Upper Semicontinuity of the Feasible Set Mapping for Linear Inequality Systems

In this paper we characterize the upper semicontinuity of the feasible set mapping at a consistent linear semi-infinite system (LSIS, in brief). In our context, no standard hypothesis is required in relation to the set indexing the constraints and, consequently, the functional dependence between the linear constraints and their associated indices has no special property. We consider, as parameter space, the set of all LSIS having the same index set, endowed with an extended metric to measure the size of the perturbations. We introduce the concept of reinforced system associated with our nominal system. Then, the upper semicontinuity property of the feasible set mapping at the nominal system is characterized looking at the feasible sets of both systems. The fact that this characterization depends only on the nominal system, not involving systems in a neighbourhood, is remarkable. We also provide a necessary and sufficient condition for the aimed property exclusively in terms of the coefficients of the system.     

Stability of the Feasible Set Mapping of Linear Systems with an Exact Constraint Set

This paper deals with the stability of the feasible set mapping of linear systems of an arbitrary number (possibly infinite) of equations and inequalities such that the variable x ranges on a certain fixed constraint set X?? ⁿ (X could represent the solution set of a given constraint system, e.g., the positive cone of ? ⁿ in the case of sign constraints). More in detail, the paper provides necessary as well as sufficient conditions for the lower and upper semicontinuity (in Berge sense), and the closedness, of the set-valued mapping which associates, with any admissible perturbation of the given (nominal) system its feasible set. The parameter space is formed by all the systems having the same structure (i.e., the same number of variables, equations and inequalities) as the nominal one, and the perturbations are measured by means of the pseudometric of the uniform convergence.     

Lower Semicontinuity of the Feasible Set Mapping of Linear Systems Relative to Their Domains

This paper deals with stability properties of the feasible set of linear inequality systems having a finite number of variables and an arbitrary number of constraints. Several types of perturbations preserving consistency are considered, affecting respectively, all of the data, the left-hand side data, or the right-hand side coefficients.     

Calmness of the Argmin Mapping in Linear Semi-Infinite Optimization

This paper characterizes the calmness property of the argmin mapping in the framework of linear semi-infinite optimization problems under canonical perturbations; i.e., continuous perturbations of the right-hand side of the constraints (inequalities) together with perturbations of the objective function coefficient vector. This characterization is new for semi-infinite problems without requiring uniqueness of minimizers. For ordinary (finitely constrained) linear programs, the calmness of the argmin mapping always holds, since its graph is piecewise polyhedral (as a consequence of a classical result by Robinson). Moreover, the so-called isolated calmness (corresponding to the case of unique optimal solution for the nominal problem) has been previously characterized. As a key tool in this paper, we appeal to a certain supremum function associated with our nominal problem, not involving problems in a neighborhood, which is related to (sub)level sets. The main result establishes that, under Slater constraint qualification, perturbations of the objective function are negligible when characterizing the calmness of the argmin mapping. This result also states that the calmness of the argmin mapping is equivalent to the calmness of the level set mapping.     

On the Stability of the Feasible Set in Linear Optimization

This paper deals with the stability of two families of linear optimization problems, each one formed by the dual problems to the members of the other family. We characterize the problems of these families that are stable in the sense that they remain consistent (inconsistent) under sufficiently small arbitrary perturbations of all the data. This characterization is established in terms of the lower semicontinuity property of the feasible set mapping and the boundedness of the optimal set of the corresponding coupled problem. Other continuity properties of the feasible set mapping are also derived. This stability theory extends some well-known theorems of Williams and Robinson on the stability of ordinary linear programming problems to linear optimization problems with infinitely many variables or constraints.     

Analytical Linear Inequality Systems and Optimization

In many interesting semi-infinite programming problems, all the constraints are linear inequalities whose coefficients are analytical functions of a one-dimensional parameter. This paper shows that significant geometrical information on the feasible set of these problems can be obtained directly from the given coefficient functions. One of these geometrical properties gives rise to a general purification scheme for linear semi-infinite programs equipped with so-called analytical constraint systems. It is also shown that the solution sets of such kind of consistent systems form a transition class between polyhedral convex sets and closed convex sets in the Euclidean space of the unknowns.     

On the Stability of the Boundary of the Feasible Set in Linear Optimization

This paper analizes the relationship between the stability properties of the closed convex sets in finite dimensions and the stability properties of their corresponding boundaries. We consider a given closed convex set represented by a certain linear inequality system whose coefficients can be arbitrarily perturbed, and we measure the size of these perturbations by means of the pseudometric of the uniform convergence. It is shown that the feasible set mapping is Berge lower semicontinuous at if and only if the boundary mapping satisfies the same property. Moreover, if the boundary mapping is semicontinuous in any sense (lower or upper; Berge or Hausdorff) at , then it is also closed at . All the mentioned stability properties are equivalent when the feasible set is a convex body.     

