On the spectral radius of ‡-shape trees

Let A(G) be the adjacency matrix of G. The characteristic polynomial of the adjacency matrix A is called the characteristic polynomial of the graph G and is denoted by φ(G, λ) or simply φ(G). The spectrum of G consists of the roots (together with their multiplicities) λ ₁(G) ? λ ₂(G) ? … ? λ _n (G) of the equation φ(G, λ) = 0. The largest root λ ₁(G) is referred to as the spectral radius of G. A ?-shape is a tree with exactly two of its vertices having maximal degree 4. We will denote by G(l ₁, l ₂, … l ₇) (l ₁ ? 0, l _i ? 1, i = 2, 3, …, 7) a ?-shape tree such that $G\left( {l_1 ,l_2 , \ldots l_7 } \right) - u - v = P_{l_1 } \cup P_{l_2 } \cup \ldots P_{l_7 }$ , where u and v are the vertices of degree 4. In this paper we prove that ${{3\sqrt 2 } \mathord{\left/ {\vphantom {{3\sqrt 2 } 2}} \right. \kern-0em} 2} < \lambda _1 \left( {G\left( {l_1 ,l_2 , \ldots l_7 } \right)} \right) < {5 \mathord{\left/ {\vphantom {5 2}} \right. \kern-0em} 2}$ .     

Successive convex approximations to cardinality-constrained convex programs: a piecewise-linear DC approach

In this paper we consider cardinality-constrained convex programs that minimize a convex function subject to a cardinality constraint and other linear constraints. This class of problems has found many applications, including portfolio selection, subset selection and compressed sensing. We propose a successive convex approximation method for this class of problems in which the cardinality function is first approximated by a piecewise linear DC function (difference of two convex functions) and a sequence of convex subproblems is then constructed by successively linearizing the concave terms of the DC function. Under some mild assumptions, we establish that any accumulation point of the sequence generated by the method is a KKT point of the DC approximation problem. We show that the basic algorithm can be refined by adding strengthening cuts in the subproblems. Finally, we report some preliminary computational results on cardinality-constrained portfolio selection problems.     

Connectivity and minimal distance spectral radius of graphs

In this article, we study how the distance spectral radius behaves when the graph is perturbed by grafting edges. As applications, we also determine the graph with k cut vertices (respectively, k cut edges) with the minimal distance spectral radius.     

一类带等式约束非光滑最优化问题的逐次二次规划方法

本文对一类带等式的非光滑最优化问题给出了一种逐次二次规划方法。这类问题的目标函数是非光滑合成函数，约束函数是非线性光滑函数。该方法通过逐次解二阶规划寻找搜索方向，使用ｌ１－罚函数的非精确线搜索得到新的迭代点。我们证明了算法的全局收敛性并给出了数值试验结果。     

Multiobjective programming problems with (G,C,ρ)-convexity

In this paper, we deal with multiobjective programming problems involving functions which are not necessarily differential. A new concept of generalized convexity, which is called (G,C,??)-convexity, is introduced. We establish not only sufficient but also necessary optimality conditions for multiobjective programming problems from a viewpoint of the new generalized convexity. When the sufficient conditions are utilized, the corresponding duality theorems are derived for general Mond-Weir type dual program.     

对称矩阵与反对称矩阵广义特征值反问题的拓广

定义了上三角等次对角线矩阵和上三角交错次对角线矩阵;讨论了矩阵方程AX-XA=0的对称解与AX XA=0的反对称解.在此基础上考虑了以下问题的可解性:给定A∈Rn×m,D∈Rm×m,分别求X,Y∈SRn×n和X,Y∈ASRn×n,使得XA=YDA.     

Improved estimation of duality gap in binary quadratic programming using a weighted distance measure

We present in this paper an improved estimation of duality gap between binary quadratic program and its Lagrangian dual. More specifically, we obtain this improved estimation using a weighted distance measure between the binary set and certain affine subspace. We show that the optimal weights can be computed by solving a semidefinite programming problem. We further establish a necessary and sufficient condition under which the weighted distance measure gives a strictly tighter estimation of the duality gap than the existing estimations.     

Characterization of Domains of Self-Adjoint Ordinary Differential Operators II

We characterize the self-adjoint domains of general even order linear ordinary differential operators in terms of real-parameter solutions of the differential equation. This for endpoints which are regular or singular and for arbitrary deficiency index. This characterization is obtained from a new decomposition of the maximal domain in terms of limit-circle solutions. These are the solutions which contribute to the self-adjoint domains in analogy with the celebrated Weyl limit-circle solutions in the second order Sturm–Liouville case. As a special case we obtain the previously known characterizations when one or both endpoints are regular.     

凹整数规划的分枝定界解法

凹整数规划是一类重要的非线性整数规划问题，也是在经济和管理中有着广泛应用的最优化问题．本文主要研究用分枝定界方法求解凹整数规划问题，这一方法的基本思想是对目标函数进行线性下逼近，然后用乘子搜索法求解连续松弛问题．数值结果表明，用这种分枝定界方法求解凹整数规划是有效的．     

Bush连续不可微函数的分形性质

本文对用递推关系确定的Bush连续不可微函数,找出了迭代函数系(IFS),从而得到它的级数表达式和所具有的自仿射分形的有关性质.最后还计算出函数图象的Hausdorff 维数的准确值.