A full-Newton step feasible interior-point algorithm for P∗(κ)-LCP based on a new search direction

Kheirfam, Behrouz; Haghighi, Masoumeh

doi:10.17535/crorr.2016.0019

Cited by 16 publications

(9 citation statements)

References 23 publications

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“…Moreover, we prove that the short-step algorithm deserves the best known iteration bound, namely, O( √ n log n ). This iteration bound is as good as the bound for LO [7,16], CQO [1], SDO [8,18] , CQSDO [2], P * (κ)-LCP [12] if κ = 0, and SDLCP [15], cases. Here, we reconsider the analysis used in [2] and [8] and we make it suited for our case.…”

Section: Introductionmentioning

confidence: 77%

“…Thus motivates researchers to extend it to other convex optimization problems and mathematical programming such as CQO, SDO, convex quadratic semidefinite optimization (CQSDO), LCP, SDLCP and so on. For an overview of these methods, we refer to the references [1,3,4,5,11,12,14,15,17,18].…”

Section: Introductionmentioning

confidence: 99%

“…Their analysis was inspired from the work of De Kelerk [8], for SDO. Furthermore, Kheirfam and Haghighi [12], presented a full-Newton step feasible IPMs for solving the P * (κ)-LCP linear complementarity problem based on a new search direction. They derive the iteration bound for their algorithm, which coincides with the best-known iteration bound for these types of algorithms.…”

Section: Introductionmentioning

confidence: 99%

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A full Nesterov-Todd step primal-dual path-following interior-point algorithm for semidefinite linear complementarity problems

Mohamed

Tabchouche

2018

Cro.Oper.Res.Rev.

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In this paper, a feasible primal-dual path-following interior-point algorithm for monotone semidefinite linear complementarity problems is proposed. At each iteration, the algorithm uses only full Nesterov-Todd feasible steps for tracing approximately the central-path and getting an approximated solution of this problem. Under a new appropriate choices of the threshold τ which defines the size of the neighborhood of the central-path and of the update barrier parameter θ, we show that the algorithm is well-defined and enjoys the locally quadratically convergence. Moreover, we prove that the short-step algorithm deserves the best known iteration bound, namely, O(√ n log n). Finally, some numerical results are reported to show the practical performance of the algorithm.

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Section: Introductionmentioning

confidence: 77%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A full Nesterov-Todd step primal-dual path-following interior-point algorithm for semidefinite linear complementarity problems

Mohamed

Tabchouche

2018

Cro.Oper.Res.Rev.

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“…• ϕ(t) = t yields a ϕ = µe − xs introduced by Roos, Terlaky, and Vial [60], used in the Mizuno-Todd-Ye PC IPA [36] and in many other proposed by Kheirfam and Haghighi [42]. Note that in all cases the functions are defined on the interval (0, ∞), hence, we have ξ = 0, except in the case of the function ϕ(t) = t− √ t, where the value of ξ is 1 2 .…”

Section: Sufficient Matrices and The Central Path For Lcpsmentioning

confidence: 99%

“…Darvay [17,18] was the first who used the square root function in order to give the search directions. In 2016, Darvay, Papp, and Takács [23] proposed an IPA for LP based on the direction using a new function, namely, ϕ(t) = t − √ t. Recently, Kheirfam and Haghighi [42] have introduced an IPA for P * (κ)-LCPs which is based on a new search direction generated by using the function…”

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confidence: 99%

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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A new long-step interior point algorithm for linear programming based on the algebraic equivalent transformation

Nagy

Varga

2022

Cent Eur J Oper Res

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In this paper, we investigate a new primal-dual long-step interior point algorithm for linear optimization. Based on the step size, interior point algorithms can be divided into two main groups, short-step, and long-step methods. In practice, long-step variants perform better, but usually, a better theoretical complexity can be achieved for the short-step methods. One of the exceptions is the large-update algorithm of Ai and Zhang. The new wide neighborhood and the main characteristics of the presented algorithm are based on their approach. In addition, we use the algebraic equivalent transformation technique of Darvay to determine new modified search directions for our method. We show that the new long-step algorithm is convergent and has the best known iteration complexity of short-step variants. We present our numerical results and compare the performance of our algorithm with two previously introduced Ai-Zhang type interior point algorithms on a set of linear programming test problems from the Netlib library.

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A full-Newton step feasible interior-point algorithm for P∗(κ)-LCP based on a new search direction

Cited by 16 publications

References 23 publications

A full Nesterov-Todd step primal-dual path-following interior-point algorithm for semidefinite linear complementarity problems

A full Nesterov-Todd step primal-dual path-following interior-point algorithm for semidefinite linear complementarity problems

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

A new long-step interior point algorithm for linear programming based on the algebraic equivalent transformation

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