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A new polynomial-time algorithm for linear programming (1984)

by N Karmarkar
Venue:Combinatorica
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Numerical optimization

by Jorge Nocedal, Stephen J. Wright , 1999
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Abstract - Cited by 3326 (28 self) - Add to MetaCart
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ATOMIC DECOMPOSITION BY BASIS PURSUIT

by Scott Shaobing Chen , David L. Donoho , Michael A. Saunders , 1995
"... The Time-Frequency and Time-Scale 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 d ..."
Abstract - Cited by 2728 (61 self) - Add to MetaCart
The Time-Frequency and Time-Scale 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 super-resolution. BP has interesting relations to ideas in areas as diverse as ill-posed problems, in abstract harmonic analysis, total variation de-noising, and multi-scale edge denoising. Basis Pursuit in highly overcomplete dictionaries leads to large-scale 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 interior-point methods. We obtain reasonable success with a primal-dual logarithmic barrier method and conjugate-gradient solver.
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...nsion in the size of linear programs solved is due to the "interior-point revolution" initiated by Karmarkar's proof that a pseudo-polynomial-time algorithm could be based on an interior-poi=-=nt method [20]-=-. Since then a very wide array of interior-point algorithms have been proposed and considerable practical [21, 23, 27] and theoretical [30] understanding is now available. In this section we describe ...

Semidefinite Programming

by Lieven Vandenberghe, Stephen Boyd - SIAM REVIEW , 1996
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Abstract - Cited by 1109 (44 self) - Add to MetaCart
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..., Ringertz [Rin91], Fan and Nekooie [FN92], Fan [Fan93], Hiriart-Urruty and Ye [HUY95], Shapiro and Fan [SF94], and Pataki [Pat94]. Interior-point methods for LPs were introduced by Karmarkar in 1984 =-=[Kar84]-=-, although many of the underlying principles are older (see, e.g., Fiacco and McCormick [FM68], Lieu and Huard [LH66], and Dikin [Dik67]). Karmarkar's algorithm, and the interior-point methods develop...

The Earth Mover's Distance as a Metric for Image Retrieval

by Yossi Rubner, et al. , 2000
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Abstract - Cited by 719 (5 self) - Add to MetaCart
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Large margin methods for structured and interdependent output variables

by Ioannis Tsochantaridis, Thorsten Joachims, Thomas Hofmann, Yasemin Altun - JOURNAL OF MACHINE LEARNING RESEARCH , 2005
"... Learning general functional dependencies between arbitrary input and output spaces is one of the key challenges in computational intelligence. While recent progress in machine learning has mainly focused on designing flexible and powerful input representations, this paper addresses the complementary ..."
Abstract - Cited by 624 (12 self) - Add to MetaCart
Learning general functional dependencies between arbitrary input and output spaces is one of the key challenges in computational intelligence. While recent progress in machine learning has mainly focused on designing flexible and powerful input representations, this paper addresses the complementary issue of designing classification algorithms that can deal with more complex outputs, such as trees, sequences, or sets. More generally, we consider problems involving multiple dependent output variables, structured output spaces, and classification problems with class attributes. In order to accomplish this, we propose to appropriately generalize the well-known notion of a separation margin and derive a corresponding maximum-margin formulation. While this leads to a quadratic program with a potentially prohibitive, i.e. exponential, number of constraints, we present a cutting plane algorithm that solves the optimization problem in polynomial time for a large class of problems. The proposed method has important applications in areas such as computational biology, natural language processing, information retrieval/extraction, and optical character recognition. Experiments from various domains involving different types of output spaces emphasize the breadth and generality of our approach.

Interior-point Methods

by Florian A. Potra, Stephen J. Wright , 2000
"... The modern era of interior-point methods dates to 1984, when Karmarkar proposed his algorithm for linear programming. In the years since then, algorithms and software for linear programming have become quite sophisticated, while extensions to more general classes of problems, such as convex quadrati ..."
Abstract - Cited by 612 (15 self) - Add to MetaCart
The modern era of interior-point methods dates to 1984, when Karmarkar proposed his algorithm for linear programming. In the years since then, algorithms and software for linear programming have become quite sophisticated, while extensions to more general classes of problems, such as convex quadratic programming, semidefinite programming, and nonconvex and nonlinear problems, have reached varying levels of maturity. We review some of the key developments in the area, including comments on both the complexity theory and practical algorithms for linear programming, semidefinite programming, monotone linear complementarity, and convex programming over sets that can be characterized by self-concordant barrier functions.
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...ethods had been considered one way or another from the 1950's, and investigated quite extensively during the 1960s (Fiacco and McCormick [5]), it was the publication of the seminal paper of Karmarkar =-=[11]-=- that placed interior-point methods at the top of the agenda for many researchers. On the theoretical side, subsequent research led to improved computational complexity bounds for linear programming (...

