Well-posedness in the generalized sense for variational inclusion and disclusion problems and well-posedness for optimization problems with constraint

In this paper, we study the well-posedness in the generalized sense for variational inclusion problems and variational disclusion problems, the well-posedness for optimization problems with variational inclusion problems, variational disclusion problems and scalar equilibrium problems as constraint.     

Well-posedness for parametric optimization problems with variational inclusion constraint

In this paper, we study the well-posedness for the parametric optimization problems with variational inclusion problems as constraint (or the perturbed problem of optimization problems with constraint). Furthermore, we consider the relation between the well-posedness for the parametric optimization problems with variational inclusion problems as constraint and the well-posedness in the generalized sense for variational inclusion problems.     

Quasi-Newton Methods for Nonlinear Least Squares Focusing on Curvatures

A new quasi-Newton method for nonlinear least squares problems is proposed. Two advantages of the method are accomplished by utilizing special geometrical properties in the problem class. First, fast convergence is established for well-conditioned problems by interpolating both the current and the previous step in each iteration. Second, high accuracy is achieved for certain difficult problems, such as ill-conditioned problems and problems with large curvatures in the tangent space. Numerical results for artificial problems and standard test problems are presented and discussed.     

On System of Generalized Vector Quasi-equilibrium Problems with Set-valued Maps

In this paper, we introduce four new types of the system of generalized vector quasi-equilibrium problems with set-valued maps which include system of vector quasi-equilibrium problems, system of vector equilibrium problems, system of variational inequality problems, and vector equilibrium problems in the literature as special cases. We prove the existence of solutions for such kinds of system of generalized vector quasi-equilibrium problems. Consequently, we derive some existence results of a solution for the system of vector quasi-equilibrium problems and the generalized Debreu type equilibrium problem for vector-valued functions.     

New fixed point theorems for mixed monotone operators and local existence-uniqueness of positive solutions for nonlinear boundary value problems

In this article we present a new fixed point theorem for a class of general mixed monotone operators, which extends the existing corresponding results. Moreover, we establish some pleasant properties of nonlinear eigenvalue problems for mixed monotone operators. Based on them the local existence-uniqueness of positive solutions for nonlinear boundary value problems which include Neumann boundary value problems, three-point boundary value problems and elliptic boundary value problems for Lane-Emden-Fowler equations is proved. The theorems for nonlinear boundary value problems obtained here are very general.     

Omni-optimizer: A generic evolutionary algorithm for single and multi-objective optimization

Due to the vagaries of optimization problems encountered in practice, users resort to different algorithms for solving different optimization problems. In this paper, we suggest and evaluate an optimization procedure which specializes in solving a wide variety of optimization problems. The proposed algorithm is designed as a generic multi-objective, multi-optima optimizer. Care has been taken while designing the algorithm such that it automatically degenerates to efficient algorithms for solving other simpler optimization problems, such as single-objective uni-optimal problems, single-objective multi-optima problems and multi-objective uni-optimal problems. The efficacy of the proposed algorithm in solving various problems is demonstrated on a number of test problems chosen from the literature. Because of its efficiency in handling different types of problems with equal ease, this algorithm should find increasing use in real-world optimization problems.     

变分不等式与互补问题、双层规划与平衡约束数学规划问题的若干进展

考虑有限维变分不等式与互补问题、双层规划以及均衡约束的数学规划问题. 在简单介绍这些问题之后,重点介绍近年来这些领域中发展迅速的几个研究方向,包括对称锥互补问题的理论与算法、变分不等式的投影收缩算法、随机变分不等式与随机互补问题的模型与方法、双层规划以及均衡约束数学规划问题的新方法. 最后提出几个进一步研究的方向.     

Well-posedness for generalized quasi-variational inclusion problems and for optimization problems with constraints

In this paper, well-posedness of generalized quasi-variational inclusion problems and of optimization problems with generalized quasi-variational inclusion problems as constraints is introduced and studied. Some metric characterizations of well-posedness for generalized quasi-variational inclusion problems and for optimization problems with generalized quasi-variational inclusion problems as constraints are given. The equivalence between the well-posedness of generalized quasi-variational inclusion problems and the existence of solutions of generalized quasi-variational inclusion problems is given under suitable conditions.     

非线性特征值问题的二次近似方法

本文研究求解非线性特征值问题的数值方法.基于矩阵值函数的二次近似,将非线性特征值问题转化为二次特征值问题,提出了求解非线性特征值问题的逐次二次近似方法,分析了该方法的收敛性.结合求解二次特征值问题的Arnoldi方法和Jacobi-Davidson方法,给出求解非线性特征值问题的一些二次近似方法.数值结果表明本文所给算法是有效的.     

伪单调型向量F-互补问题可行集的最小元问题

本文引入了几类向量F-互补问题并给出了向量F-互补问题与广义向量变分不等式之间的关系.通过定义向量F-互补问题的可行集,研究了伪单调型向量F-互补问题的可行集的最小问题,推广了已有的一些结果.     

