Abstract-Sparse approximation addresses the problem of approximately fitting a linear model with a solution having as few non-zero components as possible. While most sparse estimation algorithms rely on suboptimal formulations, this work studies the performance of exact optimization of 0-norm-based problems through Mixed-Integer Programs (MIPs). Nine different sparse optimization problems are formulated based on 1, 2 or ∞ data misfit measures, and involving whether constrained or penalized formulations. For each problem, MIP reformulations allow exact optimization, with optimality proof, for moderate-size yet difficult sparse estimation problems. Algorithmic efficiency of all formulations is evaluated on sparse deconvolution problems. This study promotes error-constrained minimization of the 0 norm as the most efficient choice when associated with 1 and ∞ misfits, while the 2 misfit is more efficiently optimized with sparsity-constrained and sparsity-penalized problems. Then, exact 0-norm optimization is shown to outperform classical methods in terms of solution quality, both for over-and underdetermined problems. Finally, numerical simulations emphasize the relevance of the different p fitting possibilities as a function of the noise statistical distribution. Such exact approaches are shown to be an efficient alternative, in moderate dimension, to classical (suboptimal) sparse approximation algorithms with 2 data misfit. They also provide an algorithmic solution to less common sparse optimization problems based on 1 and ∞ misfits. For each formulation, simulated test problems are proposed where optima have been successfully computed. Data and optimal solutions are made available as potential benchmarks for evaluating other sparse approximation methods.

An automatic method for constructing linear relaxations of constrained global optimization problems is proposed. Such a construction is based on affine and interval arithmetics and uses operator overloading. These linear programs have exactly the same numbers of variables and inequality constraints as the given problems. Each equality constraint is replaced by two inequalities. This new procedure for computing reliable bounds and certificates of infeasibility is inserted into a classical Branch and Bound algorithm based on interval analysis. Extensive computation experiments were made on 74 problems from the COCONUT database with up to 24 variables or 17 constraints; 61 of these were solved, and 30 of them for the first time, with a guaranteed upper bound on the relative error equal to 10 −8. Moreover, this sample comprises 39 examples to which the GlobSol algorithm was recently applied finding reliable solutions in 32 cases. The proposed method allows solving 31 of these, and 5 more with a CPU-time not exceeding 2 minutes.

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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.