The Sparse Approximation problem asks to find a solution $x$ such that $||y - Hx|| < \alpha$, for a given norm $||\cdot||$, minimizing the size of the support $||x||_0 := \#\{j \ |\ x_j \neq 0 \}$. 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.

中文翻译：

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稀疏近似的 MIP 和集合覆盖方法

稀疏近似问题要求找到一个解决方案 $x$，使得 $||y - Hx|| < \alpha$，对于给定的范数 $||\cdot||$，最小化支持的尺寸 $||x||_0 := \#\{j \ |\ x_j \neq 0 \}$。我们针对这个问题提出了混合整数规划 (MIP) 公式的有效不等式，并且我们证明这些族足以描述可行支持集。这导致将问题重新表述为整数规划 (IP) 模型，该模型又代表最小集覆盖公式，从而产生许多可用于加强模型的有效不等式系列。我们提出了解决稀疏逼近问题的算法，包括 MIP 的分支 \& 切割、处理覆盖 IP 的集合的两阶段算法和基于局部分支类型约束的启发式方法。