Convex Position Estimation in Wireless Sensor Networks

by Lance Doherty , Kristofer S. J. Pister , Laurent El Ghaoui
"... A method for estimating unknown node positions in a sensor network based exclusively on connectivity-induced constraints is described. Known peer-to-peer communication in the network is modeled as a set of geometric constraints on the node positions. The global solution of a feasibility problem fo ..."
Abstract - Cited by 493 (0 self) - Add to MetaCart
A method for estimating unknown node positions in a sensor network based exclusively on connectivity-induced constraints is described. Known peer-to-peer communication in the network is modeled as a set of geometric constraints on the node positions. The global solution of a feasibility problem for these constraints yields estimates for the unknown positions of the nodes in the network. Providing that the constraints are tight enough, simulation illustrates that this estimate becomes close to the actual node positions. Additionally, a method for placing rectangular bounds around the possible positions for all unknown nodes in the network is given. The area of the bounding rectangles decreases as additional or tighter constraints are included in the problem. Specific models are suggested and simulated for isotropic and directional communication, representative of broadcast-based and optical transmission respectively, though the methods presented are not limited to these simple cases.

A Technique for Drawing Directed Graphs

by Emden R. Gansner, Eleftherios Koutsofios, Stephen C. North, Kiem-phong Vo - IEEE TRANSACTIONS ON SOFTWARE ENGINEERING , 1993
"... We describe a four-pass algorithm for drawing directed graphs. The first pass finds an optimal rank assignment using a network simplex algorithm. The second pass sets the vertex order within ranks by an iterative heuristic incorporating a novel weight function and local transpositions to reduce cros ..."
Abstract - Cited by 252 (18 self) - Add to MetaCart
We describe a four-pass algorithm for drawing directed graphs. The first pass finds an optimal rank assignment using a network simplex algorithm. The second pass sets the vertex order within ranks by an iterative heuristic incorporating a novel weight function and local transpositions to reduce crossings. The third pass finds optimal coordinates for nodes by constructing and ranking an auxiliary graph. The fourth pass makes splines to draw edges. The algorithm makes good drawings and runs fast.

Second-Order Cone Programming

by F. Alizadeh, D. Goldfarb - MATHEMATICAL PROGRAMMING , 2001
"... In this paper we survey the second order cone programming problem (SOCP). First we present several applications of the problem in various areas of engineering and robust optimization problems. We also give examples of optimization problems that can be cast as SOCPs. Next we review an algebraic struc ..."
Abstract - Cited by 247 (11 self) - Add to MetaCart
In this paper we survey the second order cone programming problem (SOCP). First we present several applications of the problem in various areas of engineering and robust optimization problems. We also give examples of optimization problems that can be cast as SOCPs. Next we review an algebraic structure that is connected to SOCP. This algebra is a special case of a Euclidean Jordan algebra. After presenting duality theory, complementary slackness conditions, and definitions and algebraic characterizations of primal and dual nondegeneracy and strict complementarity we review the logarithmic barrier function for the SOCP problem and survey the path-following interior point algorithms for it. Next we examine numerically stable methods for solving the interior point methods and study ways that sparsity in the input data can be exploited. Finally we give some current and future research direction in SOCP.

The NP-completeness column: an ongoing guide

by David S. Johnson - JOURNAL OF ALGORITHMS , 1987
"... This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NP-completeness. The presentation is modeled on that used by M. R. Garey and myself in our book "Computers and Intractability: A Guide to the Theory of NP-Completeness," W. H. Freem ..."
Abstract - Cited by 239 (0 self) - Add to MetaCart
This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NP-completeness. The presentation is modeled on that used by M. R. Garey and myself in our book "Computers and Intractability: A Guide to the Theory of NP-Completeness," W. H. Freeman & Co., New York, 1979 (hereinafter referred to as "[G&J]"; previous columns will be referred to by their dates). A background equivalent to that provided by [G&J] is assumed, and, when appropriate, cross-references will be given to that book and the list of problems (NP-complete and harder) presented there. Readers who have results they would like mentioned (NP-hardness, PSPACE-hardness, polynomial-time-solvability, etc.) or open problems they would like publicized, should
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...he nature of the challenge can be illustrated by considering the recently developed polynomial-time algorithms for linear programming. The ellipsoid method [28] and its more recent successors such as =-=[25]-=- have left many in the field of mathematical programming wanting more: These algorithms run in time bounded by a polynomial in the size of the input (as measured in bits). However, even if all arithme...

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