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97
ATOMIC DECOMPOSITION BY BASIS PURSUIT
, 1995
"... 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 d ..."
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Cited by 2725 (61 self)
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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. Basis Pursuit 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.
Interiorpoint Methods
, 2000
"... The modern era of interiorpoint 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 ..."
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Cited by 607 (15 self)
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The modern era of interiorpoint 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 selfconcordant barrier functions.
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 156 (19 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.
A PrimalDual Interior Point Method Whose Running Time Depends Only on the Constraint Matrix
, 1995
"... We propose a primaldual "layeredstep " interior point (LIP) algorithm for linear programming with data given by real numbers. This algorithm follows the central path, either with short steps or with a new type of step called a "layered least squares " (LLS) ste ..."
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Cited by 57 (8 self)
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We propose a primaldual &quot;layeredstep &quot; interior point (LIP) algorithm for linear programming with data given by real numbers. This algorithm follows the central path, either with short steps or with a new type of step called a &quot;layered least squares &quot; (LLS) step. The algorithm returns an exact optimum after a finite number of stepsin particular, after O(n3:5c(A)) iterations, where c(A) is a function of the
On the Riemannian geometry defined by selfconcordant barriers and interiorpoint methods
 Found. Comput. Math
"... We consider the Riemannian geometry defined on a convex set by the Hessian of a selfconcordant barrier function, and its associated geodesic curves. These provide guidance for the construction of efficient interiorpoint methods for optimizing a linear function over the intersection of the set with ..."
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Cited by 45 (0 self)
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We consider the Riemannian geometry defined on a convex set by the Hessian of a selfconcordant barrier function, and its associated geodesic curves. These provide guidance for the construction of efficient interiorpoint methods for optimizing a linear function over the intersection of the set with an affine manifold. We show that algorithms that follow the primaldual central path are in some sense close to optimal. The same is true for methods that follow the shifted primaldual central path among certain infeasibleinteriorpoint methods. We also compute the geodesics in several simple sets. ∗ Copyright (C) by SpringerVerlag. Foundations of Computational Mathewmatics 2 (2002), 333–361.
Two Numerical Methods for Optimizing Matrix Stability
 Linear Algebra Appl
, 2001
"... Consider the ane matrix family A(x) = A 0 + k=1 x k A k , mapping a design vector x 2 R into the space of n n real matrices. ..."
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Cited by 37 (8 self)
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Consider the ane matrix family A(x) = A 0 + k=1 x k A k , mapping a design vector x 2 R into the space of n n real matrices.
InexactRestoration Algorithm for Constrained Optimization
 Journal of Optimization Theory and Applications
, 1999
"... We introduce a new model algorithm for solving nonlinear programming problems. No slack variables are introduced for dealing with inequality constraints. Each iteration of the method proceeds in two phases. In the first phase, feasibility of the current iterate is improved and in second phase the ob ..."
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Cited by 30 (7 self)
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We introduce a new model algorithm for solving nonlinear programming problems. No slack variables are introduced for dealing with inequality constraints. Each iteration of the method proceeds in two phases. In the first phase, feasibility of the current iterate is improved and in second phase the objective function value is reduced in an approximate feasible set. The point that results from the second phase is compared with the current point using a nonsmooth merit function that combines feasibility and optimality. This merit function includes a penalty parameter that changes between different iterations. A suitable updating procedure for this penalty parameter is included by means of which it can be increased or decreased along different iterations. The conditions for feasibility improvement at the first phase and for optimality improvement at the second phase are mild, and largescale implementations of the resulting method are possible. We prove that under suitable conditions, that ...
An interior point method for mathematical programs with complementarity constraints (MPCCs
 SIAM Journal on Optimization
"... Abstract. Interior point methods for nonlinear programs (NLP) are adapted for solution of mathematical programs with complementarity constraints (MPCCs). The constraints of the MPCC are suitably relaxed so as to guarantee a strictly feasible interior for the inequality constraints. The standard prim ..."
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Cited by 24 (1 self)
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Abstract. Interior point methods for nonlinear programs (NLP) are adapted for solution of mathematical programs with complementarity constraints (MPCCs). The constraints of the MPCC are suitably relaxed so as to guarantee a strictly feasible interior for the inequality constraints. The standard primaldual algorithm has been adapted with a modified step calculation. The algorithm is shown to be superlinearly convergent in the neighborhood of the solution set under assumptions of MPCCLICQ, strong stationarity and upper level strict complementarity. The modification can be easily accommodated within most nonlinear programming interior point algorithms with identical local behavior. Numerical experience is also presented and holds promise for the proposed method. Key words. Barrier method, MPECs, Complementarity, NLP. AMS subject classifications. 1. Introduction. The MPCC considered in the paper is, min f(x, w, y) x,w,y s.t. h(x, w, y) = 0 x ≥ 0