Exceptional Family of Elements for a Variational Inequality Problem and its Applications

This paper introduces a new concept of exceptional family of elements (abbreviated, exceptional family) for a finite-dimensional nonlinear variational inequality problem. By using this new concept, we establish a general sufficient condition for the existence of a solution to the problem. Such a condition is used to develop several new existence theorems. Among other things, a sufficient and necessary condition for the solvability of pseudo-monotone variational inequality problem is proved. The notion of coercivity of a function and related classical existence theorems for variational inequality are also generalized. Finally, a solution condition for a class of nonlinear complementarity problems with so-called P _* -mappings is also obtained.     

Exceptional Families and Existence Theorems for Variational Inequality Problems

This paper introduces the concept of exceptional family for nonlinear variational inequality problems. Among other things, we show that the nonexistence of an exceptional family is a sufficient condition for the existence of a solution to variational inequalities. This sufficient condition is weaker than many known solution conditions and it is also necessary for pseudomonotone variational inequalities. From the results in this paper, we believe that the concept of exceptional families of variational inequalities provides a new powerful tool for the study of the existence theory for variational inequalities.     

Solvability of Variational Inequality Problems

This paper presents the new concept of exceptional family of elements for the variational inequality problem with a continuous function over a general unbounded closed convex set. We establish a characterization theorem that can be used to derive several new existence and compactness conditions on the solution set. Our findings generalize well-known results for various types of variational inequality problems. For a pseudomonotone variational inequality problem, our new existence conditions are both sufficient and necessary.     

变分不等式问题的一个新的存在性定理

对于具有一般非空闭凸集约束的变分不等式问题 ,本文给出了一个新的例外族的定义 .通过倩同伦不变定理 ,我们证明了一个择一定理 ,这给出了所考虑问题解的一个充分性条件 .特别 ,我们建立了变分不等式问题的一个新的存在性定理 ,推广了Zhao的一个最近的存在性结果 ,进而也推广了著名Mor啨关于非线性互补问题的存在性定理 .     

Strict approximate solutions in set-valued optimization with applications to the approximate Ekeland variational principle

This work deals with strict solutions of set-valued optimization problems under the set optimality criterion. In this context, we introduce a new approximate solution concept and we obtain several properties of these solutions when the error is fixed and also for their limit behavior when the error tends to zero. Then we prove a general existence result, which is applied to obtain approximate Ekeland variational principles.     

集值变分不等式问题的例外簇

本文首先在Banach空间中证明了零调集值映射的一个Leray-Schauder型不动点定理,然后在Hilbert空间中定义了零调集值映射的变分不等式的例外簇,利用本文给出的不动点定理给出了无界集上的变分不等式问题存在解的一个充分条件．此条件弱于许多已知的关于变分不等式问题的解的存在性条件,并由此得到Hilbert空间中几个变分不等式约解的存在性定理．     

Optimal endpoints

A complete set of necessary and sufficient conditions for selecting optimal endpoints for extremals obtained from the variational Bolza problem in control notation has been developed. The method used to obtain these conditions is based on a seldom used concept of performing a dichotomy on the general optimization problem. With this concept, the problem of Bolza is decomposed into two problems, the first of which involves the selection of optimal paths with the endpoints considered fixed. The second problem involves the selection of optimal endpoints with the paths between the endpoints taken to be stationary curves. The convenience of the dichotomy in deriving the necessary and sufficient conditions for endpoints lies in its simplicity and elementary character; well-known necessary and sufficient conditions from the theory of ordinary maxima and minima are used.An endpoint necessary condition is first obtained which is simply the well-known transversality condition. An additional condition is then developed which, together with the transversality condition, leads to a set of necessary and sufficient conditions for a given extremal to be locally optimal with respect to endpoint variations. While the second condition presented is akin to the classical focal-point condition, the result is new in form and is directly applicable to the optimal control problem. In addition, it is relatively simple to apply and is easy to implement numerically when an analytical solution is not possible. It should be useful in situations where the transversality conditions yield more than one choice for an optimal endpoint.An analytic solution for a simple geodetics problem is presented to illustrate the theory. A discussion of numerical implementation of the sufficiency conditions and its application to an orbit transfer example is also included.This work was supported in part by the National Aeronautics and Space Administration, Grant No. NGR-03-002-001.     

