Semi-Infinite Programming pp 75-100 | Cite as

# Asymptotic Constraint Qualifications and Error Bounds for Semi-Infinite Systems of Convex Inequalities

## Abstract

We extend the known asymptotic constraint qualifications (ACQs) and some related constants from finite to semi-infinite convex inequality systems. We show that, in contrast with the finite case, only some of these ACQs are equivalent and only some of these constants coincide, unless we assume the”weak Pshenichnyi-Levin-Valadier property” introduced in [12]. We extend most of the global error bound results of [10] from finite systems of convex inequalities to the semi-infinite case and we show that to each semi-infinite convex inequality system with fini te-valued”sup function” one can associate an equivalent semi-infinite convex inequality system with finite-valued sup-function, admitting a global error bound. We give examples that the classical theorem of Hoffman [8] on the existence of a global error bound for each finite linear inequality system, as well as a result of [11] on global error bounds for finite différend able convex inequality systems cannot be extended to semi-infinite linear inequality systems. Finally, we give some simple sufficient conditions for the existence of a global error bound for semi-infinite linear inequality systems.

## Keywords

Error Bound Global Error Constraint Qualification Finite System Inequality System## Preview

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## References

- [1]A. Auslender, and J.-P. Crouzeix. Global regularity theorems,
*Mathematics of Operations Research*, 13:243–253, 1988.MathSciNetzbMATHCrossRefGoogle Scholar - [2]H. Bauschke, J. M. Borwein, and W. Li. Strong conical hull intersection property, bounded linear regularity, Jameson’s property (G), and error bounds in convex optimization,
*Mathematical Programming*, 86A:135–160, 1999.MathSciNetzbMATHCrossRefGoogle Scholar - [3]C. Bergthaller, and I. Singer. The distance to a polyhedron,
*Linear Algebra and its Applications*, 169:111–129, 1992.MathSciNetzbMATHCrossRefGoogle Scholar - [4]S. Deng. Global error bounds for convex inequality systems in Banach spaces,
*SIAM Journal on Control and Optimization*, 36:1240–1249, 1998.MathSciNetzbMATHCrossRefGoogle Scholar - [5]M. A. Goberna, and M. A. Lopez.
*Linear Semi-Infinite Optimization*, Wiley, 1998.zbMATHGoogle Scholar - [6]M. A. Goberna, M. A. Lopez, and M. Todorov. Stability theory for lin ear inequality systems,
*SIAM Journal on Matrix Analysis and Applications*, 17:730–743, 1996.MathSciNetzbMATHCrossRefGoogle Scholar - [7]J.-B. Hiriart-Urruty, and C. Lemaréchal.
*Convex Analysis and Minimization Algorithms. I*, Springer-Verlag, 1993.Google Scholar - [8]A. J. Hoffman. Approximate solutions of systems of linear inequalities,
*Journal of Research of the National Bureau of Standards*, 49:263–265, 1952.MathSciNetCrossRefGoogle Scholar - [9]R. B. Holmes.
*A Course on Optimization and Best Approximation*, Lecture Notes in Mathematics 257, Springer-Verlag, 1972.zbMATHGoogle Scholar - [10]D. Klatte, and W. Li. Asymptotic constraint qualifications and global error bounds for convex inequalities,
*Mathematical Programming*, 84A:137–160, 1999.MathSciNetzbMATHGoogle Scholar - [11]W. Li. Abadie’s constraint qualification, metric regularity and error bounds for convex differentiable inequalities,
*SIAM Journal on Optimization*, 7:966–978, 1997.MathSciNetzbMATHCrossRefGoogle Scholar - [12]W. Li, C. Nahak, and I. Singer. Constraint qualifications for semi-infinite systems of convex inequalities,
*SIAM Journal on Optimization*, 11:31–52, 2000.MathSciNetzbMATHCrossRefGoogle Scholar - [13]W. Li, and I. Singer. Global error bounds for convex multifunctions and applications,
*Mathematics of Operations Research*, 23:443–462, 1998.MathSciNetzbMATHCrossRefGoogle Scholar - [14]A. S. Lewis, and J.-S. Pang. Error bounds for convex inequality systems. In J.-P. Crouzeix, J.-E. Martínez-Legaz and M. Voile, editors,
*Generalized Con vexity, Generalized Monotoniaty: Recent Results*, pages 75–110, Kluwer, 1998.CrossRefGoogle Scholar - [15]J.-S. Pang. Error bounds in mathematical programming,
*Mathematical Programming*, 79A:299–332, 1997.MathSciNetzbMATHGoogle Scholar - [16]R. R. Phelps.
*Convex Functions, Monotone Operators and Differentiabil ity*(2^{nd}ed.), Lecture Notes in Mathematics 1364, Springer-Verlag, 1993.Google Scholar - [17]S. M. Robinson. An application of error bounds for convex programming in a linear space,
*SI AM Journal on Control, 13:271–273*, 1975.zbMATHCrossRefGoogle Scholar - [18]I. Singer.
*The Theory of Best Approximation and Functional Analysis*, CBMS Regional Conference Series in Applied Mathematics 13, SIAM, 1974.zbMATHCrossRefGoogle Scholar