A power penalty method for second-order cone nonlinear complementarity problems

Low-order penalty equations for semidefinite linear complementarity problems

“…By taking the advantage of the Cartesian P-property introduced by Chen and Qi in [11], we will establish the convergence of the proposed class of low-order penalty methods for Cartesian P-SDLCP with the same exponential convergence rate as shown in the context of LCPs [19], NCPs [20,21] and SOCPs [23,24] when λ tends to infinity. This indicates that the low-order penalty scheme can be successfully extended to handle SDLCPs as well.…”

“…For instance, Wang and Yang [19] proposed a power penalty method with the low-order 1/k (k ≥ 1) penalty term for solving the classic linear complementarity problems (LCPs) and established the exponential convergence rate under the cases that the involved coefficient matrix is a nonsingular M-matrix (positive definite with all off-diagonal entries non-positive) or a diagonal matrix. More recently, this power method scheme was further extended to handle more general complementarity problems such as the nonlinear complementarity problems (NCPs) with strong monotonicity [20] and with the uniform P-property [21], the parabolic linear complementarity problem arising from American options pricing problems [22], and the second-order cone complementarity problems (SOCPs) with positive definite linear mappings [23] and with the strong monotone nonlinear mappings [24]. It is well-known that the SOCP is an important extension of classic complementarity problems by substituting the involved nonnegative orthant with the so-called second order cone.…”

A new filter QP-free method for the nonlinear inequality constrained optimization problem

“…And this NCP has attracted much attention due to its various applications [ 1 – 3 ] such as the economic equilibrium problem, the restructuring problems of electricity and gas markets, and so on. Of course, there are many efficient methods for solving the NCP, which can be seen in [ 4 – 7 ]. One popular way to solve the NCP is to construct a Newton method for solving the related nonlinear equations, which is a reformulation of the KKT optimality condition.…”

A penalty approach for a box constrained variational inequality problem

“…Recently, a power penalty approach has been proposed for linear, nonlinear, and mixed nonlinear complementarity problems in both the finite-dimensional space R n and the infinite-dimensional functional spaces [9], [6], [8], [16]. This approach consists of approximating a box constrained variational inequality problem BVIP by a sequence of nonlinear penalty equations with a penalty term.…”

“…Based on the method presented in [9], [6], [8], [16] for linear, nonlinear, and mixed nonlinear complementarity problems, the present study aims to develop and analyze a penalty approach for the BVIP.…”