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101
Atomic decomposition by basis pursuit
 SIAM Journal on Scientific Computing
, 1998
"... Abstract. The timefrequency and timescale communities have recently developed a large number of overcomplete waveform dictionaries — stationary wavelets, wavelet packets, cosine packets, chirplets, and warplets, to name a few. Decomposition into overcomplete systems is not unique, and several meth ..."
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Cited by 1660 (43 self)
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Abstract. The timefrequency and timescale communities have recently developed a large number of overcomplete waveform dictionaries — stationary wavelets, wavelet packets, cosine packets, chirplets, and warplets, to name a few. Decomposition into overcomplete systems is not unique, and several methods for decomposition have been proposed, including the method of frames (MOF), Matching pursuit (MP), and, for special dictionaries, the best orthogonal basis (BOB). Basis Pursuit (BP) is a principle for decomposing a signal into an “optimal ” superposition of dictionary elements, where optimal means having the smallest l 1 norm of coefficients among all such decompositions. We give examples exhibiting several advantages over MOF, MP, and BOB, including better sparsity and superresolution. BP has interesting relations to ideas in areas as diverse as illposed problems, in abstract harmonic analysis, total variation denoising, and multiscale edge denoising. BP in highly overcomplete dictionaries leads to largescale optimization problems. With signals of length 8192 and a wavelet packet dictionary, one gets an equivalent linear program of size 8192 by 212,992. Such problems can be attacked successfully only because of recent advances in linear programming by interiorpoint methods. We obtain reasonable success with a primaldual logarithmic barrier method and conjugategradient solver.
LOQO: An interior point code for quadratic programming
, 1994
"... ABSTRACT. This paper describes a software package, called LOQO, which implements a primaldual interiorpoint method for general nonlinear programming. We focus in this paper mainly on the algorithm as it applies to linear and quadratic programming with only brief mention of the extensions to convex ..."
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Cited by 156 (9 self)
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ABSTRACT. This paper describes a software package, called LOQO, which implements a primaldual interiorpoint method for general nonlinear programming. We focus in this paper mainly on the algorithm as it applies to linear and quadratic programming with only brief mention of the extensions to convex and general nonlinear programming, since a detailed paper describing these extensions were published recently elsewhere. In particular, we emphasize the importance of establishing and maintaining symmetric quasidefiniteness of the reduced KKT system. We show that the industry standard MPS format can be nicely formulated in such a way to provide quasidefiniteness. Computational results are included for a variety of linear and quadratic programming problems. 1.
An InteriorPoint Algorithm For Nonconvex Nonlinear Programming
 COMPUTATIONAL OPTIMIZATION AND APPLICATIONS
, 1997
"... The paper describes an interiorpoint algorithm for nonconvex nonlinear programming which is a direct extension of interiorpoint methods for linear and quadratic programming. Major modifications include a merit function and an altered search direction to ensure that a descent direction for the mer ..."
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Cited by 144 (13 self)
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The paper describes an interiorpoint algorithm for nonconvex nonlinear programming which is a direct extension of interiorpoint methods for linear and quadratic programming. Major modifications include a merit function and an altered search direction to ensure that a descent direction for the merit function is obtained. Preliminary numerical testing indicates that the method is robust. Further, numerical comparisons with MINOS and LANCELOT show that the method is efficient, and has the promise of greatly reducing solution times on at least some classes of models.
Basis Pursuit
, 1994
"... The TimeFrequency and TimeScale communities have recently developed an enormous number of overcomplete signal dictionaries  wavelets, wavelet packets, cosine packets, wilson bases, chirplets, warped bases, and hyperbolic cross bases being a few examples. Basis Pursuit is a technique for decompos ..."
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Cited by 119 (15 self)
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The TimeFrequency and TimeScale communities have recently developed an enormous number of overcomplete signal dictionaries  wavelets, wavelet packets, cosine packets, wilson bases, chirplets, warped bases, and hyperbolic cross bases being a few examples. Basis Pursuit is a technique for decomposing a signal into an "optimal" superposition of dictionary elements. The optimization criterion is the l 1 norm of coefficients. The method has several advantages over Matching Pursuit and Best Ortho Basis, including superresolution and stability. 1 Introduction Over the last five years or so, there has been an explosion of awareness of alternatives to traditional signal representations. Instead of just representing objects as superpositions of sinusoids (the traditional Fourier representation) we now have available alternate dictionaries  signal representation schemes  of which the Wavelets dictionary is only the most wellknown. Wavelet dictionaries, Gabor dictionaries, Multiscale...
Testing Heuristics: We Have It All Wrong
 Journal of Heuristics
, 1995
"... The competitive nature of most algorithmic experimentation is a source of problems that are all too familiar to the research community. It is hard to make fair comparisons between algorithms and to assemble realistic test problems. Competitive testing tells us which algorithm is faster but not w ..."
