三种新变形的全一问题

该文研究三种新变形的全一问题及最小全一问题. 原始的全一问题可被形象的称为顶点点亮顶点问题, 而这三类新问题则分别被称为顶点点亮边问题,边点亮顶点问题,边点亮边问题. 顶点点亮顶点问题已经得到了广泛的研究. 比如,解的存在性问题和求解的有效算法已经被解决,一般图上的最小顶点点亮顶点问题已经被证明是NP- 完备的,树、单圈图和双圈图上的最小顶点点亮顶点问题的线性时间最优算法也已被给出等. 该文对于顶点点亮边问题,证明一个图有解当且仅当它是二部图,因此只可能有两组解和最优解. 对于边点亮顶点问题,证明一个图有解当且仅当它包含偶数个顶点,并通过将其最优问题多项式变换成最小权的完美匹配问题,得出一般图上的最小边点亮顶点问题可在多项式时间内求解. 边点亮边问题可归约成线图上的顶点点亮顶点问题.     

A study of mixed discrete bilevel programs using semidefinite and semi-infinite programming

In this paper, we study the bilevel programming problem with discrete polynomial lower level problem. We start by transforming the problem into a bilevel problem comprising a semidefinite program (SDP for short) in the lower level problem. Then, we are able to deduce some conditions of existence of solutions for the original problem. After that, we again change the bilevel problem with SDP in the lower level problem into a semi-infinite program. With the aid of the exchange technique, for simple bilevel programs, an algorithm for computing a global optimal solution is suggested, the convergence is shown, and a numerical example is given.     

General Tricomi-Rassias problem and oblique derivative problem for generalized Chaplygin equations

Many authors have discussed the Tricomi problem for some second order equations of mixed type, which has important applications in gas dynamics. In particular, Bers proposed the Tricomi problem for Chaplygin equations in multiply connected domains [L. Bers, Mathematical Aspects of Subsonic and Transonic Gas Dynamics, Wiley, New York, 1958]. And Rassias proposed the exterior Tricomi problem for mixed equations in a doubly connected domain and proved the uniqueness of solutions for the problem [J.M. Rassias, Lecture Notes on Mixed Type Partial Differential Equations, World Scientific, Singapore, 1990]. In the present paper, we discuss the general Tricomi-Rassias problem for generalized Chaplygin equations. This is one general oblique derivative problem that includes the exterior Tricomi problem as a special case. We first give the representation of solutions of the general Tricomi-Rassias problem, and then prove the uniqueness and existence of solutions for the problem by a new method. In this paper, we shall also discuss another general oblique derivative problem for generalized Chaplygin equations.     

Recovery of the unknown coefficients in a two-dimensional hyperbolic equation

In this paper, an inverse boundary value problem for a two-dimensional hyperbolic equation with overdetermination conditions is studied. To investigate the solvability of the original problem, we first consider an auxiliary inverse boundary value problem and prove its equivalence to the original problem in a certain sense. We then use the Fourier method to reduce such an equivalent problem to a system of integral equations. Furthermore, we prove the existence and uniqueness theorem for the auxiliary problem by the contraction mappings principle. Based on the equivalency of these problems, the existence and uniqueness theorem for the classical solution of the original inverse problem is proved. Some discussions on the numerical solutions for this inverse problem are presented including some numerical examples.     

Algebra complexity problems involving graph homomorphism,semigroups and the constraint satisfaction problem

In this paper, we connect the constraint satisfaction problem with other complexity problems, like the polynomial equivalence problem for combinatorial 0-simple semigroups, the graph retraction problem and the geometry problem. We show that every constraint satisfaction problem is polynomially equivalent to an easily formulated algebra complexity problem. As an application we prove that the polynomial equivalence problem (word problem) for the 2×2 matrices over the two element field is co-NP-complete.     

精确罚函数和极小极大问题

在这篇文章中我们研究了对于不等式约束的非线性规划问题如何根据极小极大问题的鞍点来找精确罚问题的解。对于一个具有不等式约束的非线性规划问题,通过罚函数,我们构造出一个极小极大问题,应用交换“极小”或“极大”次序的策略,证明了罚问题的鞍点定理。研究结果显示极小极大问题的鞍点是精确罚问题的解。     

Classification Problem for Graphs and Lattices is Wild

The classification problem for finite and infinite algebraic lattices has also been extensively addressed. A wild classification problem contains the classification problem of pairs of matrices up to simultaneous similarity. In this paper, we prove that the classification problem for graphs is wild by reducing the classification problem for finite 2-nilpotent p-groups to the classification problem for graphs.     

Split Null Point Problems and Fixed Point Problems for Demicontractive Multivalued Mappings

In this paper, we consider the split null point problem and the fixed point problem for multivalued mappings in Hilbert spaces. We introduce a Halpern-type algorithm for solving the problem for maximal monotone operators and demicontractive multivalued mappings, and establish a strong convergence result under some suitable conditions. Also, we apply our problem of main result to other split problems, that is, the split feasibility problem, the split equilibrium problem, and the split minimization problem. Finally, a numerical result for supporting our main result is also supplied.     

Simultaneous Subgradient Algorithms for the Generalized Split Variational Inclusion Problem in Hilbert Spaces

In this article, we study the generalized split variational inclusion problem. For this purpose, motivated by the projected Landweber algorithm for the split equality problem, we first present a simultaneous subgradient extragradient algorithm and give related convergence theorems for the proposed algorithm. Next, motivated by the alternating CQ-algorithm for the split equality problem, we propose another simultaneous subgradient extragradient algorithm to study the general split variational inclusion problem. As applications, we consider the split equality problem, split feasibility problem, split variational inclusion problem, and variational inclusion problem in Hilbert spaces.     

Global Method for Monotone Variational Inequality Problems with Inequality Constraints

We consider optimization methods for monotone variational inequality problems with nonlinear inequality constraints. First, we study the mixed complementarity problem based on the original problem. Then, a merit function for the mixed complementarity problem is proposed, and some desirable properties of the merit function are obtained. Through the merit function, the original variational inequality problem is reformulated as simple bounded minimization. Under certain assumptions, we show that any stationary point of the optimization problem is a solution of the problem considered. Finally, we propose a descent method for the variational inequality problem and prove its global convergence.