A predictor-corrector interior-point algorithm for P ∗ ( κ ) $P_{*}(\kappa )$ -horizontal linear complementarity problem

Kheirfam, Behrouz

doi:10.1007/s11075-013-9738-3

Cited by 26 publications

(13 citation statements)

References 10 publications

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“…Recently, Potra (2014) published a new PC IPA for sufficient LCPs using wide neighborhood with optimal iteration complexity. The AET method for determining search directions for IPAs has been also extended to LCPs (Achache 2010;Asadi and Mansouri 2013;Kheirfam 2014;Wang et al 2009) and LCPs over symmetric cones (Asadi et al 2017a, b;Mohammadi et al 2015;Wang 2012).…”

Section: Introductionmentioning

confidence: 99%

A corrector–predictor interior-point method with new search direction for linear optimization

Darvay

Illés

Kheirfam

et al. 2019

Cent Eur J Oper Res

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We introduce a feasible corrector-predictor interior-point algorithm (CP IPA) for solving linear optimization problems which is based on a new search direction. The search directions are obtained by using the algebraic equivalent transformation (AET) of the Newton system which defines the central path. The AET of the Newton system is based on the map that is a difference of the identity function and square root function. We prove global convergence of the method and derive the iteration bound that matches best iteration bounds known for these types of methods. Furthermore, we prove the practical efficiency of the new algorithm by presenting numerical results. This is the first CP IPA which is based on the above mentioned search direction.

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Section: Introductionmentioning

confidence: 99%

A corrector–predictor interior-point method with new search direction for linear optimization

Darvay

Illés

Kheirfam

et al. 2019

Cent Eur J Oper Res

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“…After that, Miao [50] generalized this IPA to P * (κ)-LCPs. Following his result, several Mizuno-Todd-Ye-type PC IPAs have been developed among others by Illés and Nagy [36] and Kheirfam [41].…”

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confidence: 95%

“…Related to this problem, Haddou, Migot, and Omer [34] have recently proposed a family of smooth concave functions which yields to IPAs with the best known iteration bound. The AET technique has been extended to other areas, too, such as LCPs [1,5,6,41,46], semidefinite programming [47,48,69], second-order cone programming [71], and symmetric optimization [40,72].…”

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confidence: 99%

“…The first PC IPAs that use the AET method for defining the search directions have been proposed by Darvay [19,20] for LP and linearly constrained convex optimization, respectively. Later on, Kheirfam [41] developed a PC IPA for P * (κ)-LCPs , which uses the direction based on the function ϕ(t) = √ t and achieved O((1+2κ) √ n log nµ 0 ) result. Wang and Bai [70] proposed IPA for sufficient LCPs based on a different technique of determining search directions, namely, considering parametric kernel functions.…”

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confidence: 99%

“…Ai and Zhang [2] solved P * (0)-LCPs using randomly generated positive semidefinite matrices. Kheirfam [41] considered nonsymmetric P * (0)-matrices and compared his IPA to the ones introduced by Illés and Nagy [36] and Wang and Bai [70] from computational point of view. Gurtuna et al [32] and Asadi et al [8] presented numerical results related to P * (κ)-LCPs, with positive handicap, by considering 2 × 2 or 3 × 3 matrices.…”

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Feasible Corrector-Predictor Interior-Point Algorithm for $P_{*} (\kappa)$-Linear Complementarity Problems Based on a New Search Direction

Darvay¹,

Illés²,

Povh³

et al. 2020

SIAM J. Optim.

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We introduce a new feasible corrector-predictor (CP) interior-point algorithm (IPA), which is suitable for solving linear complementarity problem (LCP) with P * (κ)-matrices. We use the method of algebraically equivalent transformation (AET) of the nonlinear equation of the system which defines the central path. The AET is based on the function ϕ(t) = t − √ t and plays a crucial role in the calculation of the new search direction. We prove that the algorithm has O((1 + 2κ) √ n log 9nµ 0 8) iteration complexity, where κ is an upper bound of the handicap of the input matrix. To the best of our knowledge, this is the first CP IPA for P * (κ)-LCPs which is based on this search direction. We implement the proposed CP IPA in the C++ programming language with specific parameters and demonstrate its performance on three families of LCPs. The first family consists of LCPs with P * (κ)-matrices. The second family of LCPs has the P-matrix defined by Csizmadia. Eisenberg-Nagy and de Klerk [Math. Program., 129 (2011), pp. 383-402] showed that the handicap of this matrix should be at least 2 2n−8 − 1 4. Namely, from the known complexity results for P * (κ)-LCPs it might follow that the computational performance of IPAs on LCPs with the matrix defined by Csizmadia could be very poor. Our preliminary computational study shows that an implemented variant of the theoretical version of the CP IPA (Algorithm 4.1) presented in this paper, finds a-approximate solution for LCPs with the Csizmadia matrix in a very small number of iterations. The third family of problems consists of the LCPs related to the copositivity test of 88 matrices from [C. Brás, G. Eichfelder, and J. Júdice, Comput. Optim. Appl., 63 (2016), pp. 461-493]. For each of these matrices we create a special LCP and try to solve it using our IPA. If the LCP does not have a solution, then the related matrix is strictly copositive, otherwise it is on the boundary or outside the copositive cone. For these LCPs we do not know whether the underlying matrix is P * (κ) or not, but we could reveal the real copositivity status of the input matrices in 83 out of 88 cases (accuracy ≥ 94%). The numerical test shows that our CP IPA performs well on the sets of test problems used in the paper.

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Large-step interior-point algorithm for linear optimization based on a new wide neighbourhood

Darvay

Takács

2018

Cent Eur J Oper Res

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A predictor-corrector interior-point algorithm for P ∗ ( κ ) $P_{*}(\kappa )$ -horizontal linear complementarity problem

Cited by 26 publications

References 10 publications

A corrector–predictor interior-point method with new search direction for linear optimization

A corrector–predictor interior-point method with new search direction for linear optimization

Feasible Corrector-Predictor Interior-Point Algorithm for $P_{*} (\kappa)$-Linear Complementarity Problems Based on a New Search Direction

Large-step interior-point algorithm for linear optimization based on a new wide neighbourhood

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