Results 1  10
of
168
Guaranteed minimumrank solutions of linear matrix equations via nuclear norm minimization
, 2007
"... The affine rank minimization problem consists of finding a matrix of minimum rank that satisfies a given system of linear equality constraints. Such problems have appeared in the literature of a diverse set of fields including system identification and control, Euclidean embedding, and collaborative ..."
Abstract

Cited by 218 (15 self)
 Add to MetaCart
The affine rank minimization problem consists of finding a matrix of minimum rank that satisfies a given system of linear equality constraints. Such problems have appeared in the literature of a diverse set of fields including system identification and control, Euclidean embedding, and collaborative filtering. Although specific instances can often be solved with specialized algorithms, the general affine rank minimization problem is NPhard, because it contains vector cardinality minimization as a special case. In this paper, we show that if a certain restricted isometry property holds for the linear transformation defining the constraints, the minimum rank solution can be recovered by solving a convex optimization problem, namely the minimization of the nuclear norm over the given affine space. We present several random ensembles of equations where the restricted isometry property holds with overwhelming probability, provided the codimension of the subspace is sufficiently large. The techniques used in our analysis have strong parallels in the compressed sensing framework. We discuss how affine rank minimization generalizes this preexisting concept and outline a dictionary relating concepts from cardinality minimization to those of rank minimization. We also discuss several algorithmic approaches to solving the norm minimization relaxations, and illustrate our results with numerical examples.
Informationbased complexity
, 1988
"... Computational complexity studies the intrinsic difficulty of mathematically posed problems and seeks optimal means for their solutions. This is a rich and diverse field; for the purpose of this paper we present a greatly simplified picture. Computational complexity may be divided into two branches, ..."
Abstract

Cited by 217 (34 self)
 Add to MetaCart
Computational complexity studies the intrinsic difficulty of mathematically posed problems and seeks optimal means for their solutions. This is a rich and diverse field; for the purpose of this paper we present a greatly simplified picture. Computational complexity may be divided into two branches, discrete and continuous.
A firstorder primaldual algorithm for convex problems with applications to imaging
, 2010
"... In this paper we study a firstorder primaldual algorithm for convex optimization problems with known saddlepoint structure. We prove convergence to a saddlepoint with rate O(1/N) in finite dimensions, which is optimal for the complete class of nonsmooth problems we are considering in this paper ..."
Abstract

Cited by 134 (14 self)
 Add to MetaCart
In this paper we study a firstorder primaldual algorithm for convex optimization problems with known saddlepoint structure. We prove convergence to a saddlepoint with rate O(1/N) in finite dimensions, which is optimal for the complete class of nonsmooth problems we are considering in this paper. We further show accelerations of the proposed algorithm to yield optimal rates on easier problems. In particular we show that we can achieve O(1/N 2) convergence on problems, where the primal or the dual objective is uniformly convex, and we can show linear convergence, i.e. O(1/e N) on problems where both are uniformly convex. The wide applicability of the proposed algorithm is demonstrated on several imaging problems such as image denoising, image deconvolution, image inpainting, motion estimation and image segmentation. 1
Stochastic Approximation Approach to Stochastic Programming
"... In this paper we consider optimization problems where the objective function is given in a form of the expectation. A basic difficulty of solving such stochastic optimization problems is that the involved multidimensional integrals (expectations) cannot be computed with high accuracy. The aim of th ..."
Abstract

Cited by 99 (14 self)
 Add to MetaCart
In this paper we consider optimization problems where the objective function is given in a form of the expectation. A basic difficulty of solving such stochastic optimization problems is that the involved multidimensional integrals (expectations) cannot be computed with high accuracy. The aim of this paper is to compare two computational approaches based on Monte Carlo sampling techniques, namely, the Stochastic Approximation (SA) and the Sample Average Approximation (SAA) methods. Both approaches, the SA and SAA methods, have a long history. Current opinion is that the SAA method can efficiently use a specific (say linear) structure of the considered problem, while the SA approach is a crude subgradient method which often performs poorly in practice. We intend to demonstrate that a properly modified SA approach can be competitive and even significantly outperform the SAA method for a certain class of convex stochastic problems. We extend the analysis to the case of convexconcave stochastic saddle point problems, and present (in our opinion highly encouraging) results of numerical experiments.
Primaldual subgradient methods for convex problems
, 2005
"... (after revision) In this paper we present a new approach for constructing subgradient schemes for different types of nonsmooth problems with convex structure. Our methods are primaldual since they are always able to generate a feasible approximation to the optimum of an appropriately formulated dual ..."
Abstract

