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19
Interior Methods for Constrained Optimization
 Acta Numerica
, 1992
"... Interior methods for optimization were widely used in the 1960s, primarily in the form of barrier methods. However, they were not seriously applied to linear programming because of the dominance of the simplex method. Barrier methods fell from favour during the 1970s for a variety of reasons, includ ..."
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Cited by 83 (3 self)
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Interior methods for optimization were widely used in the 1960s, primarily in the form of barrier methods. However, they were not seriously applied to linear programming because of the dominance of the simplex method. Barrier methods fell from favour during the 1970s for a variety of reasons, including their apparent inefficiency compared with the best available alternatives. In 1984, Karmarkar's announcement of a fast polynomialtime interior method for linear programming caused tremendous excitement in the field of optimization. A formal connection can be shown between his method and classical barrier methods, which have consequently undergone a renaissance in interest and popularity. Most papers published since 1984 have concentrated on issues of computational complexity in interior methods for linear programming. During the same period, implementations of interior methods have displayed great efficiency in solving many large linear programs of everincreasing size. Interior methods...
Interior methods for nonlinear optimization
 SIAM Review
, 2002
"... Abstract. Interior methods are an omnipresent, conspicuous feature of the constrained optimization landscape today, but it was not always so. Primarily in the form of barrier methods, interiorpoint techniques were popular during the 1960s for solving nonlinearly constrained problems. However, their ..."
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Cited by 76 (4 self)
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Abstract. Interior methods are an omnipresent, conspicuous feature of the constrained optimization landscape today, but it was not always so. Primarily in the form of barrier methods, interiorpoint techniques were popular during the 1960s for solving nonlinearly constrained problems. However, their use for linear programming was not even contemplated because of the total dominance of the simplex method. Vague but continuing anxiety about barrier methods eventually led to their abandonment in favor of newly emerging, apparently more efficient alternatives such as augmented Lagrangian and sequential quadratic programming methods. By the early 1980s, barrier methods were almost without exception regarded as a closed chapter in the history of optimization. This picture changed dramatically with Karmarkar’s widely publicized announcement in 1984 of a fast polynomialtime interior method for linear programming; in 1985, a formal connection was established between his method and classical barrier methods. Since then, interior methods have advanced so far, so fast, that their influence has transformed both the theory and practice of constrained optimization. This article provides a condensed, selective look at classical material and recent research about interior methods for nonlinearly constrained optimization.
Primaldual interior methods for nonconvex nonlinear programming
 SIAM Journal on Optimization
, 1998
"... Abstract. This paper concerns largescale general (nonconvex) nonlinear programming when first and second derivatives of the objective and constraint functions are available. A method is proposed that is based on finding an approximate solution of a sequence of unconstrained subproblems parameterize ..."
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Cited by 59 (5 self)
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Abstract. This paper concerns largescale general (nonconvex) nonlinear programming when first and second derivatives of the objective and constraint functions are available. A method is proposed that is based on finding an approximate solution of a sequence of unconstrained subproblems parameterized by a scalar parameter. The objective function of each unconstrained subproblem is an augmented penaltybarrier function that involves both primal and dual variables. Each subproblem is solved with a modified Newton method that generates search directions from a primaldual system similar to that proposed for interior methods. The augmented penaltybarrier function may be interpreted as a merit function for values of the primal and dual variables. An inertiacontrolling symmetric indefinite factorization is used to provide descent directions and directions of negative curvature for the augmented penaltybarrier merit function. A method suitable for large problems can be obtained by providing a version of this factorization that will treat large sparse indefinite systems.
Why a Pure Primal Newton Barrier Step May Be Infeasible
 SIAM Journal on Optimization
, 1993
"... Modern barrier methods for constrained optimization are sometimes portrayed conceptually as a sequence of inexact minimizations, with only a very few Newton iterations (perhaps just one) for each value of the barrier parameter. Unfortunately, this rosy image does not accurately reflect reality when ..."