On Metric Regularity and the Boundary of the Feasible Set in Linear Optimization

This paper deals with semi-infinite linear inequality systems in ? ⁿ and studies the stability of the boundary of their feasible sets. We analyze the equivalence between the metric regularity of the inverse of the boundary set mapping, $\mathcal{N}$ , and the stability of the feasible set mapping in the sense of the maintenance of the consistency. In doing this we provide operational formulae for distances from points to some useful sets. We also include relationships between the regularity moduli corresponding to the mappings $\mathcal{N}$ and the inverse, $\mathcal{M}$ , of the feasible set mapping, and prove their equality for finite systems and some special cases in the semi-infinite framework. Moreover, we provide conditions to assure that the metric regularity of $\mathcal{N}$ is equivalent to the lower semi-continuity of the boundary set mapping, which is important because the latter property has many characterizations. Since the boundary of a feasible set may not be convex, we cannot make use of the general theory for mappings with convex graph, as for example, the Robinson–Ursescu theorem.     

Nonsmooth Mapping of Linear Control Systems

This paper deals with linear control systems of a special form. The main goal of the paper is to find the exact analytic solution of the time-optimal control problem for an arbitrary linear control system with constant coefficients using the analytic solution of this problem for the canonical system. For this aim, we construct a certain nonsmooth mapping between the 0-controllability sets of the given systems. In other words, by means of this mapping, we investigate the equivalence of the systems with the same qualitative behavior in a neighborhood of the stationary point.     

On Linear Inequality Systems with Smooth Coefficients

A linear inequality system with infinitely many constraints is polynomial [analytical] if its index set is a compact interval of the real line and all its coefficients are polynomial [analytical] functions of the index on this interval. This paper provides sufficient conditions for a given closed convex set to be the solution set of a certain polynomial or at least analytical system.The authors are indebted to Dr. J. M. Almira for valuable comments and suggestions.     

Equivalent Sets of Consequences of Linear Inequality Systems

Different types of linear inequality systems have different consequence inequalities. Investigating several types of linear inequality systems, the present paper gives explicitly those consequences of the given system of linear inequalities that are all consistent if and only if the original system is consistent. Our results generalize the well-known Kuhn-Fourier theorem, and present important particular cases.     

极大-加混合线性不等式系统的可解性及其应用

研究极大-加混合线性不等式系统的可解性.基于极大-加线性方程系统可解的特征以及极大-加混合线性不等式系统的最大解,给出极大-加混合线性不等式系统可解的一个充分必要条件,还给出极大-加混合线性不等式系统在部分变量非负的约束条件下可解的一个充分必要条件.同时,例举一个制造系统加工工件时序规划的应用例子.     

Calmness of Linear Constraint Systems under Structured Perturbations with an Application to the Path-Following Scheme

Set-Valued and Variational Analysis - We are concerned with finite linear constraint systems in a parametric framework where the right-hand side is an affine function of the perturbation parameter....     

Stability of Linear Inequality Systems Measured by the Hausdorff Metric

In this paper, we propose a Hausdorff metric to measure the distance between two linear inequality systems on a real normed space X. For this topology, which comes through a pseudo-metric in the set of linear inequality systems, the closedness of the feasible set mapping is studied, and at the same time a characterization of the stability of the subset _c of consistent sytems is given.     

一个分段线性Hénon映射吸引集合的结构

本文采用对偶线映射的方法分析了分段线性Hénon映射(x,y)→(1-a|x|+by,x),a=8/5,b=9/25吸引集的详细结构.设A和B分别是映射在第一和第三象限内的不动鞍点,本文说明:(1)映射的吸引集是B的不稳定流形U_B的闭包ū_B,而A的不稳定流形U_A则是ū_B的一个子集;(2)吸引盆是A的稳定流形S_A的闭包S_A,其边界是B的稳定流形S_B,而S_B在A_A的极限集之内.文中还给出周期鞍点不稳定流形和不动鞍点不稳定流形之间的关系.文中的符号动力学记号可用以研究各个不变流形每段的动态以及各同宿点、异宿点的动态.     

Stability of Linear Inequality Systems in a Parametric Setting

In this paper, we propose a parametric approach to the stability theory for the solution set of a semi-infinite linear inequality system in the n-dimensional Euclidean space . The main feature of this approach is that the coefficient perturbations are modeled through the so-called mapping of parametrized systems, which assigns to each parameter, ranging in a metric space, a subset of . Each vector of this image set provides the coefficients of an inequality in and the whole image set defines the inequality system associated with the parameter. Thus, systems associated with different parameters are not required to have the same number (cardinality) of inequalities. The paper is focused mainly on the structural stability of the feasible set mapping, providing a characterization of the Berge lower-semicontinuity property. The role played by the strong Slater qualification is analyzed in detail.This research has been partially supported by Grant PB98-0975 from DGES (Spain), Grant BFM2002-04114-C02 (01–02) from MCYT (Spain), FEDER (European Union), and Bancaja-UMH (Spain).     

线性空间中的一种S凸映射及其不等式性质

本文讨论了一类定义在线性空间中的新的凸性概念－S凸映射，并给该类凸映射的部分性质。     

Stability of the Solution Set of Perturbed Nonsmooth Inequality Systems and Application

Stability properties of the solution set of generalized inequality systems with locally Lipschitz functions are obtained under a regularity condition on the generalized Jacobian and the Clarke tangent cone. From these results, we derive sufficient conditions for the optimal value function in a nonsmooth optimization problem to be continuous or locally Lipschitz at a given parameter.