Projection iterative methods for extended general variational inequalities

In this paper, we introduce and study a new class of variational inequalities involving three operators, which is called the extended general variational inequality. Using the projection technique, we show that the extended general variational inequalities are equivalent to the fixed point and the extended general Wiener-Hopf equations. This equivalent formulation is used to suggest and analyze a number of projection iterative methods for solving the extended general variational inequalities. We also consider the convergence of these new methods under some suitable conditions. Since the extended general variational inequalities include general variational inequalities and related optimization problems as special cases, results proved in this paper continue to hold for these problems.     

广义非凸变分不等式解的存在性与投影算法

本文运用Banach压缩映象原理和投影技巧研究一类新的广义非凸变分不等式问题解的存在唯一性,并在非凸集上建立一个逼近广义非凸变分不等式解的三步投影算法,在一定条件下证明了该投影算法所产生的迭代序列的收敛性.     

Algorithms for solutions of extended general mixed variational inequalities and fixed points

In this paper, we introduce and study a new class of extended general nonlinear mixed variational inequalities and a new class of extended general resolvent equations and establish the equivalence between the extended general nonlinear mixed variational inequalities and implicit fixed point problems as well as the extended general resolvent equations. Then by using this equivalent formulation, we discuss the existence and uniqueness of solution of the problem of extended general nonlinear mixed variational inequalities. Applying the aforesaid equivalent alternative formulation and a nearly uniformly Lipschitzian mapping S, we construct some new resolvent iterative algorithms for finding an element of set of the fixed points of nearly uniformly Lipschitzian mapping S which is the unique solution of the problem of extended general nonlinear mixed variational inequalities. We study convergence analysis of the suggested iterative schemes under some suitable conditions. We also suggest and analyze a class of extended general resolvent dynamical systems associated with the extended general nonlinear mixed variational inequalities and show that the trajectory of the solution of the extended general resolvent dynamical system converges globally exponentially to the unique solution of the extended general nonlinear mixed variational inequalities. The results presented in this paper extend and improve some known results in the literature.     

Generalization of an Existence Theorem for Variational Inequalities

By using the concept of exceptional family of elements, Zhao proposed a new existence theorem for variational inequalities over a general nonempty closed convex set (Ref. 1, Theorem 2.3), which is a generalization of the well-known Moré's existence theorem for nonlinear complementarity problems. The proof of Theorem 2.3 in Ref. 1 depends strongly on the condition 0∈K. Since this condition is rather strict for a general variational inequality, Zhao proposed an open question at the end of Ref. 1: Can the condition 0∈K in Theorem 2.3 be removed? In this paper, we answer this open question. Furthermore, we present the new notion of exceptional family of elements and establish a theorem of the alternative, by which we develop two new existence theorems for variational inequalities. Our results generalize the Zhao existence result.     

On General Mixed Variational Inequalities

In this paper, we introduce and consider a new class of mixed variational inequalities, which is called the general mixed variational inequality. Using the resolvent operator technique, we establish the equivalence between the general mixed variational inequalities and the fixed-point problems as well as resolvent equations. We use this alternative equivalent formulation to suggest and analyze some iterative methods for solving the general mixed variational inequalities. We study the convergence criteria of the suggested iterative methods under suitable conditions. Using the resolvent operator technique, we also consider the resolvent dynamical systems associated with the general mixed variational inequalities. We show that the trajectory of the dynamical system converges globally exponentially to the unique solution of the general mixed variational inequalities. Our methods of proofs are very simple as compared with others’ techniques. Results proved in this paper may be viewed as a refinement and important generalizations of the previous known results.     

Solution smoothness of ill-posed equations in Hilbert spaces: four concepts and their cross connections

Numerical solution of ill-posed operator equations requires regularization techniques. The convergence of regularized solutions to the exact solution can be usually guaranteed, but to also obtain estimates for the speed of convergence one has to exploit some kind of smoothness of the exact solution. We consider four such smoothness concepts in a Hilbert space setting: source conditions, approximate source conditions, variational inequalities, and approximate variational inequalities. Besides some new auxiliary results on variational inequalities the equivalence of the last three concepts is shown. In addition, it turns out that the classical concept of source conditions and the modern concept of variational inequalities are connected via Fenchel duality.     

