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Quasi-Slater and Farkas-Minkowski qualifications for semi-infinite programming with applications. (English) Zbl 1298.90119

This article is a valuable contribution to a field in the area of continuous optimization, known as semi-infinite optimization, or semi-infinite programming (SIP) which, via its possibly infinitely many inequality constraints, is neighboring and coming close to the fields of infinite programming and the areas of calculus of variations and optimal control theory. Indeed, such latter time-dependent problem classes, but also problems of approximation, of design, of maneuverability, etc., are in the wide range of applications, motivated by the real-world, where SIP problems have become of a growing importance in recent decades.
The famous Farkas-Minkowski (in short: FM) type qualification is of a central role in linear SIP and it has been advanced further by many scientists by finding and constituting optimality conditions, stability and duality for semi-infinite optimization. In this article, the authors present the concept of the quasi-Slater condition for a semi-infinite convex inequality system and show that Slater type conditions imply the FM qualification under an appropriate continuity assumption of the set-valued mapping of the index set to the model functions with their values. By applying these relations, the authors find dual characterizations, asymptotic ones and nonasymptotic ones as well, for set-containment problems and they provide sufficient conditions for ensuring strong Lagrangian duality and a Farkas lemma.
The four sections of this article are as follows: 1. Introduction, 2. Notations and preliminary results, 3. The Slater and FM qualifications, and 4. Applications, with the subsections called 4.1 Set containment characterization and 4.2 Strong Lagrangian duality and the Farkas lemma.
In fact, in the future, strong results and algorithms could be awaited further, motivated and initialized by this paper. That progress might foster and support rising contributions to science and engineering, e.g., in data mining, engineering, in economics, finance and OR, and to the living conditions in this world.

MSC:

90C34 Semi-infinite programming
90C25 Convex programming
41A29 Approximation with constraints
90C46 Optimality conditions and duality in mathematical programming
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