Abstract. This paper proposes efficient algorithms for group sparse optimization with mixed 2,1 -regularization, which arises from the reconstruction of group sparse signals in compressive sensing, and the group Lasso problem in statistics and machine learning. It is known that encoding the group information in addition to sparsity will lead to better signal recovery/feature selection. The 2,1 -regularization promotes group sparsity, but the resulting problem, due to the mixed-norm structure and possible grouping irregularity, is considered more difficult to solve than the conventional 1 -regularized problem. Our approach is based on a variable splitting strategy and the classic alternating direction method (ADM). Two algorithms are presented, one derived from the primal and the other from the dual of the 2,1 -regularized problem. The convergence of the proposed algorithms is guaranteed by the existing ADM theory. General group configurations such as overlapping groups and incomplete covers can be easily handled by our approach. Computational results show that on random problems the proposed ADM algorithms exhibit good efficiency, and strong stability and robustness.

In this paper we study practical solution methods for nding the maximum-volume ellipsoid inscribing a given full-dimensional polytope in < n de ned by a nite set of linear inequalities. Our goal is to design a general-purpose algorithmic framework that is reliable and e cient in practice. To e v aluate the merit of a practical algorithm, we consider two k ey factors: the computational cost per iteration and the typical number of iterations required for convergence. In addition, numerical stability is also an important factor. We i n vestigate some new formulations upon which w e build primal-dual type, interior-point algorithms, and we p r o vide theoretical justi cations for the proposed formulations and algorithmic framework. Extensive n umerical experiments have shown that one of the new algorithms should be the method of choice among the tested algorithms.

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