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Recently, a lot of attention has been given to penalized least squares problem formulations for sparse signal reconstruction in the presence of noise. The penalty is responsible for inducing sparsity, where the common choice used is the convex l 1 norm. While an l 0 penalty generates maximum sparsity it has been avoided due to lack of convexity. With the hope of gaining improved sparsity but more computational tractability there has been recent interest in the l q penalty. In this paper we provide a novel cyclic descent algorithm for optimizing the l q penalized least squares problem when 0 < q < 1. Optimality conditions for this problem are derived and competing ones are clarified. We illustrate with simulations comparing the reconstruction quality with three penalty functions: l 0 , l 1 and l q , 0 < q < 1.

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