非强制混合向量变分不等式解的存在性研究

本文利用例外簇方法研究非强制混合向量变分不等式的弱有效解的存在性:首先证明若混合向量变分不等式问题不存在例外簇,则混合向量变分不等式问题的弱有效解集为非空集合:利用向量值映射的渐近映射给出自反Banach空间中非强制混合向量变分不等式的弱有效解集不存在例外簇的充分条件,从而得到混合向量变分不等式问题的弱有效解的存在性结...     

A SUCCESSIVE LEAST SQUARES METHOD FOR STRUCTURED TOTAL LEAST SQUARES

A new method for Total Least Squares (TLS) problems is presented. It differs from previous approaches and is based on the solution of successive Least Squares problems.The method is quite suitable for Structured TLS (STLS) problems. We study mostly the case of Toeplitz matrices in this paper. The numerical tests illustrate that the method converges to the solution fast for Toeplitz STLS problems. Since the method is designed for general TLS problems, other structured problems can be treated similarly.     

一类Dirichlet边值逆问题

给出解析函数的一类Dirichlet边值逆问题的数学提法.依据解析函数Dirichlet边值问题和广义Dirichlet边值问题的理论,讨论了此边值逆问题的可解性.利用解析函数Dirichlet边值问题的Schwarz公式,给出了该边值逆问题的可解条件和解的表示式.     

Estimates for Solutions to Model Problems Connected with Quasistationary Approximation for the Stefan Problem

Model problems for elliptic equations are considered. The time-derivative of a solution can be contained in the boundary condition or in the conjugation condition. Suich problems appear, for example, in the study of free boundary problems for elliptic equations that can be considered as quasistationary approximations to free boundary problems for parabolic equations. Estimates for solutions to the model problems are obtained. Bibliography: 11 titles.     

A nonasymptotic method for general linear singular perturbation problems

The nonasymptotic method developed in Ref. 1 has been extended for solving general linear singularly perturbed two-point boundary-value problems. Firstly, we discuss problems with a right-hand boundary layer. Secondly, we discuss problems with an interior layer. Finally, we discuss problems with two boundary layers. Numerical experience with the method for some model problems is also reported to confirm the theoretical analysis.     

On exact and approximate boundary controllabilities for the heat equation: A numerical approach

The present article is concerned with the numerical implementation of the Hilbert uniqueness method for solving exact and approximate boundary controllability problems for the heat equation. Using convex duality, we reduce the solution of the boundary control problems to the solution of identification problems for the initial data of an adjoint heat equation. To solve these identification problems, we use a combination of finite difference methods for the time discretization, finite element methods for the space discretization, and of conjugate gradient and operator splitting methods for the iterative solution of the discrete control problems. We apply then the above methodology to the solution of exact and approximate boundary controllability test problems in two space dimensions. The numerical results validate the methods discussed in this article and clearly show the computational advantage of using second-order accurate time discretization methods to approximate the control problems.     

一类混合似变分不等式问题解的存在唯一性

本文中,我们首先给出了一类混合似变分不等式问题．接着,在Banach空间中研究了它的解的存在性和唯一性．最后,讨论了混合似变分不等式问题的扰动问题,并证明了扰动问题的解的存在唯一性定理．     

Variational problems in the theory of optimum processes

This paper presents a review of the optimization problems for control processes described by ordinary differential equations and of the variational methods for solving these problems. The following cases are studied: problems with constraints on the controls or the coordinates, problems described by equations with discontinuous right-hand sides, problems with functionals depending on intermediate coordinates, and problems with given discontinuities in the coordinates. Variational problems of synthesis of optimal systems are also discussed. The method of solution is based on the multiplier rule and the Weierstrass necessary condition for the strong minimum of a functional. In some cases, the Legendre-Clebsch necessary condition for the weak minimum of a functional is used.     

Isomorphic scheduling problems

We consider general properties of isomorphic scheduling problems that constitute a new class of pairs of mutually related scheduling problems. Any such a pair is composed of a scheduling problem with fixed job processing times and its time-dependent counterpart with processing times that are proportional-linear functions of the job starting times. In order to introduce the class formally, first we formulate a generic scheduling problem with fixed job processing times and define isomorphic problems by a one-to-one transformation of instances of the generic problem into instances of time-dependent scheduling problems with proportional-linear job processing times. Next, we prove basic properties of isomorphic scheduling problems and show how to convert polynomial algorithms for scheduling problems with fixed job processing times into polynomial algorithms for proportional-linear counterparts of the original problems. Finally, we show how are related approximation algorithms for isomorphic problems. Applying the results, we establish new worst-case results for time-dependent parallel-machine scheduling problems and prove that many single- and dedicated-machine time-dependent scheduling problems with proportional-linear job processing times are polynomially solvable.     

On general hemiequilibrium problems

In this paper, we introduce and analyze a new class of equilibrium problems known as general hemiequilibrium problems. It is shown that this class includes hemiequilibrium problems, hemivariational inequalities and complementarity problems as special cases. We use the auxiliary principle techniques to suggest some iterative-type methods for solving multivalued hemiequilibrium problems. We also analyze the convergence analysis of these new iterative methods under some mild conditions. As special cases, we obtain several new and known methods for solving variational inequalities and equilibrium problems.