Copositivity tests based on the linear complementarity problem

Brás, Carmo P.; Eichfelder, Gabriele; Júdice, Joaquim J.

doi:10.1007/s10589-015-9772-2

Cited by 12 publications

(12 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Brás, Eichfelder, and Júdice have recently [10] presented tests for copositivity based on solving associated LCPs. More precisely, the copositivity of A ∈ R n−1 ×R n−1 can be revealed by considering the solutions of (LCP), with…”

Section: Lcps Implied By the Copositivity Tests A Real Symmetric Matrixmentioning

confidence: 99%

“…where e n−1 and 0 n−1 denote the (n − 1)-dimensional vectors of all ones and zeros, respectively. They have proved (see [10,Corollary 4]) that C1. if ∃(x * , s * ) ∈ F * with x * n > 0, then A is not copositive; C2.…”

Section: Lcps Implied By the Copositivity Tests A Real Symmetric Matrixmentioning

confidence: 99%

“…If the algorithm never returns an ε-optimal solution for (LCP) with data M cop and q cop , then we conclude that F * = ∅, hence, A is strictly copositive. We evaluated this approach on matrices M 1 − M 7 from [10], for which we know the real status of copositivity. Motivated by the discussion with the authors of [10] and with their help we constructed a set of matrices related to the maximum clique problem, for which we knew by construction the real status of copositivity.…”

Section: Lcps Implied By the Copositivity Tests A Real Symmetric Matrixmentioning

confidence: 99%

“…The Arrow-Debreu competitive market equilibrium problem with linear and Leontief utility functions can be also formulated as LCP [73]. Recent work of [10] reveals that LCP can be used as the main ingredient of the NP-hard copositivity test. In general, LCP belongs to the class of NP-complete problems [12], therefore, any progress in (approximately) solving LCP would straightforwardly imply a progress in (approximately) solving several NP-hard problems.…”

mentioning

confidence: 99%

“…Additionally, we consider LCPs related to the matrix copositivity test from [10]. For these matrices we know the real status of copositivity, but the corresponding LCPs are not necessarily included in the class of sufficient LCPs.…”

mentioning

confidence: 99%

See 4 more Smart Citations

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.

View full text Add to dashboard Cite

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.

show abstract

Section: Lcps Implied By the Copositivity Tests A Real Symmetric Matrixmentioning

confidence: 99%

Section: Lcps Implied By the Copositivity Tests A Real Symmetric Matrixmentioning

confidence: 99%

Section: Lcps Implied By the Copositivity Tests A Real Symmetric Matrixmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

See 3 more Smart Citations

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.

View full text Add to dashboard Cite

show abstract

Ordering Structures and Their Applications

Eichfelder

Pilecka

2018

Applications of Nonlinear Analysis

View full text Add to dashboard Cite

On sufficient properties of sufficient matrices

Povh

Žerovnik

2021

Cent Eur J Oper Res

View full text Add to dashboard Cite

In this paper we study sufficient matrices, which play an important role in theoretical analysis of interior-point methods for linear complementarity problems. We present new characterisations of these matrices which imply new necessary and sufficient conditions for sufficiency.We use these results to develop an algorithm with exponential iteration complexity which in each iteration solves a simple instance of linear programming problem and is capable to reveal whether given symmetric matrix is sufficient or not. This algorithm demonstrates 100 % accuracy on all tested instances of matrices.Version of October 19, 2020 1. Introduction 1.1. Motivation. The linear complementarity problem (LCP) is classically formulated as a feasibility problem: given M ∈ R n×n and q ∈ R n , find vectors u, v ∈ R n , such that:We denote by • the coordinate-wise product (also known as Hadamard product) of vectors, namely uthe complementarity constraint and it is the reason why this problem is so tough and exciting.LCP has a huge modelling power. Every quadratic optimization problem with linear or binary constraints can be rewritten as LCP [13]. It has also turned out as a strong tool in copositive programming [3,8,10] and opens

show abstract

Copositivity tests based on the linear complementarity problem

Cited by 12 publications

References 22 publications

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

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

Ordering Structures and Their Applications

On sufficient properties of sufficient matrices

Contact Info

Product

Resources

About