Modulus-based matrix splitting methods for a class of horizontal nonlinear complementarity problems

Un algoritmo Newton inexacto para complementariedad horizontal

“…Estos problemas surgen de la discretización de una ecuación diferencial como la siguiente ∆x(ξ, η) + ∂ 2 w(ξ, η) ∂ξ 2 + µx(ξ, η) + νw(ξ, η) = q(ξ, η) + φ(ξ, η), con algunas condiciones de frontera y de complementariedad [25]. En la experimentación numérica usamos los parámetros establecidos en [25], A = A+µI n y B = B +νI n , con µ y ν constantes positivas, A tridiagonal por bloques, cuyas diagonales superior e inferior contienen, en cada uno de sus bloques la matriz −I m , con m 2 = n, y cuya diagonal principal contiene en cada bloque la matriz S = tridiag(− Consideramos tres términos no lineales, también propuestos en [25] que dieron lugar a tres problemas. Estas tres funciones φ 1 , φ 2 , φ 3 , se definen por, φ 1,i (x, w) = x 2 i , φ 2,i (x, w) = w 2 i , φ 3,i (x, w) = x i w i .…”

Anderson acceleration of the modulus‐based matrix splitting algorithms for horizontal nonlinear complementarity systems

“…For the HLCP(), Mezzadri and Galligani 1 constructed the standard and accelerated MS algorithms. Recently, they extended the MS algorithms to solve the HNCP(), 26 as shown in Algorithm 1 below.…”

“…In addition, the determination of the optimal iteration parameter of the MS algorithms is fundamentally important, since it is a critical step for the efficient implementation of these algorithms. In most of the literatures on this algorithm, the convergence interval of iteration parameter matrix has been given, for example, for the LCP(), 12 for the HNCP(), 26 for the HLCP() 1 and for the SOCLCP 19 . However, due to the nonlinearity of the MS algorithms, there are few studies on the optimal parameter.…”

“…In this work, to accelerate the MS algorithm of Algorithm 1 26 with AA, we firstly construct the Anderson accelerating modulus‐based MS (AAMS) algorithms for HNCP(). Then, by introducing the strong semi‐smoothness of the absolute value function, we establish the local convergence theory of the AAMS algorithms.…”

Convergence of modulus-based matrix splitting iteration method for a class of nonlinear complementarity problems

“…After that, a series of modulus-based type matrix splitting iteration methods have been proposed, for instance, the general modulus-based matrix splitting method [14], the preconditioned modulus-based matrix splitting iteration method [15], the two-step modulus-based matrix splitting iteration method [16], the accelerated modulus-based matrix splitting iteration methods [17], and so on. Since these modulus-based type matrix splitting iteration methods are usually very practical and effective, many authors extended this kind of methods to solve other complementarity problems, such as the implicit complementarity problems [6,18,19], the weakly nonlinear complementarity problems [3,4], the horizontal nonlinear complementarity problems [20,21], the quasi-complementarity problems [22], and so on.…”