A power penalty approach to a mixed quasilinear elliptic complementarity problem

Simultaneous distributed-boundary optimal control problems driven by nonlinear complementarity systems

“…Under general assumptions (a is not strongly monotone with respect to the second variable and 1 < p < +∞ is satisfied), the second goal is to employ a power penalty method to introduce a family of approximating problems without constrains (see Problem 9) and to establish a convergence result that the unique solution of Problem 2 can be approached by the approximating mixed variational inequality, Problem 9, when a penalty parameter tends to infinity. More particularly, our result, Corollary 14, extends Theorems 4.1 and 4.2 of Duan-Wang-Zhou [8]. Whereas, the third contribution of this paper is to investigate a nonlinear simultaneous distributed-boundary optimal control problem governed by Problem 2 and to deliver an existence theorem to the simultaneous distributed-boundary optimal control problem.…”

“…Indeed, it should be mentioned that if 2 = 3 = ∅ (i.e., 1 = ) and a is independent of the variable x, then Problem 1 becomes the complementarity problem with Dirichlet boundary condition which has been studied recently by Duan-Wang-Zhou [8]. More precisely, when a satisfies the hypotheses H(a ) (which requires that a is strongly monotone, see Remark 15) and inequality 2 ≤ N < p is satisfied, Duan-Wang-Zhou [8] obtained an existence theorem and a convergence result (see Theorems 4.1 and 4.2 of Duan-Wang-Zhou [8]). However, the strong monotonicity of a and inequality 2 ≤ N < p lead to inapplicability in a lot of problems.…”

An Efficient Class of Modulus-Based Matrix Splitting Methods for Nonlinear Complementarity Problems

“…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]. Most of those methods need to solve the linear complementarity subproblems; see [16][17][18][19] for several typical iteration methods.…”

Penalty approximation method for a double obstacle quasilinear parabolic variational inequality problem

“…In [26], Zhou et al explored a novel power penalty method for the approximate solution to a double obstacle complementarity problem involving a semilinear parabolic differential operator, obtained a convergence rate that is corresponded to the theoretical results. In [9], Duan et al solved a mixed quasilinear elliptic complementarity problem by a lower power penalty approximation method, derived the error estimates of the convergence of the elliptic partial differential penalty equation and showed that the lower power penalty method is efficient and robust by numerical results.…”

“…It is worth to point out that our previous work is main on the double obstacle complementarity semilinear parabolic problem [26] and the mixed quasilinear elliptic complementarity problem [9]. The double obstacle quasilinear parabolic variational inequality problem (VIP) considered in this paper is more complicated.…”