Lower Semicontinuity of the Solution Mappings to a Parametric Generalized Ky Fan Inequality

In this paper, we investigate weak vector solutions and global vector solutions to a generalized Ky Fan inequality. Under new assumptions, which are weaker than the assumption of strict C-mappings, we establish the lower semicontinuity of the solution mappings to a parametric generalized Ky Fan inequality by using a scalarization method. These results extend the corresponding ones in the literature. Some examples are given to illustrate our results.     

New Parallel Descent-like Method for Solving a Class of Variational Inequalities

To solve a class of variational inequalities with separable structures, some classical methods such as the augmented Lagrangian method and the alternating direction methods require solving two subvariational inequalities at each iteration. The most recent work (B.S. He in Comput. Optim. Appl. 42(2):195–212, 2009) improved these classical methods by allowing the subvariational inequalities arising at each iteration to be solved in parallel, at the price of executing an additional descent step. This paper aims at developing this strategy further by refining the descent directions in the descent steps, while preserving the practical characteristics suitable for parallel computing. Convergence of the new parallel descent-like method is proved under the same mild assumptions on the problem data.     

Existence and Stability of Solutions for Generalized Ky Fan Inequality Problems with Trifunctions

In this paper, an existence theorem for solutions to the generalized Ky Fan Inequality problem is obtained by means of the Kakutani-Fan-Glicksberg fixed-point theorem without imposing the condition that the dual of the ordering cone has a weak* compact base. In addition, the stability of the solution set is shown.     

Auxiliary Principle and Iterative Algorithms for Lions-Stampacchia Variational Inequalities

In this paper, we extend the auxiliary principle (Cohen in J. Optim. Theory Appl. 49:325–333, 1988) to study a class of Lions-Stampacchia variational inequalities in Hilbert spaces. Our method consists in approximating, in the subproblems, the nonsmooth convex function by a sequence of piecewise linear and convex functions, as in the bundle method for nonsmooth optimization. This makes the subproblems more tractable. We show the existence of a solution for this Lions-Stampacchia variational inequality and explain how to build a new iterative scheme and a new stopping criterion. This iterative scheme and criterion are different from those commonly used in the special case of nonsmooth optimization. We study also the convergence of iterative sequences generated by the algorithm. This work was supported by the National Natural Science Foundation of China (10671135), the Specialized Research Fund for the Doctoral Program of Higher Education (20060610005), the National Natural Science Foundation of Sichuan Education Department of China (07ZB068) and the Open Fund (PLN0703) of State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation (Southwest Petroleum University).     

Hölder Continuity for Solution Mappings of Parametric Non-convex Strong Generalized Ky Fan Inequalities

Abstract

This article focuses on a new approach to investigate the Hölder continuity for the solution mapping of a parametric non-convex strong generalized Ky Fan inequality. Based on a non-convex separation theorem, the union relation between the solution set of the parametric non-convex strong generalized Ky Fan inequality and the solution sets of a series of Ky Fan inequalities, is established. Without density results and any information on the solution mapping, a sufficient condition for the Hölder continuity of the solution mapping to the parametric non-convex strong generalized Ky Fan inequality is given by using the key union relation. Our method does not impose any convexity, monotonicity, and the single-valuedness of the solution mapping.     

Erratum to: Risk-Sensitive Control with Near Monotone Cost

The infinite horizon risk-sensitive control problem for non-degenerate controlled diffusions is analyzed under a ‘near monotonicity’ condition on the running cost that penalizes large excursions of the process.     

