No abstract

Finding solutions to least-squares problems with low cardinality has found many applications, including cardinality-constrained portfolio optimization, subset selection in Statistics, and many sparsity-enhancing inverse problems in signal processing. In general, this problem is NP-hard, and most works from a global optimization perspective consider a mixed integer programming (MIP) reformulation with binary variables, whose resolution is performed via branch-and-bound methods. We propose dedicated branch-and-bound algorithms for three possible formulations: cardinality-constrained and cardinality-penalized least-squares, and cardinality minimization under quadratic constraints. We show that the continuous relaxation problems involved at each node of the search tree are 1-norm-based optimization problems. A dedicated algorithm is built, based on the homotopy continuation principle, which efficiently computes the relaxed solutions for the three kinds of problems. The performance of the resulting global optimization procedure is then shown to compete with or improve over the CPLEX MIP solvers, especially for problems involving quadratic constraints. The proposed strategies are able to exactly solve some problems involving 500 to 1 000 unknowns in less than 1 000 seconds, for which CPLEX mostly fails.

This paper addresses the linear spectral unmixing problem, by incorporating different constraints that may be of interest in order to cope with spectral variability: sparsity (few nonzero abundances), group exclusivity (at most one nonzero abundance within subgroups of endmembers) and significance (non-zero abundances must exceed a threshold). We show how such problems can be solved exactly with mixed-integer programming techniques. Numerical simulations show that solutions can be computed for problems of limited, yet realistic, complexity, with improved estimation performance over existing methods, but with higher computing time.

We propose a new greedy sparse approximation algorithm, called SLS for Single L1 Selection, that addresses a least squares optimization problem under a cardinality constraint. The specificity and increased efficiency of SLS originate from the atom selection step, based on exploiting 1 -norm solutions. At each iteration, the regularization path of a least-squares criterion penalized by the 1 norm of the remaining variables is built. Then, the selected atom is chosen according to a scoring function defined over the solution path. Simulation results on difficult sparse deconvolution problems involving a highly correlated dictionary reveal the efficiency of the method, which outperforms popular greedy algorithms when the solution is sparse.