An algorithm for cardinality-constrained optimization with an application to the best subset selection and sparse portfolio problems

A lot of problems, from fields like sparse signal processing, statistics, portfolio selection, and machine learning, can be formulated as a cardinality constraint optimization problem. The cardinality constraint gives the problem a discrete nature, making it computationally challenging to solve as the dimension of the problem increases. In this work, we present an algorithm to solve the cardinality constraint quadratic optimization problem using the framework of the interval branch-and-bound. Interval branch-and-bound is a popular approach for finding a globally optimal solution in the field of global optimization. The proposed method is capable of solving problems of a wide range of dimensions. In particular, we solve the classical best subset selection problem in regression and compare our algorithm against another branch-and-bound method and GUROBI's quadratic mixed integer solver. Numerical results show that the proposed algorithm outperforms the first and is competitive with the second solver.