Strong Convergence of Iterative Algorithms for Variational Inequalities in Banach Spaces

Let C be a nonempty closed convex subset of a Banach space E with the dual E ^*, let T:C→E ^* be a Lipschitz continuous mapping and let S:C→C be a relatively nonexpansive mapping. In this paper, by employing the notion of generalized projection operator, we study the following variational inequality (for short, VI(T−f,C)): find x∈C such that

Hybrid Steepest Descent Methods for Zeros of Nonlinear Operators with Applications to Variational Inequalities

In this paper, the hybrid steepest descent methods are extended to develop new iterative schemes for finding the zeros of bounded, demicontinuous and φ-strongly accretive mappings in uniformly smooth Banach spaces. Two iterative schemes are proposed. Strong convergence results are established and applications to variational inequalities are given. In this research, the first author was partially supported by the National Science Foundation of China (10771141), Ph.D. Program Foundation of Ministry of Education of China (20070270004), and Science and Technology Commission of Shanghai Municipality (075105118). The third author was partially supported by Grant NSC 96-2628-E-110-014-MY3.     

Forward–Backward–Half Forward Dynamical Systems for Monotone Inclusion Problems with Application to v-GNE

Journal of Optimization Theory and Applications - In this paper, the first-order forward–backward–half forward dynamical systems associated with the inclusion problem consisting of...     

Iterative Approaches to Solving Equilibrium Problems and Fixed Point Problems of Infinitely Many Nonexpansive Mappings

Recently, O’Hara, Pillay and Xu (Nonlinear Anal. 54, 1417–1426, 2003) considered an iterative approach to finding a nearest common fixed point of infinitely many nonexpansive mappings in a Hilbert space. Very recently, Takahashi and Takahashi (J. Math. Anal. Appl. 331, 506–515, 2007) introduced an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping in a Hilbert space. In this paper, motivated by these authors’ iterative schemes, we introduce a new iterative approach to finding a common element of the set of solutions of an equilibrium problem and the set of common fixed points of infinitely many nonexpansive mappings in a Hilbert space. The main result of this work is a strong convergence theorem which improves and extends results from the above mentioned papers.     

Vector Variational Inequalities Involving Set-valued Mappings via Scalarization with Applications to Error Bounds for Gap Functions

In this paper, by using the scalarization approach of Konnov, several kinds of strong and weak scalar variational inequalities (SVI and WVI) are introduced for studying strong and weak vector variational inequalities (SVVI and WVVI) with set-valued mappings, and their gap functions are suggested. The equivalence among SVVI, WVVI, SVI, WVI is then established under suitable conditions and the relations among their gap functions are analyzed. These results are finally applied to the error bounds for gap functions. Some existence theorems of global error bounds for gap functions are obtained under strong monotonicity and several characterizations of global (respectively local) error bounds for the gap functions are derived.     

Weighted Norm Inequalities for Paraproducts and Bilinear Pseudodifferential Operators with Mild Regularity

We establish boundedness properties on products of weighted Lebesgue, Hardy, and amalgam spaces of certain paraproducts and bilinear pseudodifferential operators with mild regularity. We do so by showing that these operators can be realized as generalized bilinear Calderón–Zygmund operators.      

Hybrid Proximal-Point Methods for Common Solutions of Equilibrium Problems and Zeros of Maximal Monotone Operators

The purpose of this paper is to introduce and study two hybrid proximal-point algorithms for finding a common element of the set of solutions of an equilibrium problem and the set of solutions to the equation 0∈Tx for a maximal monotone operator T in a uniformly smooth and uniformly convex Banach space X. Strong and weak convergence results of these two hybrid proximal-point algorithms are established, respectively. The research of L.C. Ceng was partially supported by the National Science Foundation of China (10771141), Ph.D. Program Foundation of Ministry of Education of China (20070270004), Science and Technology Commission of Shanghai Municipality Grant (075105118), Innovation Program of Shanghai Municipal Education Commission (09ZZ133), and Shanghai Leading Academic Discipline Project (S30405). The research of J.C. Yao was partially supported by Grant NSC 97-2115-M-110-001. Research was carried on within the agreement between National Sun Yat-Sen University of Kaohsiung, Taiwan and Pisa University, Pisa, Italy, 2008.     