Cited by 75 (1 self)
 Add to MetaCart
(after revision) In this paper we present a new approach for constructing subgradient schemes for different types of nonsmooth problems with convex structure. Our methods are primaldual since they are always able to generate a feasible approximation to the optimum of an appropriately formulated dual problem. Besides other advantages, this useful feature provides the methods with a reliable stopping criterion. The proposed schemes differ from the classical approaches (divergent series methods, mirror descent methods) by presence of two control sequences. The first sequence is responsible for aggregating the support functions in the dual space, and the second one establishes a dynamically updated scale between the primal and dual spaces. This additional flexibility allows to guarantee a boundedness of the sequence of primal test points even in the case of unbounded feasible set. We present the variants of subgradient schemes for nonsmooth convex minimization, minimax problems, saddle point problems, variational inequalities, and stochastic optimization. In all situations our methods are proved to be optimal from the view point of worstcase blackbox lower complexity bounds.
Proxmethod with rate of convergence o(1/t) for variational inequalities with lipschitz continuous monotone operators and smooth convexconcave saddlepoint problems
 SIAM Journal on Optimization
"... Abstract. We propose a proxtype method with efficiency estimate O(ɛ−1) for approximating saddle points of convexconcave C1,1 functions and solutions of variational inequalities with monotone Lipschitz continuous operators. Application examples include matrix games, eigenvalue minimization, and com ..."
Abstract

Cited by 73 (12 self)
 Add to MetaCart
Abstract. We propose a proxtype method with efficiency estimate O(ɛ−1) for approximating saddle points of convexconcave C1,1 functions and solutions of variational inequalities with monotone Lipschitz continuous operators. Application examples include matrix games, eigenvalue minimization, and computing the Lovasz capacity number of a graph, and these are illustrated by numerical experiments with largescale matrix games and Lovasz capacity problems.
An accelerated proximal gradient algorithm for nuclear norm regularized least squares problems
, 2009
"... ..."
Method of centers for minimizing generalized eigenvalues
 Linear Algebra Appl
, 1993
"... We consider the problem of minimizing the largest generalized eigenvalue of a pair of symmetric matrices, each of which depends affinely on the decision variables. Although this problem may appear specialized, it is in fact quite general, and includes for example all linear, quadratic, and linear fr ..."
Abstract

Cited by 65 (14 self)
 Add to MetaCart
We consider the problem of minimizing the largest generalized eigenvalue of a pair of symmetric matrices, each of which depends affinely on the decision variables. Although this problem may appear specialized, it is in fact quite general, and includes for example all linear, quadratic, and linear fractional programs. Many problems arising in control theory can be cast in this form. The problem is nondifferentiable but quasiconvex, so methods such as Kelley's cuttingplane algorithm or the ellipsoid algorithm of Shor, Nemirovksy, and Yudin are guaranteed to minimize it. In this paper we describe relevant background material and a simple interior point method that solves such problems more efficiently. The algorithm is a variation on Huard's method of centers, using a selfconcordant barrier for matrix inequalities developed by Nesterov and Nemirovsky. (Nesterov and Nemirovsky have also extended their potential reduction methods to handle the same problem [NN91b].) Since the problem is quasiconvex but not convex, devising a nonheuristic stopping criterion (i.e., one that guarantees a given accuracy) is more difficult than in the convex case. We describe several nonheuristic stopping criteria that are based on the dual of a related convex problem and a new ellipsoidal approximation that is slightly sharper, in some cases, than a more general result due to Nesterov and Nemirovsky. The algorithm is demonstrated on an example: determining the quadratic Lyapunov function that optimizes a decay rate estimate for a differential inclusion.
Computational methods for sparse solution of linear inverse problems
, 2009
"... The goal of sparse approximation problems is to represent a target signal approximately as a linear combination of a few elementary signals drawn from a fixed collection. This paper surveys the major practical algorithms for sparse approximation. Specific attention is paid to computational issues, ..."
Abstract

Cited by 60 (0 self)
 Add to MetaCart
The goal of sparse approximation problems is to represent a target signal approximately as a linear combination of a few elementary signals drawn from a fixed collection. This paper surveys the major practical algorithms for sparse approximation. Specific attention is paid to computational issues, to the circumstances in which individual methods tend to perform well, and to the theoretical guarantees available. Many fundamental questions in electrical engineering, statistics, and applied mathematics can be posed as sparse approximation problems, making these algorithms versatile and relevant to a wealth of applications.
Dual averaging methods for regularized stochastic learning and online optimization
 In Advances in Neural Information Processing Systems 23
, 2009
"... We consider regularized stochastic learning and online optimization problems, where the objective function is the sum of two convex terms: one is the loss function of the learning task, and the other is a simple regularization term such as ℓ1norm for promoting sparsity. We develop extensions of Nes ..."
Abstract

Cited by 60 (3 self)
 Add to MetaCart
We consider regularized stochastic learning and online optimization problems, where the objective function is the sum of two convex terms: one is the loss function of the learning task, and the other is a simple regularization term such as ℓ1norm for promoting sparsity. We develop extensions of Nesterov’s dual averaging method, that can exploit the regularization structure in an online setting. At each iteration of these methods, the learning variables are adjusted by solving a simple minimization problem that involves the running average of all past subgradients of the loss function and the whole regularization term, not just its subgradient. In the case of ℓ1regularization, our method is particularly effective in obtaining sparse solutions. We show that these methods achieve the optimal convergence rates or regret bounds that are standard in the literature on stochastic and online convex optimization. For stochastic learning problems in which the loss functions have Lipschitz continuous gradients, we also present an accelerated version of the dual averaging method.