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

Abstract

In this paper, enlightened by the effectiveness of Anderson acceleration (AA), a well‐established technique for accelerating fixed‐point solvers, we first present the Anderson accelerating modulus‐based matrix splitting (AAMS) algorithms for a class of horizontal nonlinear complementarity problems. Then, by introducing the strong semi‐smoothness of the absolute value function, we establish the local convergence theory of the AAMS algorithms. More importantly, we provide the optimal parameter for the AAMS algorithms, which is independent of iteration and applicable to other reduced MS algorithms. Finally, the numerical experiments are conducted to clarify the efficiency and practicability of the AAMS algorithms and its optimal parameter, and the effects of two parameters (both derived from AA) of the AAMS algorithms are analyzed.

On estimation of the optimal parameter of the modulus‐based matrix splitting algorithm for linear complementarity problems on second‐order cones

“…Recently, for LCP($$$ A,b,{\mathbb{R}}_{+}^n $$$), Li et al 31 proposed a practical selection strategy of the optimal parameter of the MMS algorithm, by introducing a mapping $$$ \psi (x):= \operatorname{diag}\left(\operatorname{sign}(x)\right)\in {\mathbb{R}}^{n\times n}\kern0.3em \left(x\in {\mathbb{R}}^n\right) $$$ such that $$$ \psi (x)x=\mid x\mid, $$$ thus reformulating the implicit fixed‐point equation explicitly. Further, by using this mapping $$$ \psi (x) $$$, the estimation of the optimal parameter of the MMS algorithm for horizontal nonlinear complementarity problems is presented as well in Reference 32. However, the results in References 31 and 32 are not applicable to SOCLCP(), because the involved absolute value function $$$ ABS\left(\cdotp \right) $$$ in the implicit fixed‐point Equation (4) is based on Jordan algebra, so the mapping is different and more complicated.…”

“…Further, by using this mapping $$$ \psi (x) $$$, the estimation of the optimal parameter of the MMS algorithm for horizontal nonlinear complementarity problems is presented as well in Reference 32. However, the results in References 31 and 32 are not applicable to SOCLCP(), because the involved absolute value function $$$ ABS\left(\cdotp \right) $$$ in the implicit fixed‐point Equation (4) is based on Jordan algebra, so the mapping is different and more complicated.…”