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 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.     

Approximating Reachable Sets for n-dimensional Linear Discrete Systems

We present a technique for approximating the reachable set fromthe origin for n-dimensional discrete-time linear systems subjectto bounded control, where the state matrix is stable and diagonalizable.The procedure is based on decomposing the system into one- andtwo-dimensional subsystems for which the reachable set of eachof the subsystems can be over-estimated by a polygon. Thesepolygons are then used to create an n-dimensional polyhedronwhich contains the reachable set of the original system     

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).     

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.     

Calmness of the Feasible Set Mapping for Linear Inequality Systems

In this paper we deal with parameterized linear inequality systems in the n-dimensional Euclidean space, whose coefficients depend continuosly on an index ranging in a compact Hausdorff space. The paper is developed in two different parametric settings: the one of only right-hand-side perturbations of the linear system, and that in which both sides of the system can be perturbed. Appealing to the backgrounds on the calmness property, and exploiting the specifics of the current linear structure, we derive different characterizations of the calmness of the feasible set mapping, and provide an operative expresion for the calmness modulus when confined to finite systems. In the paper, the role played by the Abadie constraint qualification in relation to calmness is clarified, and illustrated by different examples. We point out that this approach has the virtue of tackling the calmness property exclusively in terms of the system’s data.     

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.     

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.     

Asymptotic Behavior of the Attainable Sets of Linear Periodic Control Systems

This paper deals with the attainable sets of linear periodic control systems. The asymptotic behavior of the attainable sets over a long time interval is investigated in terms of shapes of the sets. The shape of a set stands for the totality of all its images under nonsingular linear transformations. It is shown that there exist limits of the shape of attainable sets corresponding to time instants with the same residue modulo the period of the system and that the limit shapes are different if the system includes a stable subsystem.     

Description of the Structure of Irregularity Sets of Linear Differential Systems with a Linear Parameter

Differential Equations - We consider the class of linear parametric differential systems $$\dot {x}=\mu A(t)x $$ defined on the half-line $$t\geq 0 $$ , where $$\mu \in \mathbb {R} $$ is a...     

Linear k-blocking Sets

We point out the relationship between normal spreads and the linear k-blocking sets introduced in [9]. We give a characterisation of linear k-blocking sets proving that the projections and the embeddings of a PG(kt,q) in are linear k-blocking sets of . Finally, we construct some new examples. Received December 19, 1997/Revised September 19, 2000 RID="*" ID="*" Partially supported by Italian M.U.R.S.T.     

(A,B)-Invariant Polyhedral Sets of Linear Discrete-Time Systems

The problem of confining the trajectory of a linear discrete-time system in a given polyhedral domain is addressed through the concept of (A, B)-invariance. First, an explicit characterization of (A, B)-invariance of convex polyhedra is proposed. Such characterization amounts to necessary and sufficient conditions in the form of linear matrix relations and presents two major advantages compared to the ones found in the literature: it applies to any convex polyhedron and does not require the computation of vertices. Such advantages are felt particularly in the computation of the supremal (A, B)-invariant set included in a given polyhedron, for which a numerical method is proposed. The problem of computing a control law which forces the system trajectories to evolve inside an (A, B)-invariant polyhedron is treated as well. Finally, the (A, B)-invariance relations are generalized to persistently disturbed systems.     

Local Behavior of the Newton Method on Two Equivalent Systems from Linear Programming

Newton's method is a fundamental technique underlying many numerical methods for solving systems of nonlinear equations and optimization problems. However, it is often not fully appreciated that Newton's method can produce significantly different behavior when applied to equivalent systems, i.e., problems with the same solution but different mathematical formulations. In this paper, we investigate differences in the local behavior of Newton's method when applied to two different but equivalent systems from linear programming: the optimality conditions of the logarithmic barrier function formulation and the equations in the so-called perturbed optimality conditions. Through theoretical analysis and numerical results, we provide an explanation of why Newton's method performs more effectively on the latter system.     

On the Geometric Properties of Reachable and Controllable Sets for Linear Discrete Systems

A linear time-invariant discrete system with a single bounded control input is considered. Various types of constraints imposed on the control input are investigated. The set of system states that can be reached from the origin (reachable set) is studied. The set of system states from where the origin can be reached (null-controllable set) is studied as well. Various examples are presented.     

Limit Behavior of Reachable Sets of Linear Time-Invariant Systems with Integral Bounds on Control

In this paper, a linear dynamic system is considered under L _p -constraint on control. We establish the existence of the limit shape of reachable sets as time goes to infinity. Asymptotic formulas are obtained for reachable sets and their shapes. The results throw a bridge between the cases of geometric bounds on control and constraints on the total impulse of control, and create a unified picture of the structure of the limit shapes of reachable sets.