Abstract
The paper concerns the study of new classes of nonlinear and nonconvex optimization problems of the so-called infinite programming that are generally defined on infinite-dimensional spaces of decision variables and contain infinitely many of equality and inequality constraints with arbitrary (may not be compact) index sets. These problems reduce to semi-infinite programs in the case of finite-dimensional spaces of decision variables. We extend the classical Mangasarian–Fromovitz and Farkas–Minkowski constraint qualifications to such infinite and semi-infinite programs. The new qualification conditions are used for exact calculations of the appropriate normal cones to sets of feasible solutions for these programs by employing advanced tools of variational analysis and generalized differentiation. In the further development we derive first-order necessary optimality conditions for infinite and semi-infinite programs, which are new in both finite-dimensional and infinite-dimensional settings.
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Acknowledgments
The authors are indebted to Marco López and anonymous referees for their very careful reading of the paper and a number of valuable suggestions and remarks that allowed us to essentially improve the original presentation.
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Dedicated to Jon Borwein in honor of his 60th birthday.
Research was partially supported by the USA National Science Foundation under Grant DMS-1007132, by the Australian Research Council under Grant DP-12092508, and by the Portuguese Foundation of Science and Technologies under grant MAT/11109.
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Mordukhovich, B., Nghia, T.T.A. Constraint qualifications and optimality conditions for nonconvex semi-infinite and infinite programs. Math. Program. 139, 271–300 (2013). https://doi.org/10.1007/s10107-013-0672-x
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DOI: https://doi.org/10.1007/s10107-013-0672-x
Keywords
- Variational analysis and parametric optimization
- Generalized differentiation
- Optimality conditions
- Semi-infinite and infinite programming