Studying the stability of solutions to systems of linear inequalities and constructing separating hyperplanes

We consider the methods for matrix correction or correction of all parameters of systems of linear equations and inequalities. We show that the problem of matrix correction of an inconsistent system of linear inequalities with the nonnegativity condition is reduced to a linear programming problem. Some stability measure is defined for a given solution to a system of linear inequalities as the minimal possible variation of parameters under which this solution does not satisfy the system. The problem of finding the most stable solution to the system is considered. The results are applied to constructing an optimal separating hyperplane in the feature space that is the most stable to the changes of features of the objects.     

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 .     

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.     

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.     

An two phase abs method for solving over determined systems of linear inequalities

The method, called the Multi-Stage ABS algorithm, for solving the over-determined linear inequalities system and the system combined with the over-determined linear inequalities and the equations is presented. This method is characterized by translating inequalities system to an equations system with slack variables. The explicit solution with the slack variables of the equations system are given by the implicit LU algorithm, then the slack variables can be given by the ABS algorithm. Finally, the upper multiplications of the algorithm are given.     

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.     

From solvability and approximation of variational inequalities to solution of nondifferentiable optimization problems in contact mechanics

In this paper, we first gather existence results for linear and for pseudo-monotone variational inequalities in reflexive Banach spaces. We discuss the necessity of the involved coerciveness conditions and their relationship. Then, we combine Mosco convergence of convex closed sets with an approximation of pseudo-monotone bifunctions and provide a convergent approximation procedure for pseudo-monotone variational inequalities in reflexive Banach spaces. Since hemivariational inequalities in linear elasticity are pseudo-monotone, our approximation method applies to nonmonotone contact problems. We sketch how regularization of the involved nonsmooth functionals together with finite element approximation lead to an efficient numerical solution method for these nonconvex nondifferentiable optimization problems. To illustrate our theory, we give a numerical example of a 2D linear elastic block under a given nonmonotone contact law.     

基于可信性测度的模糊线性不等式组的确定性等价

在模糊优化决策问题中,约束条件包含模糊线性不等式组.如何把它们转化为确定性的等价形式,这在模糊控制和模糊决策中起着重要的作用.可信性测度是模糊数学实现公理化体系的一个基础.本文在可信性测度的基础上,探讨了模糊约束在一定的满意度水平之下转化为确定性的等价式,同时就三角形模糊变量给出了模糊线性不等式组的确定性等价的具体表达式.     

On the Determination of a Density Function by Its Autoconvolution Coefficient

We investigate eigenvalues and eigenvectors of certain linear variational eigenvalue inequalities where the constraints are defined by a convex cone as in [4], [7], [8], [10]-[12], [17]. The eigenvalues of those eigenvalue problems are of interest in connection with bifurcation from the trivial solution of nonlinear variational inequalities. A rather far reaching theory is presented for the case that the cone is given by a finite number of linear inequalities, where an eigensolution corresponds to a (+)-Kuhn-Tucker point of the Rayleigh quotient. Application to an unlaterally supported beam are discussed and numerical results are given.     

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.     

A Note on Redundancy and Linear Parametric Programming

A simple redundancy criterion for a system of linear inequalities is given. Considering a linear programming problem with linear inequality constraints the sensitivity analysis of the optimal solution of the right-hand side in combination with the redundancy criterion can help to identify redundant inequalities (constraints). The redundancy criterion is applied to a linear parametric problem which is enlarged by an auxiliary condition such as "stop the parametric procedure when a chosen constraint becomes redundant".     

On some dynamic inequalities of Steffensen type on time scales

In the present article, we investigate some new inequalities of Steffensen type on an arbitrary time scale using the diamond‐α dynamic integrals, which are defined as a linear combination of the delta and nabla integrals. The obtained inequalities extend some known dynamic inequalities on time scales and unify and extend some continuous inequalities and their discrete analogues.     

On the Gap Functions of Prevariational Inequalities

Gap functions play a crucial role in transforming a variational inequality problem into an optimization problem. Then, methods solving an optimization problem can be exploited for finding a solution of a variational inequality problem. It is known that the so-called prevariational inequality is closely related to some generalized convex functions, such as linear fractional functions. In this paper, gap functions for several kinds of prevariational inequalities are investigated. More specifically, prevariational inequalities, extended prevariational inequalities, and extended weak vector prevariational inequalities are considered. Furthermore, a class of gap functions for inequality constrained prevariational inequalities is investigated via a nonlinear Lagrangian.     

Nonconvex, lower semicontinuous piecewise linear optimization

A branch-and-cut algorithm for solving linear problems with continuous separable piecewise linear cost functions was developed in 2005 by Keha et al. This algorithm is based on valid inequalities for an SOS2 based formulation of the problem. In this paper we study the extension of the algorithm to the case where the cost function is only lower semicontinuous. We extend the SOS2 based formulation to the lower semicontinuous case and show how the inequalities introduced by Keha et al. can also be used for this new formulation. We also introduce a simple generalization of one of the inequalities introduced by Keha et al. Furthermore, we study the discontinuities caused by fixed charge jumps and introduce two new valid inequalities by extending classical results for fixed charge linear problems. Finally, we report computational results showing how the addition of the developed inequalities can significantly improve the performance of CPLEX when solving these kinds of problems.     

Convergence of the cyclical relaxation method for linear inequalities

The relaxation method for linear inequalities is studied and new bounds on convergence obtained. An asymptotically tight estimate is given for the case when the inequalities are processed in a cyclical order. An improvement of the estimate by an order of magnitude takes place if strong underrelaxation is used. Bounds on convergence usually involve the so-called condition number of a system of linear inequalities, which we estimate in terms of their coefficient matrix. Paper presented at the XI. International Symposium on Mathematical Programming, Bonn, August 23–27, 1982.     

Uniform saturation in linear inequality systems

Redundant constraints in linear inequality systems can be characterized as those inequalities that can be removed from an arbitrary linear optimization problem posed on its solution set without modifying its value and its optimal set. A constraint is saturated in a given linear optimization problem when it is binding at the optimal set. Saturation is a property related with the preservation of the value and the optimal set under the elimination of the given constraint, phenomena which can be seen as weaker forms of excess information in linear optimization problems. We say that an inequality of a given linear inequality system is uniformly saturated when it is saturated for any solvable linear optimization problem posed on its solution set. This paper characterizes the uniform saturated inequalities and other related classes of inequalities. This work was supported by the MCYT of Spain and FEDER of UE, Grant BFM2002-04114-C02-01.     

Lyapunov inequality for linear Hamiltonian systems on time scales

The principal aim of this paper is to state and prove some Lyapunov inequalities for linear Hamiltonian system on an arbitrary time scale , so that the well-known case of differential linear Hamiltonian systems and the recently developed case of discrete Hamiltonian systems are unified. Applying these inequalities, a disconjugacy criterion for Hamiltonian systems on time scales is obtained.     

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.     

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.     

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。