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247
Graph implementations for nonsmooth convex programs
 Recent Advances in Learning and Control, Lecture Notes in Control and Information Sciences
, 2008
"... Summary. We describe graph implementations, a generic method for representing a convex function via its epigraph, described in a disciplined convex programming framework. This simple and natural idea allows a very wide variety of smooth and nonsmooth convex programs to be easily specified and effi ..."
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Cited by 263 (10 self)
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Summary. We describe graph implementations, a generic method for representing a convex function via its epigraph, described in a disciplined convex programming framework. This simple and natural idea allows a very wide variety of smooth and nonsmooth convex programs to be easily specified and efficiently solved, using interiorpoint methods for smooth or cone convex programs. Key words: Convex optimization, nonsmooth optimization, disciplined convex programming, optimization modeling languages, semidefinite program
Total variation models for variable lighting face recognition and uneven background correction
 IEEE Transactions on Pattern Analysis and Machine Intelligence
, 2005
"... In this paper, we present the logarithmic total variation (LTV) model for face recognition under varying illumination, including natural lighting condition, where we can hardly know the strength, the directions, and the number of light sources. The proposed LTV model has the capability to factorize ..."
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Cited by 70 (6 self)
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In this paper, we present the logarithmic total variation (LTV) model for face recognition under varying illumination, including natural lighting condition, where we can hardly know the strength, the directions, and the number of light sources. The proposed LTV model has the capability to factorize a single face image and obtain the illumination invariant facial structure, which is then used for face recognition. The merit of this model is that neither does it require any lighting assumption nor does it need any training process. Besides, there is only one parameter which could be easily set. The LTV model is able to reach very high recognition rates on both Yale and CMU PIE face databases as well as on a face database containing 765 subjects under outdoor lighting conditions. Keywords: I.5.4.d Face and gesture recognition; I.5.4.m Signal processing; I.4 Image Processing and Computer Vision; I.5.2.c Pattern analysis;
Secondorder cone programming methods for total variationbased image restoration
 SIAM Journal of Scientific Computing
, 2004
"... Abstract. In this paper we present optimization algorithms for image restoration based on the total variation (TV) minimization framework of L. Rudin, S. Osher and E. Fatemi (ROF). Our approach formulates TV minimization as a secondorder cone program which is then solved by interiorpoint algorithm ..."
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Cited by 66 (11 self)
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Abstract. In this paper we present optimization algorithms for image restoration based on the total variation (TV) minimization framework of L. Rudin, S. Osher and E. Fatemi (ROF). Our approach formulates TV minimization as a secondorder cone program which is then solved by interiorpoint algorithms that are efficient both in practice (using nested dissection and domain decomposition) and in theory (i.e., they obtain solutions in polynomial time). In addition to the original ROF minimization model, we show how to apply our approach to other TV models including ones that are not solvable by PDE based methods. Numerical results on a varied set of images are presented to illustrate the effectiveness of our approach.
An implementable proximal point algorithmic framework for nuclear norm minimization
, 2010
"... The nuclear norm minimization problem is to find a matrix with the minimum nuclear norm subject to linear and second order cone constraints. Such a problem often arises from the convex relaxation of a rank minimization problem with noisy data, and arises in many fields of engineering and science. In ..."
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Cited by 40 (5 self)
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The nuclear norm minimization problem is to find a matrix with the minimum nuclear norm subject to linear and second order cone constraints. Such a problem often arises from the convex relaxation of a rank minimization problem with noisy data, and arises in many fields of engineering and science. In this paper, we study inexact proximal point algorithms in the primal, dual and primaldual forms for solving the nuclear norm minimization with linear equality and second order cone constraints. We design efficient implementations of these algorithms and present comprehensive convergence results. In particular, we investigate the performance of our proposed algorithms in which the inner subproblems are approximately solved by the gradient projection method or the accelerated proximal gradient method. Our numerical results for solving randomly generated matrix completion problems and real matrix completion problems show that our algorithms perform favorably in comparison to several recently proposed stateoftheart algorithms. Interestingly, our proposed algorithms are connected with other algorithms that have been studied in the literature. Key words. Nuclear norm minimization, proximal point method, rank minimization, gradient projection method, accelerated proximal gradient method.
A convex programming approach for generating guaranteed passive approximations to tabulated frequencydata
 IEEE Trans. on ComputerAided Design of Integrated Circuits and Systems
, 2004
"... Abstract—In this paper,we present a methodology for generating guaranteed passive timedomain models of subsystems described by tabulated frequencydomain data obtained through measurement or through physical simulation. Such descriptions are commonly used to represent on and offchip interconnect ..."
