Optimality Conditions for Convex Semi-infinite Programming Problems with Finitely Representable Compact Index Sets
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In the present paper, we analyze a class of convex semi-infinite programming problems with arbitrary index sets defined by a finite number of nonlinear inequalities. The analysis is carried out by employing the constructive approach, which, in turn, relies on the notions of immobile indices and their immobility orders. Our previous work showcasing this approach includes a number of papers dealing with simpler cases of semi-infinite problems than the ones under consideration here. Key findings of the paper include the formulation and the proof of implicit and explicit optimality conditions under assumptions, which are less restrictive than the constraint qualifications traditionally used. In this perspective, the optimality conditions in question are also compared to those provided in the relevant literature. Finally, the way to formulate the obtained optimality conditions is demonstrated by applying the results of the paper to some special cases of the convex semi-infinite problems.
KeywordsConvex programming Semi-infinite programming (SIP) Nonlinear programming (NLP) Convex set Finitely representable set Constraint qualifications (CQ) Immobile index Optimality conditions
Mathematics Subject Classification90C25 90C30 90C34
The authors are sincerely grateful to two anonymous reviewers for their valuable comments, suggestions, and corrections that allowed to improve the presentation of the results. Our special thanks to the Editors for the useful advices that helped to prepare the revised version. This work was partially supported by the state research program “Convergence” of Republic Belarus: Task 1.3.0 “Development of the methods for solving the uncorrect problems of the control theory for distributed dynamical systems”, and by Portuguese funds through CIDMA—Center for Research and Development in Mathematics and Applications, and FCT—Portuguese Foundation for Science and Technology, within the Project UID/MAT/04106/2013.
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