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Zhaosong Lu (zhaosongsfu.ca) Abstract: In this paper we consider sparse approximation problems, that is, general $l_0$ minimization problems with the $l_0$``norm'' of a vector being a part of constraints or objective function. In particular, we first study the firstorder optimality conditions for these problems. We then propose penalty decomposition (PD) methods for solving them in which a sequence of penalty subproblems are solved by a block coordinate descent (BCD) method. Under some suitable assumptions, we establish that any accumulation point of the sequence generated by the PD methods satisfies the firstorder optimality conditions of the problems. Furthermore, for the problems in which the $l_0$ part is the only nonconvex part, we show that such an accumulation point is a local minimizer of the problems. In addition, we show that any accumulation point of the sequence generated by the BCD method is a saddle point of the penalty subproblem. Moreover, for the problems in which the $l_0$ part is the only nonconvex part, we establish that such an accumulation point is a local minimizer of the penalty subproblem. Finally, we test the performance of our PD methods by applying them to sparse logistic regression, sparse inverse covariance selection, and compressed sensing problems. The computational results demonstrate that our methods generally outperform the existing methods in terms of solution quality and/or speed. Keywords: $l_0$ minimization, penalty decomposition methods, block coordinate descent method, compressed sensing, sparse logistic regression, sparse inverse covariance selection Category 1: Convex and Nonsmooth Optimization (Nonsmooth Optimization ) Category 2: Nonlinear Optimization (Constrained Nonlinear Optimization ) Category 3: Integer Programming ((Mixed) Integer Nonlinear Programming ) Citation: Manuscript, Department of Mathematics, Simon Fraser University, 8888 University Drive, Burnaby, BC, V5A 1S6, Canada, May 2012 Download: [PDF] Entry Submitted: 05/11/2012 Modify/Update this entry  
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