Finite-dimensional linear approximations of solutions to general irregular nonlinear operator equations and equations with quadratic operators

A general scheme for improving approximate solutions to irregular nonlinear operator equations in Hilbert spaces is proposed and analyzed in the presence of errors. A modification of this scheme designed for equations with quadratic operators is also examined. The technique of universal linear approximations of irregular equations is combined with the projection onto finite-dimensional subspaces of a special form. It is shown that, for finite-dimensional quadratic problems, the proposed scheme provides information about the global geometric properties of the intersections of quadrics.     

Multi-valued variational inequalities with K-pseudomonotone operators

In this paper, we first employ the 1961 celebrated Fan lemma to derive a very general existence result for multi-valued variational inequalities involving multi-valued K-pseudomonotone operators. It will be seen that this result improves and unifies existence results of variational inequalities for monotone operators. Next, we establish some uniqueness results for multi-valued variational inequalities by introducing the concepts of strict, , and strong K-pseudomonotonicity of multi-valued operators, respectively. These uniqueness results appear to be new even if the underlying space is finite-dimensional.This work was partially supported by the National Science Council Grant NSC 82-0208-M-110-023. The author would like to express his sincere thanks to the referees for their valuable comments and suggestions that improved this paper substantially.     

Solving a class of variational inequalities with inexact oracle operators

Consider a class of variational inequality problems of finding ${x^*\in S}Consider a class of variational inequality problems of finding x^* ? S{x^*\in S}, such that

