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Finding solutions to least-squares problems with low cardinality has found many applications, including portfolio optimization, subset selection in statistics, and inverse problems in signal processing. Although most works consider local approaches that scale with high-dimensional problems, some others address its global optimization via mixed integer programming (MIP) reformulations. We propose dedicated branch-and-bound methods for the exact resolution of moderate-size, yet difficult, sparse optimization problems, through three possible formulations: cardinality-constrained and cardinality-penalized least-squares, and cardinality minimization under quadratic constraints. A specific tree exploration strategy is built. Continuous relaxation problems involved at each node are reformulated as-norm-based optimization problems, for which a dedicated algorithm is designed. The obtained certified solutions are shown to better estimate sparsity patterns than standard methods on simulated variable selection problems involving highly correlated variables. Problem instances selecting up to 24 components among 100 variables, and up to 15 components among 1000 variables, can be solved in less than 1000 s. Unguaranteed solutions obtained by limiting the computing time to 1s are also shown to provide competitive estimates. Our algorithms strongly outperform the CPLEX MIP solver as the dimension increases, especially for quadratically-constrained problems. The source codes are made freely available online.

Global optimization for sparse solution of least squares problems