凸函数的若干新性质及应用

本文证明了凸函数的若干新性质 ,讨论了这些性质在求解线性与非线性不等式组和线性规划中的应用 ,为线性与非线性不等式组、线性规划的求解提供了一种新方法 .     

Bifractional inequalities and convex cones

In this paper we give a systematic study of a class of linear inequalities related to convex cones in linear spaces. In particular, Chebyshev and Andersson type inequalities are discussed. Some classical and new inequalities are derived from the results.     

Existence of solutions of certain systems of non-linear inequalities

In recent years there has been increasing interest in the problem of the existence of solutions to systems of linear inequalities, however little has been done in the general case of non-linear inequalities. In this paper a certain class of non-linear inequalities is considered. Several existence theorems are established which generalize certain recent results in linear inequalities.     

线性不等式组的简单对偶非线性方法

将线性不等式组问题转化为一个形式简单的对偶空间非线性极值问题,本提出了一类新的求解线性不等式组的方法-简单对偶非线性方法,它在理论上是多项式算法,并可以从任意点启动,可以应用共轭梯度方法有效地求解大规模线性不等式组问题。本给出了不同的算法实现,数值实验结果表明,简单对偶非线性方法是有效的。     

半序线性空间上线性泛函的两个Gr\"{u}ss型不等式$^{}$

证明了半序线性空间上线性泛函的两个Gr\"{u}ss型不等式, 并由此给出了Karamata型积分不等式的一种推广形式, 得到了一个新的Gr\"{u}ss型积分不等式及关于傅立叶系数的两个不等式. 最后利用所得结论研究了关于矩阵及线性算子的一些Gr\"{u}ss型不等式.     

New valid inequalities for the fixed-charge and single-node flow polytopes

The most effective software packages for solving mixed 0–1linear programs use strong valid linear inequalities derived from polyhedral theory. We introduce a new procedure which enables one to take known valid inequalities for the knapsack polytope, and convert them into valid inequalities for the fixed-charge and single-node flow polytopes. The resulting inequalities are very different from the previously known inequalities (such as flow cover and flow pack inequalities), and define facets under certain conditions.     

Nonnegative solvability of linear equations in certain ordered rings

In the integers and in certain densely ordered rings that are not fields, projections of the solution set of finitely many homogeneous weak linear inequalities may be defined by finitely many congruence inequalities, where a congruence inequality combines a weak inequality with a system of congruences. These results extend well-known facts about systems of weak linear inequalities over ordered fields and imply corresponding analogues of Farkas' Lemma on nonnegative solvability of systems of linear equations.

     

Gap inequalities for non-convex mixed-integer quadratic programs

Laurent and Poljak introduced a very general class of valid linear inequalities, called gap inequalities, for the max-cut problem. We show that an analogous class of inequalities can be defined for general non-convex mixed-integer quadratic programs. These inequalities dominate some inequalities arising from a natural semidefinite relaxation.     

Alexander polynomials of alternating knots of genus two III

In the previous paper, the author gave linear inequalities on the coefficients of the Alexander polynomials of alternating knots of genus two, which are best possible as linear inequalities on the coefficients of them. In this paper, we give infinitely many Alexander polynomials which satisfy the linear inequalities, but they are not realized by alternating knots.     

Partial linear characterizations of the asymmetric travelling salesman polytope

We consider the linear programming formulation of the asymmetric travelling salesman problem. Several new inequalities are stated which yield a sharper characterization in terms of linear inequalities of the travelling salesman polytope, i.e., the convex hull of tours. In fact, some of the new inequalities as well as some of the well-known subtour elimination constraints are indeed facets of the travelling salesman polytope, i.e., belong to the class of inequalities that uniquely characterize the convex hull of tours to an-city problem.