Eigenvalue problems for anisotropic discrete boundary value problems

In this paper, we prove the existence of a continuous spectrum for a family of discrete boundary value problems. The main existence results are obtained by using critical point theory. The equations studied in the paper represent a discrete variant of some recent anisotropic variable exponent problems, which deserve as models in different fields of mathematical physics.     

集值映射空间上的Tightness和Fan Tightness

本文讨论了连续集值映射空间在赋予点态收敛拓扑和紧开拓扑下的tightness和fan tightness, 将关于连续单值映射空间的某些结果推广到连续集值映射空间.     

On inertial proximal algorithm for split variational inclusion problems

ABSTRACT

In this paper, a hybrid proximal algorithm with inertial effect is introduced to solve a split variational inclusion problem in real Hilbert spaces. Under mild conditions on the parameters, we establish weak convergence results for the proposed algorithm. Unlike the earlier iterative methods, we do not impose any conditions on the sequence generated by the proposed algorithm. Also, we extend our results to find a common solution of a split variational inclusion problem and a fixed-point problem. Finally, some numerical examples are given to discuss the convergence and superiority of the proposed iterative methods.     

求解弱线性双层规划问题的一种全局优化方法

双层规划在经济、交通、生态、工程等领域有着广泛而重要的应用.目前对双层规划的研究主要是基于强双层规划和弱双层规划.然而,针对弱双层规划的求解方法却鲜有研究.研究求解弱线性双层规划问题的一种全局优化方法,首先给出弱线性双层规划问题与其松弛问题在最优解上的关系,然后利用线性规划的对偶理论和罚函数方法,讨论该松弛问题和它的罚问题之间的关系.进一步设计了一种求解弱线性双层规划问题的全局优化方法,该方法的优势在于它仅仅需要求解若干个线性规划问题就可以获得原问题的全局最优解.最后,用一个简单算例说明了所提出的方法是可行的.     

A trust region algorithm for solving bilevel programming problems

In this paper, we present a new trust region algorithm for a nonlinear bilevel programming problem by solving a series of its linear or quadratic approximation subproblems. For the nonlinear bilevel programming problem in which the lower level programming problem is a strongly convex programming problem with linear constraints, we show that each accumulation point of the iterative sequence produced by this algorithm is a stationary point of the bilevel programming problem.     

Penalty function theory for general convex programming problems

A class of penalty functions for solving convex programming problems with general constraint sets is considered. Convergence theorems for penalty methods are established by utilizing the concept of infimal convergence of a sequence of functions. It is shown that most existing penalty functions are included in our class of penalty functions.     

Epi-Convergence and Lower and Epi-Upper Semicontinuous Approximations in Distribution

The paper introduces an extension of the epi-convergence, the lower semicontinuous approximation and the epi-upper semicontinuous approximation of random real functions in distribution. The new notions could be helpful tools for sensitivity analyzes of stochastic optimization problems. The research is evoked by S. Vogel and continues the research started by Vogel and the author.     

An improved sequential quadratic programming algorithm for solving general nonlinear programming problems

In this paper, a class of general nonlinear programming problems with inequality and equality constraints is discussed. Firstly, the original problem is transformed into an associated simpler equivalent problem with only inequality constraints. Then, inspired by the ideals of the sequential quadratic programming (SQP) method and the method of system of linear equations (SLE), a new type of SQP algorithm for solving the original problem is proposed. At each iteration, the search direction is generated by the combination of two directions, which are obtained by solving an always feasible quadratic programming (QP) subproblem and a SLE, respectively. Moreover, in order to overcome the Maratos effect, the higher-order correction direction is obtained by solving another SLE. The two SLEs have the same coefficient matrices, and we only need to solve the one of them after a finite number of iterations. By a new line search technique, the proposed algorithm possesses global and superlinear convergence under some suitable assumptions without the strict complementarity. Finally, some comparative numerical results are reported to show that the proposed algorithm is effective and promising.     