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Cited by 37 (1 self)
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Abstract—In this paper,we present a methodology for generating guaranteed passive timedomain models of subsystems described by tabulated frequencydomain data obtained through measurement or through physical simulation. Such descriptions are commonly used to represent on and offchip interconnect effects,package parasitics,and passive devices common in highfrequency integrated circuit applications. The approach,which incorporates passivity constraints via convex optimization algorithms,is guaranteed to produce a passivesystem model that is optimal in the sense of having minimum error in the frequency band of interest over all models with a prescribed set of system poles. We demonstrate that this algorithm is computationally practical for generating accurate highorder models of data sets representing realistic, complicated multiinput,multioutput systems. Index Terms—Behavior modeling,convex optimization,convex programming,interconnect modeling,rational fitting,system identification. I.
Living on the edge: A geometric theory of phase transitions in convex optimization
, 2013
"... Recent empirical research indicates that many convex optimization problems with random constraints exhibit a phase transition as the number of constraints increases. For example, this phenomenon emerges in the `1 minimization method for identifying a sparse vector from random linear samples. Indee ..."
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Cited by 36 (4 self)
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Recent empirical research indicates that many convex optimization problems with random constraints exhibit a phase transition as the number of constraints increases. For example, this phenomenon emerges in the `1 minimization method for identifying a sparse vector from random linear samples. Indeed, this approach succeeds with high probability when the number of samples exceeds a threshold that depends on the sparsity level; otherwise, it fails with high probability. This paper provides the first rigorous analysis that explains why phase transitions are ubiquitous in random convex optimization problems. It also describes tools for making reliable predictions about the quantitative aspects of the transition, including the location and the width of the transition region. These techniques apply to regularized linear inverse problems with random measurements, to demixing problems under a random incoherence model, and also to cone programs with random affine constraints. These applications depend on foundational research in conic geometry. This paper introduces a new summary parameter, called the statistical dimension, that canonically extends the dimension of a linear subspace to the class of convex cones. The main technical result demonstrates that the sequence of conic intrinsic volumes of a convex cone concentrates sharply near the statistical dimension. This fact leads to an approximate version of the conic kinematic formula that gives bounds on the probability that a randomly oriented cone shares a ray with a fixed cone.
Distance weighted discrimination
 J. Am. Statist. Assoc
, 2007
"... Abstract High Dimension Low Sample Size statistical analysis is becoming increasingly important in a wide range of applied contexts. In such situations, it is seen that the popular Support Vector Machine suffers from "data piling" at the margin, which can diminish generalizability. This l ..."
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Cited by 36 (9 self)
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Abstract High Dimension Low Sample Size statistical analysis is becoming increasingly important in a wide range of applied contexts. In such situations, it is seen that the popular Support Vector Machine suffers from "data piling" at the margin, which can diminish generalizability. This leads naturally to the development of Distance Weighted Discrimination, which is based on Second Order Cone Programming, a modern computationally intensive optimization method.
Robust Convex Quadratically Constrained Programs
 Mathematical Programming
, 2002
"... In this paper we study robust convex quadratically constrained programs, a subset of the class of robust convex programs introduced by BenTal and Nemirovski [4]. Unlike [4], our focus in this paper is to identify uncertainty structures that allow the corresponding robust quadratically constrained p ..."
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Cited by 36 (2 self)
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In this paper we study robust convex quadratically constrained programs, a subset of the class of robust convex programs introduced by BenTal and Nemirovski [4]. Unlike [4], our focus in this paper is to identify uncertainty structures that allow the corresponding robust quadratically constrained programs to be reformulated as secondorder cone programs. We propose three classes of uncertainty sets that satisfy this criterion and present examples where these classes of uncertainty sets are natural. 1 Problem formulation A generic quadratically constrained program (QCP) is defined as follows.
Secondorder cone programming relaxation of sensor network localization
 SIAM J. Optimization
, 2007
"... Abstract. The sensor network localization problem has been much studied. Recently Biswas and Ye proposed a semidefinite programming (SDP) relaxation of this problem which has various nice properties and for which a number of solution methods have been proposed. Here, we study a secondorder cone pro ..."
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Cited by 36 (2 self)
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Abstract. The sensor network localization problem has been much studied. Recently Biswas and Ye proposed a semidefinite programming (SDP) relaxation of this problem which has various nice properties and for which a number of solution methods have been proposed. Here, we study a secondorder cone programming (SOCP) relaxation of this problem, motivated by its simpler structure and its potential to be solved faster than SDP. We show that the SOCP relaxation, though weaker than the SDP relaxation, has nice properties that make it useful as a problem preprocessor. In particular, sensors that are uniquely positioned among interior solutions of the SOCP relaxation are accurate up to the square root of the distance error. Thus, these sensors, which are easily identified, are accurately positioned. In our numerical simulation, the interior solution found can accurately position up to 80–90 % of the sensors. We also propose a smoothing coordinate gradient descent method for finding an interior solution that is faster than an interiorpoint method. Key words. sensor network localization, semidefinite program, secondorder cone program, approximation algorithm, error bound