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Cited by 21 (3 self)
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Modern barrier methods for constrained optimization are sometimes portrayed conceptually as a sequence of inexact minimizations, with only a very few Newton iterations (perhaps just one) for each value of the barrier parameter. Unfortunately, this rosy image does not accurately reflect reality when the barrier parameter is reduced at a reasonable rate. We present local analysis showing why a pure Newton step in a longstep barrier method for nonlinearly constrained optimization may be seriously infeasible, even when taken from an apparently favorable point. The features described are illustrated numerically and connected to known theoretical results for convex problems satisfying selfconcordancy assumptions. We also indicate the contrasting nature of an approximate step to the desired minimizer of the barrier function. 1. Introduction 1.1. Background Interior methods, most commonly based on barrier functions, have been applied with great practical success in recent years to many con...
The InteriorPoint Revolution in Constrained Optimization
 of Appl. Optim
, 1998
"... Interior methods are a central, striking feature of the constrained optimization landscape today, but it was not always so. Primarily in the form of barrier methods, interiorpoint techniques were widely used during the 1960s to solve nonlinearly constrained problems. However, their use for linear ..."
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Cited by 20 (0 self)
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Interior methods are a central, striking feature of the constrained optimization landscape today, but it was not always so. Primarily in the form of barrier methods, interiorpoint techniques were widely used during the 1960s to solve nonlinearly constrained problems. However, their use for linear programming was not even contemplated because of the total dominance of the simplex method. During the 1970s, barrier methods were superseded by newly emerging, apparently more efficient alternatives such as augmented Lagrangian and sequential quadratic programming methods. By the early 1980s, barrier methods were almost universally regarded as a closed chapter in the history of optimization. This picture changed dramatically in the mid1980s. In 1984, Karmarkar announced a fast polynomialtime interior method for linear programming; in 1985, a formal connection was established between his method and classical barrier methods. Since then, the new incarnations of interior methods ha...
The interiorpoint revolution in optimization: history, recent developments, and lasting consequences
 Bull. Amer. Math. Soc. (N.S
, 2005
"... Abstract. Interior methods are a pervasive feature of the optimization landscape today, but it was not always so. Although interiorpoint techniques, primarily in the form of barrier methods, were widely used during the 1960s for problems with nonlinear constraints, their use for the fundamental pro ..."
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Cited by 17 (1 self)
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Abstract. Interior methods are a pervasive feature of the optimization landscape today, but it was not always so. Although interiorpoint techniques, primarily in the form of barrier methods, were widely used during the 1960s for problems with nonlinear constraints, their use for the fundamental problem of linear programming was unthinkable because of the total dominance of the simplex method. During the 1970s, barrier methods were superseded, nearly to the point of oblivion, by newly emerging and seemingly more efficient alternatives such as augmented Lagrangian and sequential quadratic programming methods. By the early 1980s, barrier methods were almost universally regarded as a closed chapter in the history of optimization. This picture changed dramatically in 1984, when Narendra Karmarkar announced a fast polynomialtime interior method for linear programming; in 1985, a formal connection was established between his method and classical barrier methods. Since then, interior methods have continued to transform both the theory and practice of constrained optimization. We present a condensed,
Complete Orthogonal Decomposition for Weighted Least Squares
 SIAM J. Matrix Anal. Appl
, 1995
"... Consider a fullrank weighted leastsquares problem in which the weight matrix is highly illconditioned. Because of the illconditioning, standard methods for solving leastsquares problems, QR factorization and the nullspace method for example, break down. G. W. Stewart established a norm bound fo ..."
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Cited by 14 (4 self)
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Consider a fullrank weighted leastsquares problem in which the weight matrix is highly illconditioned. Because of the illconditioning, standard methods for solving leastsquares problems, QR factorization and the nullspace method for example, break down. G. W. Stewart established a norm bound for such a system of equations, indicating that it may be possible to find an algorithm that gives an accurate solution. S. A. Vavasis proposed a new definition of stability that is based on this result. He also defined the NSH algorithm for solving this leastsquares problem and showed that it satisfies his definition of stability. In this paper, we propose a complete orthogonal decomposition algorithm to solve this problem and show that it is also stable. This new algorithm is simpler and more efficient than the NSH method. 1 Introduction We consider solving the problem min y2R n kD \Gamma1=2 (Ay \Gamma b) k (1) for y, where D is a symmetric positive definite m \Theta m matrix, A is an ...
