The Picard–HSS iteration method for absolute value equations

On the global convergence of the inexact semi-smooth Newton method for absolute value equation

“…For instance, linear complementarity problem, linear programming or convex quadratic programming can be equivalently reformulated in the form of (1) and thus solved as absolute value equations; see [8,12,17,19]. As far as we know, since Mangasarian and Meyer [14] established existence results for this class of absolute value equations (1), the interest for this subject has increased substantially; see [11,13,20] and reference therein.Several algorithms have been designed to solve the systems of AVEs involving smooth, semismooth and Picard techniques; see [1,7,10,21,22]. In [9], Mangasarian applied the nonsmooth Newton method for solving AVE obtaining global Q-linear convergence and showing its numerical effectiveness.…”

“…Several algorithms have been designed to solve the systems of AVEs involving smooth, semismooth and Picard techniques; see [1,7,10,21,22]. In [9], Mangasarian applied the nonsmooth Newton method for solving AVE obtaining global Q-linear convergence and showing its numerical effectiveness.…”

Solving absolute value equation using complementarity and smoothing functions

“…In the special case where the problem is uniquely solvable, a family of Newton methods has been proposed first in [15], then completed with global and quadratic convergence in [2], an inexact version in [1] and other related methods [5,20,30]. Also, Picard-HSS iteration methods and nonlinear HSS-like methods have been considered for instance in [28,22,31]. It is of a great interest to consider methods that remain valid in the general case.…”

Scaled norm minimization method for computing the parameters of the HSS and the two‐parameter HSS preconditioners

“…8 Due to its promising performance and elegant mathematical properties, the HSS scheme immediately attracted wide attention and was used to solve different kinds of problems, such as saddle-point problems, [9][10][11][12][13][14][15] complex linear systems, [16][17][18][19][20] certain singular problems, 21,22 and nonlinear problems. 23,24 As it is used as a solver, the HSS method was also used as a preconditioner to accelerate the convergence speed of the Krylov subspace methods. [25][26][27][28] This preconditioner has the form of…”