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It has been recently shown that the <inline-formula> <tex-math notation="LaTeX">$l_{0}$ </tex-math></inline-formula>-norm problem can be reformulated into a mixed integer quadratic programming (MIQP) problem. CPLEX, a commercial optimization software package that can solve integer programming problems, is used to find the global solution to this MIQP problem for sparse signal estimation. However, CPLEX uses an exhaustive approach to search a feasible space to this MIQP problem. Thus, its running time grows exponentially as the problem dimension grows. This means that CPLEX quickly becomes computationally intractable for higher dimension problems. In this paper, we aim to propose a fast first-order-type method for solving this MIQP problem based on the alternating direction method. We conduct extensive simulations to demonstrate that: 1) our method is used to estimate a sparse signal by solving this problem and 2) our method is computationally tractable for problem dimensions up to the order of 1 million.

Fast Mixed Integer Quadratic Programming for Sparse Signal Estimation