TY - JOUR

T1 - A new class of neural networks for NCPs using smooth perturbations of the natural residual function

AU - Alcantara, Jan Harold

AU - Chen, Jein Shan

N1 - Funding Information:
The author's work is supported by Ministry of Science and Technology, Taiwan.
Publisher Copyright:
© 2022 Elsevier B.V.

PY - 2022/6

Y1 - 2022/6

N2 - We present a new class of neural networks for solving nonlinear complementarity problems (NCPs) based on some family of real-valued functions (denoted by ℱ) that can be used to construct smooth perturbations of the level curve defined by ΦNR(x,y)=0, where ΦNR is the natural residual function (also called the “min” function). We introduce two important subclasses of ℱ, which deserve particular attention because of their significantly different theoretical and numerical properties. One of these subfamilies yields a smoothing function for ΦNR, while the other subfamily only yields a smoothing curve for ΦNR(x,y)=0. We also propose a simple framework for generating functions from these subclasses. Using the smoothing approach, we build two types of neural networks and provide sufficient conditions to guarantee asymptotic and exponential stability of equilibrium solutions. Finally, we present extensive numerical experiments to validate the theoretical results and to illustrate the difference in numerical performance of functions from the two subclasses. Numerical comparisons with existing neural networks for NCPs are also demonstrated.

AB - We present a new class of neural networks for solving nonlinear complementarity problems (NCPs) based on some family of real-valued functions (denoted by ℱ) that can be used to construct smooth perturbations of the level curve defined by ΦNR(x,y)=0, where ΦNR is the natural residual function (also called the “min” function). We introduce two important subclasses of ℱ, which deserve particular attention because of their significantly different theoretical and numerical properties. One of these subfamilies yields a smoothing function for ΦNR, while the other subfamily only yields a smoothing curve for ΦNR(x,y)=0. We also propose a simple framework for generating functions from these subclasses. Using the smoothing approach, we build two types of neural networks and provide sufficient conditions to guarantee asymptotic and exponential stability of equilibrium solutions. Finally, we present extensive numerical experiments to validate the theoretical results and to illustrate the difference in numerical performance of functions from the two subclasses. Numerical comparisons with existing neural networks for NCPs are also demonstrated.

KW - Complementarity problem

KW - Neural network

KW - Smoothing method

KW - Stability

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UR - https://www.mendeley.com/catalogue/a3e5a9e8-f8a1-39c0-8d8c-60742bccdd47/

U2 - 10.1016/j.cam.2022.114092

DO - 10.1016/j.cam.2022.114092

M3 - Article

AN - SCOPUS:85123253033

SN - 0377-0427

VL - 407

JO - Journal of Computational and Applied Mathematics

JF - Journal of Computational and Applied Mathematics

M1 - 114092

ER -