TY - JOUR

T1 - A Low-Cost Alternating Projection Approach for a Continuous Formulation of Convex and Cardinality Constrained Optimization

AU - Krejić, N.

AU - Krulikovski, E. H. M.

AU - Raydan, M.

N1 - 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).

PY - 2023/12

Y1 - 2023/12

N2 - 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.

AB - 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.

KW - 65K05

KW - 90C30

KW - 91G10

KW - 91G15

KW - Cardinality constraints

KW - Dykstra’s algorithm

KW - Efficient frontier

KW - Portfolio optimization

KW - Projected gradient methods

UR - http://www.scopus.com/inward/record.url?scp=85173766513&partnerID=8YFLogxK

U2 - 10.1007/s43069-023-00257-w

DO - 10.1007/s43069-023-00257-w

M3 - Article

AN - SCOPUS:85173766513

VL - 4

JO - SN Operations Research Forum

JF - SN Operations Research Forum

IS - 4

M1 - 73

ER -