A uniqueness theorem for two-point boundary value problems

We prove a uniqueness theorem for solutions of two-point boundary value problems, which says that when the nonlinearity is sublinear and Lipschitzian, and one of the boundary values is fixed, then for sufficiently large values of the other boundary value there is a unique solution of the problem. The proof is based on the Banach's fixed-point theorem, and the argument used to show that the relevant operator is a contraction makes use of a generalization of the Riemann-Lebesgue lemma     

集值映射空间上的基数函数

讨论了集值映射空间在赋予点态收敛拓扑或紧开拓扑下的权数,特征,网络权,稠密度等基数函数,利用自然映射,诱导映射和嵌入等方法将单值连续映射空间的有关结论推广到集值映射空间类上.     

Spectral analysis of fourth order differential operators III

This paper extends the results of the two previous papers in several directions. For one we allow slower decay of the coefficients, but higher order differentiability. For this an expansion for the diagonalizing transformations is derived. Secondly unbounded coefficients are permitted. This requires further transformations in order to achieve Levinson's form, but also a modification with the usual M‐matrix approach. While the standard results carry over to weakly singular coefficients, very singular coefficients will generally lead to discrete spectra (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Approximations of linear Dirichlet problems with singularities

A priori estimates for elliptic linear Dirichlet problems are revisited and precisely determined. Such estimates are used for the analysis of Dirichlet problems with singular coefficients and data. Elliptic Dirichlet problems with singularities are reformulated in appropriate generalized function algebras and uniquely solved.     

Parabolic Differential Equations with Unbounded Coefficients – A Generalization of the Parametrix Method

We consider uniformly parabolic differential equations with unbounded first- and zero-order coefficients. A fundamental solution is constructed based on the classical parametrix method of E. Levi. From this the existence and uniqueness of the corresponding Cauchy problem is derived. Our approach does not require differentiable coefficients, as is usually assumed in the unbounded case. It only requires Hölder continuous coefficients. In this respect, our new proof also extends known results. We briefly discuss applications which make essential use of this extension.     

Fractional Brownian Heavy Traffic Approximations of Multiclass Feedforward Queueing Networks

We consider multiclass feedforward queueing networks under first in first out and priority service disciplines driven by long-range dependent arrival and service time processes. We show that in critical loading the normalized workload, queue length and sojourn time processes can converge to a multi-dimensional reflected fractional Brownian motion. This weak heavy traffic approximation is deduced from a deterministic pathwise approximation of the network behavior close to constant critical load in terms of the solution of a Skorokhod problem. Since we model the doubly infinite time interval, our results directly cover the stationary case.AMS subject classification: primary 90B15, secondary 60K25, 68M20     

Exact algorithm for solving a special fixed-charge linear programming problem

A branch-and-bound algorithm (A) for solving a fixed-charge linear programming problem (P) involving identical fixed charges, one equality constraint, and explicit bounds on the variables is presented. Problem (P) can serve as a mathematical model for profit optimization in sawn timber production. Some theoretical considerations upon a fixed-charge problem (P), arising from (P) by permitting the fixed charges to be different for each variable, are carried out. A basic algorithm (A₀) is stated, and it is proved that Algorithm (A₀) finds an optimal solution of Problem (P) [resp., (P)] within a finite number of steps. Algorithm (A₀), combined with bounds developed with regard to Problem (P), yields Algorithm (A), which operates on a subset of all vertices of the feasible region. Finally, computational results concerning the numerical solution of Problem (P) by Algorithm (A) are stated.A part of this work was carried out in connection with the project Optimierung der Schnittholzproducktion auf Zerspaneranlagen, which was done at the Institute of Mathematics of the University of Klagenfurt in cooperation with the firm J. Offner, Holzindustrie GmbH, Wolfsberg. This project was partially supported by Forschungsförderungsfonds für die gewerbliche Wirtschaft. The author would like to thank Professor H. Stettner, C. Nowak, and H. Woschitz for their support and G. Stoiser for his help in achieving the numerical results.