MIP and Set Covering approaches for Sparse Approximation
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The Sparse Approximation problem asks to find a solution x such that ||y - Hx|| < α, for a given norm ||·||, minimizing the size of the support ||x||_0 := #{j | x_j ≠ 0 }. We present valid inequalities for Mixed Integer Programming (MIP) formulations for this problem and we show that these families are sufficient to describe the set of feasible supports. This leads to a reformulation of the problem as an Integer Programming (IP) model which in turn represents a Minimum Set Covering formulation, thus yielding many families of valid inequalities which may be used to strengthen the models up. We propose algorithms to solve sparse approximation problems including a branch & cut for the MIP, a two-stages algorithm to tackle the set covering IP and a heuristic approach based on Local Branching type constraints. These methods are compared in a computational experimentation with the goal of testing their practical potential.
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