Duality for nonconvex absolute value programming and a characterization of linear max-min programs

In this paper a dual problem for nonconvex linear programs with absolute value functionals is constructed by means of a max-min problem involving bivalent variables. A relationship between the classical linear max-min problem and a linear program with absolute value functionals is developed. This program is then used to compute the duality gap between some max-min and min-max linear problems.     

拟圆的最大-最小不等式性质

利用区域的最大-最小不等式性质的拟共形不变性和圆的最大-最小不等式性质,得到了拟圆的最大-最小不等式性质.     

An interval version of separation by semispaces in max-min convexity

In this paper, we study separation of a closed box from a max-min convex set by max-min semispaces. This can be regarded as an interval extension of the known separation results. We give a constructive proof of the separation in the case when the box satisfies a certain condition, and we show that the separation is never possible when the condition is not satisfied. We also study the separation of two max-min convex sets by a box and by a box and a semispace.     

On the solution of some max-min problems

A random search-type method for finding max-min of multivariable functions is suggested. The solution of such problems is reduced to the construction, on the basis of random tests, of certain regression curves and to the determination of their roots. One-stage and multistage max-min problems are considered. Various modifications of the method are given.     

模糊正则文法的逼近性

为了扩大现有研究的应用范围,基于max-min型模糊正则文法引入了max-*型模糊正则文法(其中*为特定t-模)的概念,讨论了max-*型模糊正则文法可以通过max-min型模糊正则文法在任意给定的逼近精度来逼近.研究表明,当t-模*满足一定的条件时,max-*型模糊正则文法与max-min型模糊正则文法在生成模糊语言能力方面是等价的.     

A queueing analysis of max-min fairness, proportional fairness and balanced fairness

We compare the performance of three usual allocations, namely max-min fairness, proportional fairness and balanced fairness, in a communication network whose resources are shared by a random number of data flows. The model consists of a network of processor-sharing queues. The vector of service rates, which is constrained by some compact, convex capacity set representing the network resources, is a function of the number of customers in each queue. This function determines the way network resources are allocated. We show that this model is representative of a rich class of wired and wireless networks. We give in this general framework the stability condition of max-min fairness, proportional fairness and balanced fairness and compare their performance on a number of toy networks.     

非线性极大极小系统全局优化算法的分析

非线性极大极小系统的全局优化可用于柔性制造和智能交通的决策与控制.实现了非线性极大极小系统的全局优化算法的仿真,并进行了计算时间分析.数值实验表明了全局优化算法的可行性.算法的计算时间主要由系统的优化极大射影矩阵数目决定,而优化极大射影矩阵数目与系统解析式中单极大式的系数紧密相关,系数取值越分散,简约极大射影矩阵的效果越好,计算效率越高.     

On topological types of ordered median functions

An ordered median functions is a continuous piecewise-linear function. It is well known that in finite dimensional spaces every continuous piecewise-linear function admits a max-min representation in terms of its linear functions. An explicit representation of an ordered median function in max-min form is given by the authors and will appear in a forthcoming issue of this journal. Based on this representation, we give a topological classification of ordered median functions through their simplicial complex of ascent (resp. descent) cones.     

On the max-min pursuit game

The max-min linear pursuit game with quadratic performance measure is examined by means of an approximation by a finite sequence of ‘open-loop’ max-min control stages. It is shown that under mild conditions the finite approximation provides a satisfactory representation of the actual game. The limiting behaviour (as the number of stages approaches infinity) is also discussed.     

Shifting algorithms for tree partitioning with general weighting functions

Recently two shifting algorithms were designed for two optimum tree partitioning problems: The problem of max-min q-partition [4] and the problem of min-max q-partition [1]. In this work we investigate the applicability of these two algorithms to max-min and min-max partitioning of a tree for various different weighting functions. We define the families of basic and invariant weighting functions. It is shown that the first shifting algorithm yields a max-min q-partition for every basic weighting function. The second shifting algorithm yields a min-max q-partition for every invariant weighting function. In addition a modification of the second algorithm yields a min-max q-partition for the noninvariant diameter weighting function.     

On the bottleneck assignment problem

In this paper, a new approach for solving the bottleneck assignment problem is presented. The problem is treated as a special class of permutation problems which we call max-min permutation problems. By defining a suitable neighborhood system in the space of permutations and designating certain permutations as critical solutions, it is shown that any critical solution yields a global optimum. This theorem is then used as a basis to develop a general method to solve max-min permutation problems.This work was carried out by the junior author while holding a Purdue University Fellowship.     

