19 research outputs found
Necessary and sufficient conditions for unique solvability of absolute value equations: A Survey
In this survey paper, we focus on the necessary and sufficient conditions for
the unique solvability and unsolvability of the absolute value equations (AVEs)
during the last twenty years (2004 to 2023). We discussed unique solvability
conditions for various types of AVEs like standard absolute value equation
(AVE), Generalized AVE (GAVE), New generalized AVE (NGAVE), Triple AVE (TAVE)
and a class of NGAVE based on interval matrix, P-matrix, singular value
conditions, spectral radius and -property. Based on the unique
solution of AVEs, we also discussed unique solvability conditions for linear
complementarity problems (LCP) and horizontal linear complementarity problems
(HLCP)
A Note on the Paper "The unique solution of the absolute value equations"
In this note, we give the possible revised version of the unique solvability
conditions for the two incorrect results that appeared in the published paper
by Wu et al. (Appl Math Lett 76:195-200, 2018)
The Unique Solvability Conditions for the Generalized Absolute Value Equations
This paper investigates the conditions that guarantee unique solvability and
unsolvability for the generalized absolute value equations (GAVE) given by . Further, these conditions are also valid to determine
the unique solution of the generalized absolute value matrix equations (GAVME)
. Finally, certain aspects related to the solvability
and unsolvability of the absolute value equations (AVE) have been deliberated
upon
Improved Harmony Search Algorithm with Chaos for Absolute Value Equation
In this paper, an improved harmony search with chaos (HSCH) is presented for solving NP-hard absolute value equation (AVE) Ax - |x| = b, where A is an arbitrary square matrix whose singular values exceed one. The simulation results in solving some given AVE problems demonstrate that the HSCH algorithm is valid and outperforms the classical HS algorithm (HS) and HS algorithm with differential mutation operator (HSDE)
Optimal parameter for the SOR-like iteration method for solving the system of absolute value equations
The SOR-like iteration method for solving the absolute value equations~(AVE)
of finding a vector such that with is investigated. The convergence conditions of the SOR-like iteration method
proposed by Ke and Ma ([{\em Appl. Math. Comput.}, 311:195--202, 2017]) are
revisited and a new proof is given, which exhibits some insights in determining
the convergent region and the optimal iteration parameter. Along this line, the
optimal parameter which minimizes with and the approximate optimal parameter which
minimizes are explored.
The optimal and approximate optimal parameters are iteration-independent and
the bigger value of is, the smaller convergent region of the iteration
parameter is. Numerical results are presented to demonstrate that the
SOR-like iteration method with the optimal parameter is superior to that with
the approximate optimal parameter proposed by Guo, Wu and Li ([{\em Appl. Math.
Lett.}, 97:107--113, 2019]). In some situation, the SOR-like itration method
with the optimal parameter performs better, in terms of CPU time, than the
generalized Newton method (Mangasarian, [{\em Optim. Lett.}, 3:101--108, 2009])
for solving the AVE.Comment: 23 pages, 7 figures, 7 table