Power Series Variants of Karmarkar-Type Algorithms

Karmarkar, Narendra; Lagarias, Jeffrey C.; Slutsman, Lev; Wang, Pyng

doi:10.1002/j.1538-7305.1989.tb00316.x

Cited by 26 publications

(5 citation statements)

References 19 publications

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“…Further developments on this implementation were reported in Choi, Monma, and Shanno [4]. In doing so it makes use of the work of Monteiro, Adler, and Resende [28] and Karmarkar, Lagarias, Slutsman, and Wang [14]. This method is developed by considering the Newton direction on the optimality conditions for the logarithmic barrier problem.…”

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confidence: 99%

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On the Implementation of a Primal-Dual Interior Point Method

1992

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This paper gives an approach to implementing a second-order primal-dual interior point method. It uses a Taylor polynomial of second order to approximate a primal-dual trajectory. The computations for the second derivative are combined with the computations for the centering direction. Computations in this approach do not require that primal and dual solutions be feasible. Expressions are given to compute all the higher-order derivatives of the trajectory of interest. The implementation ensures that a suitable potential function is reduced by a constant amount at each iteration.There are several salient features of this approach. An adaptive heuristic for estimating the centering parameter is given. The approach used to compute the step length is also adaptive. A new practical approach to compute the starting point is given. This approach treats primal and dual problems symmetrically.Computational results on a subset of problems available from netlib are given. On mutually tested problems the results show that the proposed method requires approximately 40 percent fewer iterations than the implementation proposed in Lustig, Marsten, and Shanno Tech. Rep. TR J-89-11, Georgia Inst. of Technology, Atlanta, 1989]. It requires approximately 50 percent fewer iterations than the dual affine scaling method in Adler, Karmarkar, Resende, and Veiga [Math. Programming, 44 (1989), pp. 297-336], and 35 percent fewer iterations than the second-order dual affine scaling method in the same paper. The new approach for estimating the centering parameter and finding the step length and the starting point have contributed to the reduction in the number of iterations. However, the contribution due to the use of second derivative is most significant.On the tested problems, on the average the implementation shown was found to be approximately two times faster than OB1 (version 02/90) described in Lustig, Marsten, and Shanno and 2.5 times faster than MINOS 5.3 described in Murtagh and Saunders A comparison with the results reported in the literature (on mutually tested problems) shows that the method developed in this paper takes approximately 50 percent fewer iterations than the dual affine scaling method as implemented by Adler, Karmarkar, Resende, and Veiga 1 ], 40 percent fewer iterations than the primal-dual method implemented in Lustig, Marsten, and Shanno [18], and 55 percent fewer iterations than the logarithmic barrier function method implemented in Gill, Murray, and Saunders [8]. It requires 35 percent fewer iterations than the second-order dual affine scaling method implemented in Adler, Karmarkar, Resende, and Veiga 1 and 20 percent fewer iterations than the "optimal three-dimensional method" implemented by Domich, Boggs, Donaldson, and Witzgall [5].An efficient preliminary implementation of the proposed approach was developed. On average, it was found to be two times faster than OB1 (version 02/1990) [18]. On average, it was also found to be 2.5 times faster than MINOS 5.3.

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“…The recursion (4.5) is the same as the recursion given in Monteiro, Adler, and Resende [28] and Karmarkar et al [14] (see also Megiddo [21] and Bayer and Lagarias [2]). The derivatives resulting from the computations would be the same if :k 0 and k 0 is assumed, and in the latter paper if no centering and reparameterization is done.…”

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confidence: 99%

On the Implementation of a Primal-Dual Interior Point Method

1992

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“…Therefore, conventional linear programming procedures can be used to solve system (33). Well‐established methods are the simplex (Dantzig, ) and Karmakar's (Karmarkar et al ., ) algorithms.…”

Section: Optimal Alpha Spending For Continuous Sequential Analysismentioning

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Optimal Alpha Spending for Sequential Analysis with Binomial Data

Silva¹,

Kulldorff

Yih

2020

Journal of the Royal Statistical Society Series B: Statistical Methodology

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Summary For sequential analysis hypothesis testing, various alpha spending functions have been proposed. Given a prespecified overall alpha level and power, we derive the optimal alpha spending function that minimizes the expected time to signal for continuous as well as group sequential analysis. If there is also a restriction on the maximum sample size or on the expected sample size, we do the same. Alternatively, for fixed overall alpha, power and expected time to signal, we derive the optimal alpha spending function that minimizes the expected sample size. The method constructs alpha spending functions that are uniformly better than any other method, such as the classical Wald, Pocock or O’Brien–Fleming methods. The results are based on exact calculations using linear programming. All numerical examples were run by using the R Sequential package.

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“…[MAR88,KLSW89,JSS90]) have suggested alternatives to the Newton direction. Most of these suggestions may be traced to the idea of "extrapolation" first described by Fiacco and McCormick [FM68].…”

Section: A Predictor-corrector Approachmentioning

confidence: 99%

Solving Reduced KKT Systems in Barrier Methods for Linear and Quadratic Programming

Gill¹,

Murray²,

Ponceleón³

et al. 1991

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In barrier methods for constrained optimization, the main work lies in solving large linear systems Kp = r, where K is symmetric and indefinite.For linear programs, these KKT systems are usually reduced to smaller positive-definite systems AH AT is typically more ill-conditioned than K.In order to improve the numerical properties of barrier implementations, we discuss the use of "reduced KKT systems", whose dimension and condition lie somewhere in between those of K and AH −1 A T . The approach applies to linear programs and to positive semidefinite quadratic programs whose Hessian H is at least partially diagonal.We have implemented reduced KKT systems in a primal-dual algorithm for linear programming, based on the sparse indefinite solver MA27 from the Harwell Subroutine Library. Some features of the algorithm are presented, along with results on the netlib LP test set.

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Power Series Variants of Karmarkar-Type Algorithms

Cited by 26 publications

References 19 publications

On the Implementation of a Primal-Dual Interior Point Method

On the Implementation of a Primal-Dual Interior Point Method

Optimal Alpha Spending for Sequential Analysis with Binomial Data

Solving Reduced KKT Systems in Barrier Methods for Linear and Quadratic Programming

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