A penalty method for a mixed nonlinear complementarity problem

We propose a power penalty method for a mixed nonlinear complementarity problem (MNCP) and show that the solution to the penalty equation converges to that of the MNCP exponentially as the penalty parameter approaches infinity, provided that the mapping involved in the MNCP is both continuous and ξ-monotone. Furthermore, a convergence theorem is established when the monotonicity assumption on the mapping is removed. To demonstrate the usefulness and the convergence rates of this method, we design a non-trivial test MNCP problem arising in shape-preserving bi-harmonic interpolation and apply our method to this test problem. The numerical results confirm our theoretical findings.     

A power penalty method for linear complementarity problems

We propose a power penalty approach to a linear complementarity problem (LCP) in Rⁿ based on approximating the LCP by a nonlinear equation. We prove that the solution to this equation converges to that of the LCP at an exponential rate when the penalty parameter tends to infinity.     

A complementarity approach to a quasistatic multi-rigid-body contact problem

In this paper, we study the problem of predicting the quasistatic planar motion of a passive rigid body in frictional contact with a set of active rigid bodies. The active bodies can be thought of as the links of a mechanism or robot manipulator whose positions can be actively controlled by actuators. The passive body can be viewed as a grasped object, which moves only in response to contact forces and other external forces such as those due to gravity. We formulate this problem as a certain uncoupled complementarity problem, and show that it belongs to the class of NP-complete problems. Finally, numerical results of our proposed linear programming-based solution algorithm for this class of problems are presented and compared to the only other currently available solution algorithm.The research of this author was based on work supported by the National Science Foundation under grants DDM-9104078 and CCR-9213739.The research of this author was partially supported by the National Science Foundation under grant IRI-9304734, the Texas Advanced Technology Program under grant 999903-095, and the Texas Advanced Research Program under grant 999903-078.     

A power penalty approach to a Nonlinear Complementarity Problem

We propose a novel power penalty approach to a Nonlinear Complementarity Problem (NCP) in which the NCP is approximated by a nonlinear equation containing a power penalty term. We show that the solution to the penalty equation converges to that of the NCP at an exponential rate when the function involved is continuous and ξ-monotone. A higher convergence rate is also obtained when the function becomes Lipschitz continuous. Numerical results are presented to confirm the theoretical findings.     

Large solutions to an anisotropic quasilinear elliptic problem

In this paper we consider existence, asymptotic behavior near the boundary and uniqueness of positive solutions to the problem ${\rm div}_x (|\nabla_x u|^{p-2}\nabla_xu)(x,y) + {\rm div}_y (|\nabla_y u|^{q-2}\nabla_y u) (x, y) = u^r(x, y)$ in a bounded domain ${\Omega \subset \mathbb{R}^N \times \mathbb{R}^M}In this paper we consider existence, asymptotic behavior near the boundary and uniqueness of positive solutions to the problem

