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A Low-Cost Alternating Projection Approach for a Continuous Formulation of Convex and Cardinality Constrained Optimization
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Orientador(es)
Resumo(s)
We consider convex constrained optimization problems that also include a cardinality constraint. In general, optimization problems with cardinality constraints are difficult mathematical programs which are usually solved by global techniques from discrete optimization. We assume that the region defined by the convex constraints can be written as the intersection of a finite collection of convex sets, such that it is easy and inexpensive to project onto each one of them (e.g., boxes, hyper-planes, or half-spaces). Taking advantage of a recently developed continuous reformulation that relaxes the cardinality constraint, we propose a specialized penalty gradient projection scheme combined with alternating projection ideas to compute a solution candidate for these problems, i.e., a local (possibly non-global) solution. To illustrate the proposed algorithm, we focus on the standard mean-variance portfolio optimization problem for which we can only invest in a preestablished limited number of assets. For these portfolio problems with cardinality constraints, we present a numerical study on a variety of data sets involving real-world capital market indices from major stock markets. In many cases, we observe that the proposed scheme converges to the global solution. On those data sets, we illustrate the practical performance of the proposed scheme to produce the effective frontiers for different values of the limited number of allowed assets.
Descrição
Funding Information: Open access funding provided by FCT|FCCN (b-on). The first author was financially supported by the Serbian Ministry of Education, Science, and Technological Development and Serbian Academy of Science and Arts, grant no. F10. The second author was financially supported by Fundação para a Ciência e a Tecnologia (FCT) (Portuguese Foundation for Science and Technology) under the scope of the projects UIDB/MAT/00297/2020, UIDP/MAT/00297/2020 (Centro de Matemática e Aplicações), and UI/297/2020-5/2021. The third author was financially supported by Fundação para a Ciência e a Tecnologia (FCT) (Portuguese Foundation for Science and Technology) under the scope of the projects UIDB/MAT/00297/2020, UIDP/MAT/00297/2020 (Centro de Matemática e Aplicações). Publisher Copyright: © 2023, The Author(s).
Palavras-chave
65K05 90C30 91G10 91G15 Cardinality constraints Dykstra’s algorithm Efficient frontier Portfolio optimization Projected gradient methods Economics, Econometrics and Finance (miscellaneous) Computer Science Applications Control and Optimization Applied Mathematics