This paper introduces thelocally Farkas-Minkowski (LFM) linear inequality systems in a finite dimensional Euclidean space. These systems are those ones that satisfy that any consequence of the system that is active at some solution point is also a consequence of some finite subsystem. This class includes the Farkas-Minkowski systems and verifies most of the properties that these systems possess. Moreover, it contains the locally polyhedral systems, which are the natural external representation of quasi-polyhedral sets. TheLFM systems appear to be the natural external representation of closed convex sets. A characterization based on their properties under the union of systems is provided. In linear semi-infinite programming, theLFM property is the more general constraint qualification such that the Karush-Kuhn-Tucker condition characterizes the optimal points. Furthermore, the pair of Haar dual problems has no duality gap.
[1]
Marco A. López,et al.
Optimality theory for semi-infinite linear programming ∗
,
1995
.
[2]
Virginia N. Vera de Serio,et al.
Quasipolyhedral sets in linear semiinfinite inequality systems
,
1997
.
[3]
V. Klee.
Some characterizations of convex polyhedra
,
1959
.
[4]
Marco A. López,et al.
A theory of linear inequality systems
,
1988
.
[5]
Marco A. López,et al.
Locally polyhedral linear inequality systems
,
1998
.
[6]
W W Cooper,et al.
DUALITY, HAAR PROGRAMS, AND FINITE SEQUENCE SPACES.
,
1962,
Proceedings of the National Academy of Sciences of the United States of America.
[7]
Ky Fan,et al.
On infinite systems of linear inequalities
,
1968
.
[8]
Kenneth O. Kortanek,et al.
Semi-Infinite Programming: Theory, Methods, and Applications
,
1993,
SIAM Rev..