InteriorPoint Methodology for 3D PET Reconstruction
, 2000
"... Interiorpoint methods have been successfully applied to a wide variety of linear and nonlinear programming applications. This paper presents a class of algorithms, based on pathfollowing interiorpoint methodology, for performing regularized maximumlikelihood (ML) reconstructions on threedimensi ..."
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Cited by 14 (0 self)
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Interiorpoint methods have been successfully applied to a wide variety of linear and nonlinear programming applications. This paper presents a class of algorithms, based on pathfollowing interiorpoint methodology, for performing regularized maximumlikelihood (ML) reconstructions on threedimensional (3D) emission tomography data. The algorithms solve a sequence of subproblems that converge to the regularized maximum likelihood solution from the interior of the feasible region (the nonnegative orthant). We propose two methods, a primal method which updates only the primal image variables and a primaldual method which simultaneously updates the primal variables and the Lagrange multipliers. A parallel implementation permits the interiorpoint methods to scale to very large reconstruction problems. Termination is based on welldefined convergence measures, namely, the KarushKuhnTucker firstorder necessary conditions for optimality. We demonstrate the rapid convergence of the pathfollowing interiorpoint methods using both data from a small animal scanner and Monte Carlo simulated data. The proposed methods can readily be applied to solve the regularized, weighted least squares reconstruction problem.
On the convergence of the Newton/logbarrier method
 Preprint ANL/MCSP681 0897, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, Ill
, 1997
"... Abstract. In the Newton/logbarrier method, Newton steps are taken for the logbarrier function for a xed value of the barrier parameter until a certain convergence criterion is satis ed. The barrier parameter is then decreased and the Newton process is repeated. A naive analysis indicates that Newt ..."
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Cited by 12 (2 self)
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Abstract. In the Newton/logbarrier method, Newton steps are taken for the logbarrier function for a xed value of the barrier parameter until a certain convergence criterion is satis ed. The barrier parameter is then decreased and the Newton process is repeated. A naive analysis indicates that Newton's method does not exhibit superlinear convergence to the minimizer of each instance of the logbarrier function until it reaches a very small neighborhood of the minimizer. By partitioning according to the subspace of active constraint gradients, however, we show that this neighborhood is actually quite large, thus explaining why reasonably fast local convergence can be attained in practice. Moreover, we show that the overall convergence rate of the Newton/logbarrier algorithm is superlinear in the number of function/derivative evaluations, provided that the nonlinear program is formulated with a linear objective and that the schedule for decreasing the barrier parameter is related in a certain way to the convergence criterion for each Newton process. 1.
Properties of the LogBarrier Function on Degenerate Nonlinear Programs
 Math. Oper. Res
, 1999
"... . We examine the sequence of local minimizers of the logbarrier function for a nonlinear program near a solution at which secondordersufficient conditions and the MangasarianFromovitz constraint qualifications are satisfied, but the active constraint gradients are not necessarily linearly independ ..."
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Cited by 11 (0 self)
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. We examine the sequence of local minimizers of the logbarrier function for a nonlinear program near a solution at which secondordersufficient conditions and the MangasarianFromovitz constraint qualifications are satisfied, but the active constraint gradients are not necessarily linearly independent. When a strict complementarity condition is satisfied, we show uniqueness of the local minimizer of the barrier function in the vicinity of the nonlinear program solution, and obtain a semiexplicit characterization of this point. When strict complementarity does not hold, we obtain several other interesting characterizations, in particular, an estimate of the distance between the minimizers of the barrier function and the nonlinear program in terms of the barrier parameter, and a result about the direction of approach of the sequence of minimizers of the barrier function to the nonlinear programming solution. 1. Introduction We consider the nonlinear programming problem min f(x) subjec...