Abstract
In this paper, we present an efficient method for finding a numerical solution for nonlinear complementarity problems (NCPs). We first reformulate an NCP as an equivalent system of fixed-point equations and then present a modulus-based matrix splitting iteration method. We prove the convergence of the proposed method with theorems with the relevant conditions. Our preliminary numerical results show that the method is feasible and effective.
Highlights
The nonlinear compressed sensing theory has been widely applied in asteroseimology for significant detection. e polynomial structure has been employed in many applications cases, such as quadratic measurements in sparse signal recovery and nonlinear compressed sensing with polynomial measurements
Nonlinear complementarity problem (NCP) can be derived from discrete simulations of the Bratu obstacle problem and free boundary problems with nonlinear source terms. is problem has received much attention during the past two decades and has been studied extensively with applicable numerical methods to obtain an approximated solution. ere are different kinds of numerical methods that have been developed, including the classical linearized projected relaxation method [8], multilevel method [9], domain decomposition method [10, 11], penalty method [12,13,14], and semismooth Newton method [15]
We have presented an efficient class of modulus-based matrix splitting methods for NCPs, which are based on an implicit system of fixed-point equations
Summary
We consider the following nonlinear complementarity problem with a nonlinear source term, namely, finding a u ∈ Rn such that u ≥ 0, v F(u) ≥ 0,. E polynomial structure has been employed in many applications cases, such as quadratic measurements in sparse signal recovery and nonlinear compressed sensing with polynomial measurements In this regard, under some constraint qualifications, the original problem turns out to finding the sparsest solutions to a special NCP. Many scholars have developed various kinds of modulus-based matrix splitting iteration methods; see, for example, [21,22,23] for LCP and [24, 25] for NCP. Motivated by the ongoing work in this field, in this paper, we extend the new modulusbased matrix splitting method for a kind of NCP and establish its convergence through theorems.
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