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Cited by 119 (2 self)
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The competitive nature of most algorithmic experimentation is a source of problems that are all too familiar to the research community. It is hard to make fair comparisons between algorithms and to assemble realistic test problems. Competitive testing tells us which algorithm is faster but not why. Because it requires polished code, it consumes time and energy that could be spent doing more experiments. This paper argues that a more scientific approach of controlled experimentation, similar to that used in other empirical sciences, avoids or alleviates these problems. We have confused research and development; competitive testing is suited only for the latter. Most experimental studies of heuristic algorithms resemble track meets more than scientific endeavors. Typically an investigator has a bright idea for a new algorithm and wants to show that it works better, in some sense, than known algorithms. This requires computational tests, perhaps on a standard set of benchmark p...
A trust region method based on interior point techniques for nonlinear programming
 Mathematical Programming
, 1996
"... Jorge Nocedal z An algorithm for minimizing a nonlinear function subject to nonlinear inequality constraints is described. It applies sequential quadratic programming techniques to a sequence of barrier problems, and uses trust regions to ensure the robustness of the iteration and to allow the direc ..."
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Cited by 103 (17 self)
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Jorge Nocedal z An algorithm for minimizing a nonlinear function subject to nonlinear inequality constraints is described. It applies sequential quadratic programming techniques to a sequence of barrier problems, and uses trust regions to ensure the robustness of the iteration and to allow the direct use of second order derivatives. This framework permits primal and primaldual steps, but the paper focuses on the primal version of the new algorithm. An analysis of the convergence properties of this method is presented. Key words: constrained optimization, interior point method, largescale optimization, nonlinear programming, primal method, primaldual method, SQP iteration, barrier method, trust region method.
The Mathematics Of Eigenvalue Optimization
, 2003
"... Optimization problems involving the eigenvalues of symmetric and nonsymmetric matrices present a fascinating mathematical challenge. Such problems arise often in theory and practice, particularly in engineering design, and are amenable to a rich blend of classical mathematical techniques and contemp ..."
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Cited by 92 (13 self)
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Optimization problems involving the eigenvalues of symmetric and nonsymmetric matrices present a fascinating mathematical challenge. Such problems arise often in theory and practice, particularly in engineering design, and are amenable to a rich blend of classical mathematical techniques and contemporary optimization theory. This essay presents a personal choice of some central mathematical ideas, outlined for the broad optimization community. I discuss the convex analysis of spectral functions and invariant matrix norms, touching briey on semide nite representability, and then outlining two broader algebraic viewpoints based on hyperbolic polynomials and Lie algebra. Analogous nonconvex notions lead into eigenvalue perturbation theory. The last third of the article concerns stability, for polynomials, matrices, and associated dynamical systems, ending with a section on robustness. The powerful and elegant language of nonsmooth analysis appears throughout, as a unifying narrative thread.
Implementation of Interior Point Methods for Large Scale Linear Programming
 in Interior Point Methods in Mathematical Programming
, 1996
"... In the past 10 years the interior point methods (IPM) for linear programming have gained extraordinary interest as an alternative to the sparse simplex based methods. This has initiated a fruitful competition between the two types of algorithms which has lead to very efficient implementations on bot ..."
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Cited by 70 (22 self)
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In the past 10 years the interior point methods (IPM) for linear programming have gained extraordinary interest as an alternative to the sparse simplex based methods. This has initiated a fruitful competition between the two types of algorithms which has lead to very efficient implementations on both sides. The significant difference between interior point and simplex based methods is reflected not only in the theoretical background but also in the practical implementation. In this paper we give an overview of the most important characteristics of advanced implementations of interior point methods. First, we present the infeasibleprimaldual algorithm which is widely considered the most efficient general purpose IPM. Our discussion includes various algorithmic enhancements of the basic algorithm. The only shortcoming of the "traditional" infeasibleprimaldual algorithm is to detect a possible primal or dual infeasibility of the linear program. We discuss how this problem can be solve...
Continuation and Path Following
, 1992
"... CONTENTS 1 Introduction 1 2 The Basics of PredictorCorrector Path Following 3 3 Aspects of Implementations 7 4 Applications 15 5 PiecewiseLinear Methods 34 6 Complexity 41 7 Available Software 44 References 48 1. Introduction Continuation, embedding or homotopy methods have long served as useful ..."
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Cited by 70 (6 self)
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CONTENTS 1 Introduction 1 2 The Basics of PredictorCorrector Path Following 3 3 Aspects of Implementations 7 4 Applications 15 5 PiecewiseLinear Methods 34 6 Complexity 41 7 Available Software 44 References 48 1. Introduction Continuation, embedding or homotopy methods have long served as useful theoretical tools in modern mathematics. Their use can be traced back at least to such venerated works as those of Poincar'e (18811886), Klein (1882 1883) and Bernstein (1910). Leray and Schauder (1934) refined the tool and presented it as a global result in topology, viz., the homotopy invariance of degree. The use of deformations to solve nonlinear systems of equations Partially supported by the National Science Foundation via grant # DMS9104058 y Preprint, Colorado State University, August 2 E. Allgower and K. Georg may be traced back at least to Lahaye (1934). The classical embedding methods were the