f(x*)T (z-x*) 3 0,    "z ? S,f(x^*)^\top (z-x^*)\geq 0,\quad \forall z\in S,      6. Numerical approximation of variational inequalities with one-sided irregular obstacles      S. Finzi Vita 《Numerical Functional Analysis & Optimization》2013,34(1):51-71 For a class of variational inequalities with the obstacle satisfying a one-sided Holder condition, we. show the uniform convergence in C '? (for a suitable ae]0,1]) of the finite element approximate solutions to the exact solution of the problem.      7. Modified variational inequalities to semilinear wave equations      Yukiyoshi Ebihara 《Nonlinear Analysis: Theory, Methods & Applications》1983,7(8):821-826       8. Reduction of affine variational inequalities      Stephen M. Robinson 《Computational Optimization and Applications》2016,65(2):493-509       9. Spectral problems for variational inequalities with discontinuous operators      D. K. Potapov 《Mathematical Notes》2013,93(1-2):288-296 Some spectral problems for variational inequalities with discontinuous nonlinear operators are considered. The variational method is used to prove the assumption that such problems are solvable. The general results are applied to the corresponding elliptic variational inequalities with discontinuous nonlinearities.      10. On a generic class of regular one-parametric variational inequalities      Walter Gómez Bofill 《Mathematical Methods of Operations Research》2000,51(2):179-201       11. On a class of variational inequalities involving gradient operators      Andrzej Szulkin 《Journal of Mathematical Analysis and Applications》1984,100(2):486-499       12. On solvability of variational inequalities with discontinuous semimonotone operators      V. N. Pavlenko 《Ukrainian Mathematical Journal》1993,45(3):475-480 By using the method of monotone operators, a theorem on the existence of the solution with a special property is obtained for an elliptic variational inequality with discontinuous semimonotone operator; this theorem is then used to prove the existence of a semicorrect solution of a variational inequality with a differential semilinear high-order operator of elliptic type with a nonsymmetric linear part and discontinuous nonlinearity.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 3, pp. 443–447, March, 1993.      13. Generalized vector variational inequalities with star-pseudomonotone and discontinuous operators      Bui Trong Kien  Ngai Ching Wong  Jen-Chih Yao 《Nonlinear Analysis: Theory, Methods & Applications》2008 It is well known that a vector variational inequality can be a very efficient model for use in studying vector optimization problems. By using the Ky Fan fixed point theorem and the scalarization method we will prove some existence theorems for strong solutions for generalized vector variational inequalities where discontinuous and star-pseudomonotone operators are involved. Our results can be applied to the study of the existence of solutions of vector optimal problems. Some examples are given and analyzed.      14. Backward stochastic dynamics with a subdifferential operator and non-local parabolic variational inequalities      Alain Bensoussan  Yiqun Li  Sheung Chi Phillip Yam 《Stochastic Processes and their Applications》2018,128(2):644-688 In this article, we provide the first systematic study on the unique existence of the solution of backward stochastic dynamical variational inequalities on a general complete filtered probability space. We also build up a comprehensive analysis of the correspondence between these stochastic variational inequalities (resp. backward stochastic dynamics) and the weak solutions (instead of viscosity ones due to the intrinsic non-local nature of the integral of the gradient involved) of a class of non-local parabolic variational inequalities (resp. parabolic partial differential equations), which is barely touched in the existing literature due to its unconventional setting.      15. Nonlinear variational inequalities for jump-diffusion processes and irregular obstacles with a financial application      《Nonlinear Analysis: Theory, Methods & Applications》1998,34(6):889-905       16. Perturbed unstable variational inequalities with unbounded operators on approximately given sets      Ya. I. Alber  A. I. Notik 《Set-Valued Analysis》1993,1(4):393-402 The stability and convergence of the solutions of perturbed and regularized variational inequality to the solutions of the primary (unstable a priori) variational inequality with proper monotone operator are investigated. All the objects of inequality: the operatorA, the right-hand partf and the set of constrains are to be perturbed. At the same time no assumptions of boundedness and smoothness of the operatorA are used. The connection between the parameters of perturbations, which guarantees strong convergence of approximate solutions, is established. It is proved that the existence of the solution to the unperturbed variational inequality is necessary and sufficient condition for convergence of the regularized perturbed inequality solutions.This research was supported in part by the Ministry of Science Grant 3481-1-91 and by the Ministry of Absorption Center for Absorption in Science.      17. Homogenization of nonlinear equations and variational inequalities with obstacles      G. V. Sandrakov 《Doklady Mathematics》2006,73(2):178-181       18. Block monotone iterative algorithms for variational inequalities with nonlinear operators      Ming-hui Ren  Jin-ping Zeng 《应用数学学报(英文版)》2008,24(2):221-232 Some block iterative methods for solving variational inequalities with nonlinear operators are proposed. Monotone convergence of the algorithms is obtained. Some comparison theorems are also established. Compared with the research work in given by Pao in 1995 for nonlinear equations and research work in given by Zeng and Zhou in 2002 for elliptic variational inequalities, the algorithms proposed in this paper are independent of the boundedness of the derivatives of the nonlinear operator.      19. An operator splitting method for monotone variational inequalities with a new perturbation strategy      Zhili Ge  Deren Han  Qin Ni  David Z. W. Wang 《Optimization Letters》2018,12(1):103-122 In variational inequalities arising from applications such as engineering, economics and transportation, partial mappings are usually unknown, e.g., the demand function in traffic assignment problem. As a consequence, classical methods can not deal with this class of problems. On the other hand, the recently developed methods require restrictive conditions such as strong monotonicity of some mappings, which excludes many interesting applications. In this paper, we propose an operator splitting method with a new perturbation strategy for solving variational inequality problems with partially unknown mappings. Under the mild condition that the underlying mapping is monotone, we prove the global convergence of the method. We also report some preliminary numerical results which show that the new algorithm is also interesting from the numerical point of view.      20. Quasi‐subdifferential operator approach to elliptic variational and quasi‐variational inequalities          下载免费PDF全文 M. Kubo  Y. Murase 《Mathematical Methods in the Applied Sciences》2016,39(18):5626-5635 We prove an existence theorem for an abstract operator equation associated with a quasi‐subdifferential operator and then apply it to concrete elliptic variational and quasi‐variational inequalities. Copyright © 2016 John Wiley & Sons, Ltd.

Numerical approximation of variational inequalities with one-sided irregular obstacles

For a class of variational inequalities with the obstacle satisfying a one-sided Holder condition, we. show the uniform convergence in C '? (for a suitable ae]0,1]) of the finite element approximate solutions to the exact solution of the problem.     

