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We present an alternating direction dual augmented Lagrangian method for solving semidefinite programming (SDP) problems in standard form. At each iteration, our basic algorithm minimizes the augmented Lagrangian function for the dual SDP problem sequentially, first with respect to the dual variables corresponding to the linear constraints, and then with respect to the dual slack variables, while in each minimization keeping the other variables fixed, and then finally it updates the Lagrange multipliers (i.e., primal variables). Convergence is proved by using a fixed-point argument. For SDPs with inequality constraints and positivity constraints, our algorithm is extended to separately minimize the dual augmented Lagrangian function over four sets of variables. Numerical results for frequency assignment, maximum stable set and binary integer quadratic programming problems demonstrate that our algorithms are robust and very efficient due to their ability or exploit special structures, such as sparsity and constraint orthogonality in these problems.
We present two algorithms to solve the Group Lasso problem (Yuan and Lin in, J R Stat Soc Ser B (Stat Methodol) 68(1):49–67, 2006). First, we propose a general version of the Block Coordinate Descent (BCD) algorithm for the Group Lasso that employs an efficient approach for optimizing each subproblem exactly.We show that it exhibits excellent performance when the groups are of moderate size. For groups of large size, we propose an extension of ISTA/FISTA SIAM (Beck and Teboulle in, SIAM J Imag Sci 2(1):183–202, 2009) based on variable step-lengths that can be viewed as a simplified version of BCD. By combining the two approaches we obtain an implementation that is very competitive and often outperforms other state-of-the-art approaches for this problem. We show how these methods fit into the globally convergent general block coordinate gradient descent framework in Tseng and Yun (Math Program 117(1):387–423, 2009). We also show that the proposed approach is more efficient in practice than the one implemented in Tseng and Yun (Math Program 117(1):387–423, 2009). In addition, we apply our algorithms to the Multiple Measurement Vector (MMV) recovery problem, which can be viewed as a special case of the Group Lasso problem, and compare their performance to other methods in this particular instance.