divx (|?x u|p-2?xu)(x,y) + divy (|?y u|q-2?y u) (x, y) = ur(x, y){\rm div}_x (|\nabla_x u|^{p-2}\nabla_xu)(x,y) + {\rm div}_y (|\nabla_y u|^{q-2}\nabla_y u) (x, y) = u^r(x, y)      8. On a jumping problem for quasilinear elliptic equations      Annamaria Canino 《Mathematische Zeitschrift》1997,226(2):193-210       9. On a quasilinear elliptic eigenvalue problem with constraint   总被引：1，自引：0，他引：1   CHEN Jianqing  CHEN Shaowei & LI YongqingDepartment of Mathematics  Fujian Normal University  Fuzhou  China  Institute of Mathematics  AMSS  Chinese Academy of Sciences  Beijing  China 《中国科学A辑(英文版)》2004,47(4):523-537 Via construction of pseudo gradient vector field and descending flow argument, we prove the existence of one positive, one negative and one sign-changing solutions for a quasilinear elliptic eigenvalue problem with constraint.      10. A dual exact penalty formulation for the linear complementarity problem      P. K. Subramanian 《Journal of Optimization Theory and Applications》1988,58(3):525-538 We consider a dual exact penalty formulation for the monotone linear complementarity problem. Tihonov regularization is then used to reduce the solution of the problem to the solution of a sequence of positive-definite, symmetric quadratic programs. A modified form of an SOR method due to Mangasarian is proposed to solve these quadratic programs. We also indicate how to obtain approximate solutions to predefined tolerance by solving a single quadratic program, in special cases.This research was sponsored by US Army Contract DAAG29-80-C-0041, by National Science Foundation Grants DCR-8420963 and MCS-8102684, and AFSOR Grant AFSOR-ISSA-85-0880.      11. A degenerate Neumann problem for quasilinear elliptic integro-differential operators      D.K. Palagachev  P.R. Popivanov  K. Taira 《Mathematische Zeitschrift》1999,230(4):679-694 This paper is devoted to the study of the following degenerate Neumann problem for a quasilinear elliptic integro-differential operator Here is a second-order elliptic integro-differential operator of Waldenfels type and is a first-order Ventcel' operator with a(x) and b(x) being non-negative smooth functions on such that on . Classical existence and uniqueness results in the framework of H?lder spaces are derived under suitable regularity and structure conditions on the nonlinear term f(x,u,Du). Received April 22, 1997; in final form March 16, 1998      12. Constant boundary value problem for a quasilinear elliptic system      Yaotian Shen  Shusen Yan 《manuscripta mathematica》1993,79(1):99-112 We study the constant boundary value problem for a class of quasilinear elliptic systems. The existence of a nonconstant solution is proved by discussing the minimization problem for the relaxed energy. As a by-product, we show that generally the Pohozaev identity does not hold for H1-weak solutions.      13. A unified approach to monotone iterative technique for quasilinear elliptic problems      S. Köksal  V. Lakshmikantham 《Applicable analysis》2013,92(3-4):391-403 A unified approach Tor the miinatone Itera!ive technique is discussed relative to quasilinear elliptic boundary value problems when the nonlinear term involved admits a splitting of the difference of two monotone functions. This setting includes several results in one framework and is applicable to a variety of nonlinear problems.      14. Some remarks on a quasilinear elliptic boundary value problem      Nguyên Phuong Các 《Nonlinear Analysis: Theory, Methods & Applications》1984,8(7):697-709       15. A mixed finite-element method for quasilinear degenerate fourth-order elliptic equations      M. M. Karchevskii  A. D. Lyashko  M. R. Timerbaev 《Differential Equations》2000,36(7):1050-1057       16. Locating the peaks of least-energy solutions to a quasilinear elliptic Neumann problem      Yi Li  Chunshan Zhao 《Journal of Mathematical Analysis and Applications》2007,336(2):1368-1383 In this paper we study the shape of least-energy solutions to the quasilinear problem εmΔmu−um−1+f(u)=0 with homogeneous Neumann boundary condition. We use an intrinsic variation method to show that as ε→0+, the global maximum point Pε of least-energy solutions goes to a point on the boundary ∂Ω at the rate of o(ε) and this point on the boundary approaches to a point where the mean curvature of ∂Ω achieves its maximum. We also give a complete proof of exponential decay of least-energy solutions.      17. A general theorem on error estimates with application to a quasilinear elliptic optimal control problem      Eduardo Casas  Fredi Tr?ltzsch 《Computational Optimization and Applications》2012,53(1):173-206 A theorem on error estimates for smooth nonlinear programming problems in Banach spaces is proved that can be used to derive optimal error estimates for optimal control problems. This theorem is applied to a class of optimal control problems for quasilinear elliptic equations. The state equation is approximated by a finite element scheme, while different discretization methods are used for the control functions. The distance of locally optimal controls to their discrete approximations is estimated.      18. A semilinear elliptic problem on unbounded domains with reverse penalty      Kyril Tintarev 《Nonlinear Analysis: Theory, Methods & Applications》2006 In the well-known work of P.-L. Lions [The concentration–compactness principle in the calculus of variations, The locally compact case, part 1. Ann. Inst. H. Poincaré, Analyse Non Linéaire 1 (1984) 109–1453] existence of positive solutions to the equation -Δu+u=b(x)up-1$-\mathrm{\Delta }u+u=b(x)u^{p-1}$, u>0$u>0$, u∈H1(RN)$u\in H^1({\mathbb{R}}^N)$, p∈(2,2N/(N-2))$p\in (2,2N/(N-2))$ was proved under assumption b(x)?b∞?lim|x|→∞b(x)$b(x)?b_{\infty }?{\lim }_{\left|x\right|\to \infty }b(x)$. In this paper we prove the existence for certain functions b   satisfying the reverse inequality b(x)<b∞$b(x)<b_{\infty }$. For any periodic lattice L   in RN${\mathbb{R}}^N$ and for any b∈C(RN)$b\in C({\mathbb{R}}^N)$ satisfying b(x)<b∞$b(x)<b_{\infty }$, b∞>0$b_{\infty }>0$, there is a finite set Y⊂L$Y\subset L$ and a convex combination bY$b^Y$ of b(·-y)$b(·-y)$, y∈Y$y\in Y$, such that the problem -Δu+u=bY(x)up-1$-\mathrm{\Delta }u+u=b^Y(x)u^{p-1}$ has a positive solution u∈H1(RN)$u\in H^1({\mathbb{R}}^N)$.      19. A penalty approach to a discretized double obstacle problem with derivative constraints      Song Wang 《Journal of Global Optimization》2015,62(4):775-790       20. A variational approach to singular quasilinear elliptic problems with sign changing nonlinearities      Ravi P. Agarwal  Kanishka Perera  Donal O'Regan 《Applicable analysis》2013,92(10):1201-1206 We obtain positive solutions of singular p-Laplacian problems with sign changing nonlinearities using variational methods.