Some Connections between Dirac-Fock and Electron-Positron Hartree-Fock

We study the ground state solutions of the Dirac-Fock model in the case of weak electronic repulsion, using bifurcation theory. They are solutions of a minmax problem. Then we investigate a max-min problem coming from the electronpositron field theory of Bach-Barbaroux-Helffer-Siedentop. We show that given a radially symmetric nuclear charge, the ground state of Dirac-Fock solves this maxmin problem for certain numbers of electrons. But we also exhibit a situation in which the max-min level does not correspond to a solution of the Dirac-Fock equations together with its associated self-consistent projector.submitted 19/03/04, accepted 30/07/04     

Controllability of Some Nonlinear Systems in Hilbert Spaces

In this paper, several abstract results concerning the controllability of semilinear evolution systems are obtained. First, approximate controllability conditions for semilinear systems are obtained by means of a fixed-point theorem of the Rothe type; in this case, the compactness of the linear operator is assumed. Next, the exact controllability of semilinear systems with nonlinearities having small Lipschitz constants is derived by means of the Banach fixed-point theorem; in this case, the compactness of the operators is not assumed. In both cases, it is proven that the controllability of the linear system implies the controllability of the associated semilinear system. Finally, these abstract results are applied to the controllability of the semilinear wave and heat equations.     

Relative controllability of stochastic nonlinear systems with delay in control

The existence of solutions to systems is a natural premise to carry our study about controllability. Under the basic and readily verified conditions to guarantee the existence of the solutions to a system, in this paper, we prove the relative controllability (approximate controllability ) of the stochastic differential systems with delay in control. Sufficient conditions are given firstly for the relative controllability and relative approximate controllability in finite dimensional spaces, and these results are then generalized to infinite-dimensional Hilbert spaces. Finally, examples are given to illustrate the effectiveness of the proposed methods.     

On the controllability of infinite-dimensional linear systems

Two types of controllability, namely weak controllability and strong controllability, are defined for general linear systems. Both types of controllability are useful in control theory. If the system is finite-dimensional, the two types of controllability are equivalent to each other and to the standard concept of complete state-controllability.     

A controllability minimum principle

A controllability minimum principle and two associated transversality conditions are presented, dealing with the controllability of nonlinear systems. The theorems represent necessary conditions for a control function to generate a system path which lies in the boundary of the set of points that are controllable to a target. The theorems presented here are controllability counterparts to Pontryagin's maximum principle, and undoubtedly these results will seem familiar or may have occurred to other researchers in the area of optimal control. The purpose of this paper is to make the distinction explicit and to establish the validity of these controllability theorems on their own merits. The theorems are demonstrated using a simple example and the principal result (a controllability minimum principle) is shown to be equivalent to the Kalman controllability criterion for linear systems.     

Novel Controllability Conditions for a Class of Singularly-Perturbed Systems with Small State Delays

A singularly-perturbed linear time-dependent controlled system with small pointwise and distributed delays in the state variable is considered. Two simpler parameter-free systems, the slow and fast ones, can be associated with the original system. It was established in the literature that the Euclidean space controllability of the original system, valid for all sufficiently small values of the parameter of singular perturbations, follows from the controllability properties of the slow and fast systems. It also was established that such a connection between the controllability properties of the original system and the slow and fast systems is correct, in general, only in one direction. Namely, the controllability of the slow and fast systems provides the controllability of the original system, while the controllability of the original system not always yields the controllability of both the slow and fast systems. In this paper, we consider the original system such that the respective fast system is uncontrollable, meaning that the previously established controllability conditions are not applicable to this original system. In this case, novel parameter-free sufficient conditions for the Euclidean space controllability of the original system, robust with respect to the small parameter of singular perturbations, are derived. Illustrative examples are presented.     

多元函数的最值定理及其应用

给出多元函数最值存在的两个定理,然后通过一些实例说明它们在有界开集或无界闭集上的极值问题中的运用.     

Boundary Controllability for the Quasilinear Wave Equation

We study the boundary exact controllability for the quasilinear wave equation in high dimensions. Our main tool is the geometric analysis. We derive the existence of long time solutions near an equilibrium, prove the locally exact controllability around the equilibrium under some checkable geometrical conditions. We then establish the globally exact controllability in such a way that the state of the quasilinear wave equation moves from an equilibrium in one location to an equilibrium in another location under some geometrical conditions. The Dirichlet action and the Neumann action are studied, respectively. Our results show that exact controllability is geometrical characters of a Riemannian metric, given by the coefficients and equilibria of the quasilinear wave equation. A criterion of exact controllability is given, which based on the sectional curvature of the Riemann metric. Some examples are presented to verify the global exact controllability.     

边值系统的能控性与能观测性问题

把抽象系统的能控性和能观测性推广到由强连续双半群描述的抽象边值系统,给出了相应的边值系统能控性的充要条件,并研究了能观测性与能控性之间的对偶关系.最后作为例子,研究了双曲系统能控性.文中所得的结果可用于讨论现代物理系统中出现一类边值系统的能控性与能观测性问题.