Nonconvex Variational Inequalities Research Articles

New Classes of Exponentially General Nonconvex Variational Inequalities

In this paper, some new classes of exponentially general nonconvex variational inequalities are introduced and investigated. Several special cases are discussed as applications of these nonconvex variational inequalities. Projection technique is used to establish the equivalence between the non covex variational inequalities and fixed point problem. This equivalent formulation is used to discuss the existence of the solution. Several inertial type methods are suggested and analyzed for solving exponentially general nonconvex variational inequalities. using the technique of the projection operator and dynamical systems. Convergence analysis of the iterative methods is analyzed under suitable and appropriate weak conditions. In this sense, our result can be viewed as improvement and refinement of the previously known results. Our methods of proof are very simple as compared with other techniques.

Split Systems of Nonconvex Variational Inequalities and Fixed Point Problems on Uniformly Prox-Regular Sets

In this paper, we studied variational inequalities and fixed point problems in nonconvex cases. By the projection method over prox-regularity sets, the convergence of the suggested iterative scheme was established under some mild rules.

Auxiliary principle technique for solving regularized nonconvex set-valued mixed variational inequalities

In this paper, we introduce a new class of regularized nonconvex set-valued mixed variational inequalities where the underlying set is uniformly prox-regular. By using the auxiliary principle technique, we propose a predictor-corrector method for solving such class of nonconvex variational inequalities and study the strong convergence of the sequences generated by the proposed algorithm. As a consequence of our main results, we provide the correct version of algorithms and results presented in the literature.

Split systems of general nonconvex variational inequalities and fixed point problems

The purpose of this paper is to introduce and study split systems of general nonconvex variational inequalities. Taking advantage of the projection technique over uniformly prox-regularity sets and utilizing two nonlinear operators, we propose and analyze an iterative scheme for solving the split systems of general nonconvex variational inequalities and fixed point problems. We prove that the sequence generated by the suggested iterative algorithm converges strongly to a common solution of the foregoing split problem and fixed point problem. The result presented in this paper extends and improves some well-known results in the literature. Numerical example illustrates the theoretical result.

Nonconvex Policy Search Using Variational Inequalities.

Policy search is a class of reinforcement learning algorithms for finding optimal policies in control problems with limited feedback. These methods have been shown to be successful in high-dimensional problems such as robotics control. Though successful, current methods can lead to unsafe policy parameters that potentially could damage hardware units. Motivated by such constraints, we propose projection-based methods for safe policies. These methods, however, can handle only convex policy constraints. In this letter, we propose the first safe policy search reinforcement learner capable of operating under nonconvex policy constraints. This is achieved by observing, for the first time, a connection between nonconvex variational inequalities and policy search problems. We provide two algorithms, Mann and two-step iteration, to solve the above problems and prove convergence in the nonconvex stochastic setting. Finally, we demonstrate the performance of the algorithms on six benchmark dynamical systems and show that our new method is capable of outperforming previous methods under a variety of settings.

Convergence analysis of an iterative algorithm for the extended regularized nonconvex variational inequalities

In this paper, we suggest and analyze a new system of extended regularized nonconvex variational inequalities and prove the equivalence between the aforesaid system and a fixed point problem. We introduce a new perturbed projection iterative algorithm with mixed errors to find the solution of the system of extended regularized nonconvex variational inequalities. Furthermore, under moderate assumptions, we research the convergence analysis of the suggested iterative algorithm.

Linear and nonlinear solvers for variational phase-field models of brittle fracture

The variational approach to fracture is effective for simulating the nucleation and propagation of complex crack patterns, but is computationally demanding. The model is a strongly nonlinear non-convex variational inequality that demands the resolution of small length scales. The current standard algorithm for its solution, alternate minimization, is robust but converges slowly and demands the solution of large, ill-conditioned linear subproblems. In this paper, we propose several advances in the numerical solution of this model that improve its computational efficiency. We reformulate alternate minimization as a nonlinear Gauss-Seidel iteration and employ over-relaxation to accelerate its convergence; we compose this accelerated alternate minimization with Newton's method, to further reduce the time to solution; and we formulate efficient preconditioners for the solution of the linear subproblems arising in both alternate minimization and in Newton's method. We investigate the improvements in efficiency on several examples from the literature; the new solver is 5–6× faster on a majority of the test cases considered.

