Abstract
The paper studies a nonlinear optimization problem under resource allocation constraints. Using quasi-gradient duality it is shown that the feasible set of the problem is a singleton (in the case of a single resource) or the set of Pareto efficient solutions of an associated vector maximization problem (in the case of \(k>1\) resources). As a result, a nonlinear optimization problem under resource allocation constraints reduces to an optimization over the efficient set. The latter problem can further be converted into a quasiconvex maximization over a compact convex subset of \(\mathbb{R }^k_+.\) Alternatively, it can be approached as a bilevel program and converted into a monotonic optimization problem in \(\mathbb{R }^k_+.\) In either approach the converted problem falls into a common class of global optimization problems for which several practical solution methods exist when the number \(k\) of resources is relatively small, as it often occurs.
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Acknowledgments
The authors are grateful to Prof. Hoang Tuy for several suggestions and advices which have helped to improve the presentation of a first draft of the paper. Also the authors would like to thank the referees for several useful remarks.
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Thach, P.T., Thang, T.V. Problems with resource allocation constraints and optimization over the efficient set. J Glob Optim 58, 481–495 (2014). https://doi.org/10.1007/s10898-013-0055-0
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DOI: https://doi.org/10.1007/s10898-013-0055-0
Keywords
- Duality
- Resource allocation constraint
- Optimization over efficient set
- Bilevel programming
- Monotonic optimization