Yin Zhang We extend a recently proposed alternating minimization algorithm to the case of recovering blurry multichannel (color) images corrupted by impulsive rather than Gaussian noise. The algorithm minimizes the sum of a multichannel extension of total variation (TV), either isotropic or anisotropic, and a data fidelity term measured in the L1-norm. We derive the algorithm by … Read more
Compressive (or compressed) sensing (CS) is an emerging methodology in computational signal processing that has recently attracted intensive research activities. At present, the basic CS theory includes recoverability and stability: the former quantifies the central fact that a sparse signal of length n can be exactly recovered from much less than n measurements via L1-minimization … Read more
We propose and test a simple algorithmic framework for recovering images from blurry and noisy observations based on total variation (TV) regularization when a blurring point-spread function is given. Using a splitting technique, we construct an iterative procedure of alternately solving a pair of easy subproblems associated with an increasing sequence of penalty parameter values. … Read more
We consider solving minimization problems with $\ell_1$-regularization: $$\min \|x\|_1 + \mu f(x),$$ particularly for $f(x) = \frac{1}{2}\|Ax-b\|_M^2$ where $A \in \R^{m \times n}$ with $m < n$. Our goal is to construct efficient and robust algorithms for solving large-scale problems with dense data, and our approach is based on two powerful algorithmic ideas, operator-splitting and ... Read more
Most research in robust optimization has so far been focused on inequality-only, convex conic programming with simple linear models for uncertain parameters. Many practical optimization problems, however, are nonlinear and non-convex. Even in linear programming, coefficients may still be nonlinear functions of uncertain parameters. In this paper, we propose robust formulations that extend the robust-optimization … Read more
In this paper we study practical solution methods for finding the maximum-volume ellipsoid inscribing a given full-dimensional polytope in $\Re^n$ defined by a finite set of linear inequalities. Our goal is to design a general-purpose algorithmic framework that is reliable and efficient in practice. To evaluate the merit of a practical algorithm, we consider two … Read more
In this paper, we introduce a transformation that converts a class of linear and nonlinear semidefinite programming (SDP) problems into nonlinear optimization problems. For those problems of interest, the transformation replaces matrix-valued constraints by vector-valued ones, hence reducing the number of constraints by an order of magnitude. The class of transformable problems includes instances of … Read more
The authors of this paper recently introduced a transformation \cite{BuMoZh99-1} that converts a class of semidefinite programs (SDPs) into nonlinear optimization problems free of matrix-valued constraints and variables. This transformation enables the application of nonlinear optimization techniques to the solution of certain SDPs that are too large for conventional interior-point methods to handle efficiently. Based … Read more