier (i.e., incomplete frequency) measurements. Signals are reconstructed as minimizers of

The matrix completion problem is to recover a low-rank matrix from a subset of its entries. The main solution strategy for this problem has been based on nuclear-norm minimization which requires computing singular value decompositionsa task that is increasingly costly as matrix sizes and ranks increase. To improve the capacity of solving large-scale problems, we propose a low-rank factorization model and construct a nonlinear successive over-relaxation (SOR) algorithm that only requires solving a linear least squares problem per iteration. Extensive numerical experiments show that the algorithm can reliably solve a wide range of problems at a speed at least several times faster than many nuclear-norm minimization algorithms. In addition, convergence of this nonlinear SOR algorithm to a stationary point is analyzed.