An <i>O</i>$(\sqrtn L)$ Iteration Primal-dual Path-following Method, Based on Wide Neighborhoods and Large Updates, for Monotone LCP

Ai, Wenbao; Zhang, Shuzhong

doi:10.1137/040604492

Cited by 97 publications

(65 citation statements)

References 14 publications

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“…They also managed to extend these results to P * (κ)-nonlinear complementarity problems (NCPs), of which the LCP is a special case, as well as to other important optimization problems such as semidefinite and second order cone optimization problems. This iteration bound has since been matched, as already mentioned, in [4] and numerous other papers, some of which will be mentioned in what follows.…”

Section: Introductionmentioning

confidence: 98%

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Unified Analysis of Kernel-Based Interior-Point Methods for $P_*(\kappa)$-Linear Complementarity Problems

Lešaja¹,

Roos²

2010

SIAM J. Optim.

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Abstract. We present an interior-point method for the P * (κ)-linear complementarity problem (LCP) that is based on barrier functions which are defined by a large class of univariate functions called eligible kernel functions. This class is fairly general and includes the classical logarithmic function and the self-regular functions, as well as many non-self-regular functions as special cases. We provide a unified analysis of the method and give a general scheme on how to calculate the iteration bounds for the entire class. We also calculate the iteration bounds of both long-step and short-step versions of the method for several specific eligible kernel functions. For some of them we match the best known iteration bounds for the long-step method, while for the short-step method the iteration bounds are of the same order of magnitude. As far as we know, this is the first paper that provides a unified approach and comprehensive treatment of interior-point methods for P * (κ)-LCPs based on the entire class of eligible kernel functions.

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Section: Introductionmentioning

confidence: 98%

“…Most of these methods are higher order methods with the exception of a few methods that are first order methods. We mention the first order method of Ai and Zhang [4] for monotone LCPs with O( √ n log n log n ) iteration complexity. Potra [21] proposed higher order method with the same complexity.…”

Section: Introductionmentioning

confidence: 99%

Unified Analysis of Kernel-Based Interior-Point Methods for $P_*(\kappa)$-Linear Complementarity Problems

Lešaja¹,

Roos²

2010

SIAM J. Optim.

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“…Later, Li and Terlaky [9] generalized Ai and Zhang's algorithm [1] for LCPs to SDO problems and proved that the iteration complexity of their algorithm is the same as that of Ai and Zhang [1] for LCPs. Feng and Fang [5], using Ai and Zhang's wide neighborhood [1], suggested a wide-neighborhood interior-point algorithm for SDO problems.…”

Section: Introductionmentioning

confidence: 99%

“…In theory, the iteration bound for wide-neighborhood IPMs (large-update IPMs) is worse than that proved for small-neighborhood IPMs (small-update IPMs). In 2005, Ai and Zhang [1] introduced a new wide neighborhood around the central path of linear complementarity problems (LCPs) and proposed the first wide-neighborhood interior-point algorithm for LCPs [1], in which their algorithm enjoys the low iteration bound O ( √ nL) . Later, Li and Terlaky [9] generalized Ai and Zhang's algorithm [1] for LCPs to SDO problems and proved that the iteration complexity of their algorithm is the same as that of Ai and Zhang [1] for LCPs.…”

Section: Introductionmentioning

confidence: 99%

“…In 2005, Ai and Zhang [1] introduced a new wide neighborhood around the central path of linear complementarity problems (LCPs) and proposed the first wide-neighborhood interior-point algorithm for LCPs [1], in which their algorithm enjoys the low iteration bound O ( √ nL) . Later, Li and Terlaky [9] generalized Ai and Zhang's algorithm [1] for LCPs to SDO problems and proved that the iteration complexity of their algorithm is the same as that of Ai and Zhang [1] for LCPs. Feng and Fang [5], using Ai and Zhang's wide neighborhood [1], suggested a wide-neighborhood interior-point algorithm for SDO problems.…”

Section: Introductionmentioning

confidence: 99%

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A Mehrotra predictor-corrector interior-point algorithm for semidefinite optimization

Pirhaji¹,

Mansouri²

2017

Turk J Math

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This paper proposes a second-order Mehrotra-type predictor-corrector feasible interior-point algorithm for semidefinite optimization problems. In each iteration, the algorithm computes the Newton search directions through a new form of combination of the predictor and corrector directions. Using the Ai-Zhang wide neighborhood for linear complementarity problems, it is shown that the complexity bound of the algorithm is O (√ n log ε −1 ) for the Nesterov-Todd search direction and O ( n log ε −1 ) for the Helmberg-Kojima-Monteiro search directions.

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Self-Regular Interior-Point Methods for Semidefinite Optimization

Salahi

Terlaky

2011

International Series in Operations Research &Amp; Management Science

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An O$(\sqrtn L)$ Iteration Primal-dual Path-following Method, Based on Wide Neighborhoods and Large Updates, for Monotone LCP

Cited by 97 publications

References 14 publications

Unified Analysis of Kernel-Based Interior-Point Methods for $P_*(\kappa)$-Linear Complementarity Problems

Unified Analysis of Kernel-Based Interior-Point Methods for $P_*(\kappa)$-Linear Complementarity Problems

A Mehrotra predictor-corrector interior-point algorithm for semidefinite optimization

Self-Regular Interior-Point Methods for Semidefinite Optimization

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