SYSTEM OF GENERALIZED NONLINEAR REGULARIZED NONCONVEX VARIATIONAL INEQUALITIES

In this work, we suggest a new it system of generalized nonlinear regularized nonconvex variational inequalities in a real Hilbert space and establish an equivalence relation between this system and fixed point problems. By using the equivalence relation we suggest a new perturbed projection iterative algorithms with mixed errors for finding a solution set of system of generalized nonlinear regularized nonconvex variational inequalities.

Calculus Rules forV-Proximal Subdifferentials in Smooth Banach Spaces

In 2010, Bounkhel et al. introduced new proximal concepts (analytic proximal subdifferential, geometric proximal subdifferential, and proximal normal cone) in reflexive smooth Banach spaces. They proved, inp-uniformly convex andq-uniformly smooth Banach spaces, the density theorem for the new concepts of proximal subdifferential and various important properties for both proximal subdifferential concepts and the proximal normal cone concept. In this paper, we establish calculus rules (fuzzy sum rule and chain rule) for both proximal subdifferentials and we prove the Bishop-Phelps theorem for the proximal normal cone. The limiting concept for both proximal subdifferentials and for the proximal normal cone is defined and studied. We prove that both limiting constructions coincide with the Mordukhovich constructions under some assumptions on the space. Applications to nonconvex minimisation problems and nonconvex variational inequalities are established.

Approximate solutions for nonconvex set-valued optimization and vector variational inequality

This note deals with approximate solutions in vector optimization involving a generalized cone-invex set-valued mapping. First, a new class of generalized cone-invex set-valued maps, called cone-subinvex set-valued maps, is introduced. Then the sufficient optimality condition and two types dual theorems are established for weakly approximate minimizers under the assumption of cone-subinvexity. Finally, it also reveals the closed relationships between a weakly approximate minimizer of a cone-subinvex set-valued optimization problem and a weakly approximate solution of a kind of vector variational inequality.

Sensitivity Analysis for General Nonlinear Nonconvex Variational Inequalities

In this communication, we proved that the parametric general nonlinear nonconvex variational inequalities are equivalent to the parametric general Wiener-Hopf equations. We use this alternative equivalence formulation to studied the sensitivity analysis for general nonlinear nonconvex variational inequalities without assuming the differentiability of the given data.

On a system of regularized nonconvex variational inequalities

On a system of regularized nonconvex variational inequalities

A General System of Regularized Non-convex Variational Inequalities

In this communication, we introduced a general system of regularized non-convex variational inequalities (GSRNVI) and established an equivalence between this system and fixed point problems. By using this equivalence we define a projection iterative algorithm for solving GSRNVI, we also proved existence and uniqueness of GSRNVI. The convergence analysis of the suggested iterative algorithm is studied.

Some remarks on regularized nonconvex variational inequalities

In this paper, we investigate and analyze the nonconvex variational inequalities introduced by Noor in (Optim. Lett. 3:411-418, 2009) and (Comput. Math. Model. 21:97-108, 2010) and prove that the algorithms and results in the above mentioned papers are not valid. To overcome the problems in the above cited papers, we introduce and consider a new class of variational inequalities, named regularized nonconvex variational inequalities, instead of the class of nonconvex variational inequalities introduced in the above mentioned papers. We also consider a class of nonconvex Wiener-Hopf equations and establish the equivalence between the regularized nonconvex variational inequalities and the fixed point problems as well as the nonconvex Wiener-Hopf equations. By using the obtained equivalence formulations, we prove the existence of a unique solution for the regularized nonconvex variational inequalities and propose some projection iterative schemes for solving the regularized nonconvex variational inequalities. We also study the convergence analysis of the suggested iterative schemes under some certain conditions. MSC:47H05, 47J20, 49J40, 90C33.