Resolvent Methods for Solving System of General Variational Inclusions

In this paper, we introduce and consider some new systems of extended general variational inclusions involving seven different operators. Using the resolvent operator technique, we show that the new systems of extended general variational inclusions are equivalent to the fixed point problems. This equivalent formulation is used to suggest and analyze some new iterative methods for this system of extended general variational inclusions. We also study the convergence analysis of the new iterative method under certain mild conditions. Several special cases are also discussed. Results obtained in this paper can be viewed as pure mathematical contribution to variational analysis.     

Stabilized sequential quadratic programming for optimization and a stabilized Newton-type method for variational problems

The stabilized version of the sequential quadratic programming algorithm (sSQP) had been developed in order to achieve fast convergence despite possible degeneracy of constraints of optimization problems, when the Lagrange multipliers associated to a solution are not unique. Superlinear convergence of sSQP had been previously established under the strong second-order sufficient condition for optimality (without any constraint qualification assumptions). We prove a stronger superlinear convergence result than the above, assuming the usual second-order sufficient condition only. In addition, our analysis is carried out in the more general setting of variational problems, for which we introduce a natural extension of sSQP techniques. In the process, we also obtain a new error bound for Karush–Kuhn–Tucker systems for variational problems that holds under an appropriate second-order condition.     

Projection iterative methods for solving some systems of general nonconvex variational inequalities

In this article, we introduce and consider a new system of general nonconvex variational inequalities involving four different operators. We use the projection operator technique to establish the equivalence between the system of general nonconvex variational inequalities and the fixed points problem. This alternative equivalent formulation is used to suggest and analyse some new explicit iterative methods for this system of nonconvex variational inequalities. We also study the convergence analysis of the new iterative method under certain mild conditions. Since this new system includes the system of nonconvex variational inequalities, variational inequalities and related optimization problems as special cases, results obtained in this article continue to hold for these problems. Our results can be viewed as a refinement and an improvement of the previously known results for variational inequalities.     

Projection methods for nonconvex variational inequalities

In this paper, we introduce and consider a new class of variational inequalities, which is called the nonconvex variational inequalities. We establish the equivalence between the nonconvex variational inequalities and the fixed-point problems using the projection technique. This equivalent formulation is used to discuss the existence of a solution of the nonconvex variational inequalities. We also use this equivalent alternative formulation to suggest and analyze a new iterative method for solving the nonconvex variational inequalities. We also discuss the convergence of the iterative method under suitable conditions. Our method of proof is very simple as compared with other techniques.     

New Condition Characterizing the Solutions of Variational Inequality Problems

This paper is devoted to the study of a new necessary condition in variational inequality problems: approximated gradient projection (AGP). A feasible point satisfies such condition if it is the limit of a sequence of the approximated solutions of approximations of the variational problem. This condition comes from optimization where the error in the approximated solution is measured by the projected gradient onto the approximated feasible set, which is obtained from a linearization of the constraints with slack variables to make the current point feasible. We state the AGP condition for variational inequality problems and show that it is necessary for a point being a solution even without constraint qualifications (e.g., Abadie’s). Moreover, the AGP condition is sufficient in convex variational inequalities. Sufficiency also holds for variational inequalities involving maximal monotone operators subject to the boundedness of the vectors in the image of the operator (playing the role of the gradients). Since AGP is a condition verified by a sequence, it is particularly interesting for iterative methods. Research of R. Gárciga Otero was partially supported by CNPq, FAPERJ/Cientistas do Nosso Estado, and PRONEX Optimization. Research of B.F. Svaiter was partially supported by CNPq Grants 300755/2005-8 and 475647/2006-8 and by PRONEX Optimization.     

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.     

Exceptional family of elements for generalized variational inequalities

This paper introduces a new concept of exceptional family of elements for a finite-dimensional generalized variational inequality problem. Based on the topological degree theory of set-valued mappings, an alternative theorem is obtained which says that the generalized variational inequality has either a solution or an exceptional family of elements. As an application, we present a sufficient condition to ensure the existence of a solution to the variational inequality. The set-valued mapping is assumed to be upper semicontinuous with nonempty compact convex values.