Spectral problems for variational inequalities with discontinuous operators

Some spectral problems for variational inequalities with discontinuous nonlinear operators are considered. The variational method is used to prove the assumption that such problems are solvable. The general results are applied to the corresponding elliptic variational inequalities with discontinuous nonlinearities.     

On solvability of variational inequalities with discontinuous semimonotone operators

By using the method of monotone operators, a theorem on the existence of the solution with a special property is obtained for an elliptic variational inequality with discontinuous semimonotone operator; this theorem is then used to prove the existence of a semicorrect solution of a variational inequality with a differential semilinear high-order operator of elliptic type with a nonsymmetric linear part and discontinuous nonlinearity.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 3, pp. 443–447, March, 1993.     

Generalized vector variational inequalities with star-pseudomonotone and discontinuous operators

It is well known that a vector variational inequality can be a very efficient model for use in studying vector optimization problems. By using the Ky Fan fixed point theorem and the scalarization method we will prove some existence theorems for strong solutions for generalized vector variational inequalities where discontinuous and star-pseudomonotone operators are involved. Our results can be applied to the study of the existence of solutions of vector optimal problems. Some examples are given and analyzed.     

Backward stochastic dynamics with a subdifferential operator and non-local parabolic variational inequalities

In this article, we provide the first systematic study on the unique existence of the solution of backward stochastic dynamical variational inequalities on a general complete filtered probability space. We also build up a comprehensive analysis of the correspondence between these stochastic variational inequalities (resp. backward stochastic dynamics) and the weak solutions (instead of viscosity ones due to the intrinsic non-local nature of the integral of the gradient involved) of a class of non-local parabolic variational inequalities (resp. parabolic partial differential equations), which is barely touched in the existing literature due to its unconventional setting.     

Perturbed unstable variational inequalities with unbounded operators on approximately given sets

The stability and convergence of the solutions of perturbed and regularized variational inequality to the solutions of the primary (unstable a priori) variational inequality with proper monotone operator are investigated. All the objects of inequality: the operatorA, the right-hand partf and the set of constrains are to be perturbed. At the same time no assumptions of boundedness and smoothness of the operatorA are used. The connection between the parameters of perturbations, which guarantees strong convergence of approximate solutions, is established. It is proved that the existence of the solution to the unperturbed variational inequality is necessary and sufficient condition for convergence of the regularized perturbed inequality solutions.This research was supported in part by the Ministry of Science Grant 3481-1-91 and by the Ministry of Absorption Center for Absorption in Science.     

Block monotone iterative algorithms for variational inequalities with nonlinear operators

Some block iterative methods for solving variational inequalities with nonlinear operators are proposed. Monotone convergence of the algorithms is obtained. Some comparison theorems are also established. Compared with the research work in given by Pao in 1995 for nonlinear equations and research work in given by Zeng and Zhou in 2002 for elliptic variational inequalities, the algorithms proposed in this paper are independent of the boundedness of the derivatives of the nonlinear operator.     

An operator splitting method for monotone variational inequalities with a new perturbation strategy

In variational inequalities arising from applications such as engineering, economics and transportation, partial mappings are usually unknown, e.g., the demand function in traffic assignment problem. As a consequence, classical methods can not deal with this class of problems. On the other hand, the recently developed methods require restrictive conditions such as strong monotonicity of some mappings, which excludes many interesting applications. In this paper, we propose an operator splitting method with a new perturbation strategy for solving variational inequality problems with partially unknown mappings. Under the mild condition that the underlying mapping is monotone, we prove the global convergence of the method. We also report some preliminary numerical results which show that the new algorithm is also interesting from the numerical point of view.     

Quasi‐subdifferential operator approach to elliptic variational and quasi‐variational inequalities

We prove an existence theorem for an abstract operator equation associated with a quasi‐subdifferential operator and then apply it to concrete elliptic variational and quasi‐variational inequalities. Copyright © 2016 John Wiley & Sons, Ltd.