System of implicit nonconvex variationl inequality problems A projection method approach

In this paper, we consider a new system of implicit nonconvex variational inequality problems in setting of prox-regular subsets of two different Hilbert spaces. Using projection method, we establish the equivalence between the system of implicit nonconvex variational inequality problems and a system of relations. Using this equivalence formulation, we suggest some iterative algorithms for finding the approximate solution of the system of implicit nonconvex variational inequality problems and its special case. Further, we establish some theorems for the existence and iterative approximation of the solutions of the system of implicit nonconvex variational inequality problems and its special case. The results presented in this paper are new and different form the previously known results for nonconvex variational inequality problems. These results also generalize, unify and improve the previously known results of this area.

Iterative algorithms for systems of extended regularized nonconvex variational inequalities and fixed point problems

This paper points out some fatal errors in the equivalent formulations used in Noor 2011 [Noor MA. Projection iterative methods for solving some systems of general nonconvex variational inequalities. Applied Analysis. 2011;90:777–786] and consequently in Noor 2009 [Noor MA. System of nonconvex variational inequalities. Journal of Advanced Research Optimization. 2009;1:1–10], Noor 2010 [Noor MA, Noor KI. New system of general nonconvex variational inequalities. Applied Mathematics E-Notes. 2010;10:76–85] and Wen 2010 [Wen DJ. Projection methods for a generalized system of nonconvex variational inequalities with different nonlinear operators. Nonlinear Analysis. 2010;73:2292–2297]. Since these equivalent formulations are the main tools to suggest iterative algorithms and to establish the convergence results, the algorithms and results in the aforementioned articles are not valid. It is shown by given some examples. To overcome with the problems in these papers, we consider a new system of extended regularized nonconvex variational inequalities, and establish the existence and uniqueness result for a solution of the aforesaid system. We suggest and analyse a new projection iterative algorithm to compute the unique solution of the system of extended regularized nonconvex variational inequalities which is also a fixed point of a nearly uniformly Lipschitzian mapping. Furthermore, the convergence analysis of the proposed iterative algorithm under some suitable conditions is studied. As a consequence, we point out that one can derive the correct version of the algorithms and results presented in the above mentioned papers.

Sensitivity analysis of general nonconvex variational inequalities

In this paper, we show that the parametric general nonconvex variational inequalities are equivalent to the parametric Wiener-Hopf equations. We use this alternative equivalent formulation to study the sensitivity analysis for the nonconvex variational inequalities without assuming the differentiability of the given data. Our results can be considered as a significant extension of previously known results for the variational inequalities.

Predictor–Corrector Methods for General Regularized Nonconvex Variational Inequalities

This paper is devoted to the study of a new class of nonconvex variational inequalities, named general regularized nonconvex variational inequalities. By using the auxiliary principle technique, a new modified predictor–corrector iterative algorithm for solving general regularized nonconvex variational inequalities is suggested and analyzed. The convergence of the iterative algorithm is established under the partially relaxed monotonicity assumption. As a consequence, the algorithm and results presented in the paper overcome incorrect algorithms and results existing in the literature.

Solvability of a system of nonconvex general variational inequalities

Using the prox-regularity notion, we introduce a system of nonconvex general variational inequalities and a general three step algorithm for approximate solvability of this system. We establish the convergence of three-step projection method for a general system of nonconvex variational inequality problem. We obtain as a particular case some known results.

Projection Method Approach for General Regularized Non-convex Variational Inequalities

In this paper, we investigate or analyze non-convex variational inequalities and general non-convex variational inequalities. Two new classes of non-convex variational inequalities, named regularized non-convex variational inequalities and general regularized non-convex variational inequalities, are introduced, and the equivalence between these two classes of non-convex variational inequalities and the fixed point problems are established. A projection iterative method to approximate the solutions of general regularized non-convex variational inequalities is suggested. Meanwhile, the existence and uniqueness of solution for general regularized non-convex variational inequalities is proved, and the convergence analysis of the proposed iterative algorithm under certain conditions is studied.