On a quasilinear elliptic eigenvalue problem with constraint

Via construction of pseudo gradient vector field and descending flow argument, we prove the existence of one positive, one negative and one sign-changing solutions for a quasilinear elliptic eigenvalue problem with constraint.     

A dual exact penalty formulation for the linear complementarity problem

We consider a dual exact penalty formulation for the monotone linear complementarity problem. Tihonov regularization is then used to reduce the solution of the problem to the solution of a sequence of positive-definite, symmetric quadratic programs. A modified form of an SOR method due to Mangasarian is proposed to solve these quadratic programs. We also indicate how to obtain approximate solutions to predefined tolerance by solving a single quadratic program, in special cases.This research was sponsored by US Army Contract DAAG29-80-C-0041, by National Science Foundation Grants DCR-8420963 and MCS-8102684, and AFSOR Grant AFSOR-ISSA-85-0880.     

A degenerate Neumann problem for quasilinear elliptic integro-differential operators

This paper is devoted to the study of the following degenerate Neumann problem for a quasilinear elliptic integro-differential operator Here is a second-order elliptic integro-differential operator of Waldenfels type and is a first-order Ventcel' operator with a(x) and b(x) being non-negative smooth functions on such that on . Classical existence and uniqueness results in the framework of H?lder spaces are derived under suitable regularity and structure conditions on the nonlinear term f(x,u,Du). Received April 22, 1997; in final form March 16, 1998     

Constant boundary value problem for a quasilinear elliptic system

We study the constant boundary value problem for a class of quasilinear elliptic systems. The existence of a nonconstant solution is proved by discussing the minimization problem for the relaxed energy. As a by-product, we show that generally the Pohozaev identity does not hold for H¹-weak solutions.     

A unified approach to monotone iterative technique for quasilinear elliptic problems

A unified approach Tor the miinatone Itera!ive technique is discussed relative to quasilinear elliptic boundary value problems when the nonlinear term involved admits a splitting of the difference of two monotone functions. This setting includes several results in one framework and is applicable to a variety of nonlinear problems.     

Locating the peaks of least-energy solutions to a quasilinear elliptic Neumann problem

In this paper we study the shape of least-energy solutions to the quasilinear problem εmΔmu−um−1+f(u)=0 with homogeneous Neumann boundary condition. We use an intrinsic variation method to show that as ε→⁰⁺, the global maximum point Pε of least-energy solutions goes to a point on the boundary ∂Ω at the rate of o(ε) and this point on the boundary approaches to a point where the mean curvature of ∂Ω achieves its maximum. We also give a complete proof of exponential decay of least-energy solutions.     

A general theorem on error estimates with application to a quasilinear elliptic optimal control problem

A theorem on error estimates for smooth nonlinear programming problems in Banach spaces is proved that can be used to derive optimal error estimates for optimal control problems. This theorem is applied to a class of optimal control problems for quasilinear elliptic equations. The state equation is approximated by a finite element scheme, while different discretization methods are used for the control functions. The distance of locally optimal controls to their discrete approximations is estimated.     

A semilinear elliptic problem on unbounded domains with reverse penalty

In the well-known work of P.-L. Lions [The concentration–compactness principle in the calculus of variations, The locally compact case, part 1. Ann. Inst. H. Poincaré, Analyse Non Linéaire 1 (1984) 109–1453] existence of positive solutions to the equation -Δu+u=b(x)u^p-1$-\mathrm{\Delta }u+u=b(x)u^{p-1}$, u>0$u>0$, u∈H¹(R^N)$u\in H^1({\mathbb{R}}^N)$, p∈(2,2N/(N-2))$p\in (2,2N/(N-2))$ was proved under assumption b(x)?_b∞?_lim|x|→∞b(x)$b(x)?b_{\infty }?{\lim }_{\left|x\right|\to \infty }b(x)$. In this paper we prove the existence for certain functions b   satisfying the reverse inequality b(x)<_b∞$b(x)<b_{\infty }$. For any periodic lattice L   in R^N${\mathbb{R}}^N$ and for any b∈C(R^N)$b\in C({\mathbb{R}}^N)$ satisfying b(x)<_b∞$b(x)<b_{\infty }$, _b∞>0$b_{\infty }>0$, there is a finite set Y⊂L$Y\subset L$ and a convex combination b^Y$b^Y$ of b(·-y)$b(·-y)$, y∈Y$y\in Y$, such that the problem -Δu+u=b^Y(x)u^p-1$-\mathrm{\Delta }u+u=b^Y(x)u^{p-1}$ has a positive solution u∈H¹(R^N)$u\in H^1({\mathbb{R}}^N)$.     

A variational approach to singular quasilinear elliptic problems with sign changing nonlinearities

We obtain positive solutions of singular p-Laplacian problems with sign changing nonlinearities using variational methods.