Renormalization and Dimensional Regularization for a Scalar Field with Gauss–Bonnet-Type Coupling to Curvature

We consider a scalar field with a Gauss–Bonnet-type coupling to the curvature in a curved space–time. For such a quadratic coupling to the curvature, the metric energy–momentum tensor does not contain derivatives of the metric of orders greater than two. We obtain the metric energy–momentum tensor and find the geometric structure of the first three counterterms to the vacuum averages of the energy–momentum tensors for an arbitrary background metric of an N-dimensional space–time. In a homogeneous isotropic space, we obtain the first three counterterms of the n-wave procedure, which allow calculating the renormalized values of the vacuum averages of the energy–momentum tensors in the dimensions N=4,5. Using dimensional regularization, we establish that the geometric structures of the counterterms in the n-wave procedure coincide with those in the effective action method.     

Modified Extragradient Methods for a System of Variational Inequalities in Banach Spaces

In this paper, we introduce a new system of general variational inequalities in Banach spaces. We establish the equivalence between this system of variational inequalities and fixed point problems involving the nonexpansive mapping. This alternative equivalent formulation is used to suggest and analyze a modified extragradient method for solving the system of general variational inequalities. Using the demi-closedness principle for nonexpansive mappings, we prove the strong convergence of the proposed iterative method under some suitable conditions.     

Noncoercive Variational–Hemivariational Inequalities: Existence,Approximation by Double Regularization,and Application to Nonmonotone Contact Problems

Journal of Optimization Theory and Applications - We study noncoercive nonlinear variational–hemivariational inequalities that encompass semicoercive nonlinear monotone variational...     

Algorithms for the design of network topologies with balanced disjoint rings

We define the Balanced Disjoint Rings (BDR) problem as developing a method to partition of the nodes in a network to form a given number of disjoint rings of minimum total link length in such a way that there is almost the same number of nodes in each ring. The BDR problem has potential applications in the design of survivable network structures in telecommunications as well as in the identification of local distribution/collection routes in logistics. The BDR problem can also be considered a generalization of the traveling salesman problem since we are interested in multiple tours instead of a single tour. We develop an efficient heuristic solution methodology that involves various GRASP-based randomized solution construction routines that allow a multi-start framework and a n effective combination of cyclic-exchange and single-move neighborhoods in a local search improvement procedure. The algorithms perform very well in our numerical studies, providing encouraging optimality and lower bound gaps with very reasonable runtimes.     

Pseudotransient Continuation for Solving Systems of Nonsmooth Equations with Inequality Constraints

This paper investigates a pseudotransient continuation algorithm for solving a system of nonsmooth equations with inequality constraints. We first transform the inequality constrained system of nonlinear equations to an augmented nonsmooth system, and then employ the pseudotransient continuation algorithm for solving the corresponding augmented nonsmooth system. The method gets its global convergence properties from the dynamics, and inherits its local convergence properties from the semismooth Newton method. Finally, we illustrate the behavior of our approach by some numerical experiments.     

Well-posedness for a Class of Variational–Hemivariational Inequalities with Perturbations

In this paper, we consider an extension of well-posedness for a minimization problem to a class of variational–hemivariational inequalities with perturbations. We establish some metric characterizations for the well-posed variational–hemivariational inequality and give some conditions under which the variational–hemivariational inequality is strongly well-posed in the generalized sense. Under some mild conditions, we also prove the equivalence between the well-posedness of variational–hemivariational inequality and the well-posedness of corresponding inclusion problem.     

Comparison Results for a Kind of Impulsive Parabolic Equations with Application to Population Dynamics

In this paper,for a semi-linear parabolic partial differential equations with impulsive effects,theexistence-comparison theorem and comparison principles are established using the method of upper and lowersolutions.These results are applied to obtain the stability results of the steady-state solutions in a reaction-diffusion equations modelling two competing species with instantaneous stocking.