  We tackle the problem of accelerating column generation (CG) approaches to
set cover formulations in operations research. At each iteration of CG we
generate a dual solution that approximately solves the LP over all columns
consisting of a subset of columns in the nascent set. We refer to this linear
program (LP) as the Family Restricted Master Problem (FRMP), which provides a
tighter bound on the master problem at each iteration of CG, while preserving
efficient inference. For example, in the single source capacitated facility
location problem (SSCFLP) the family of a column $l$ associated with facility
$f$ and customer set $N_l$ contains the set of columns associated with $f$ and
the customer set that lies in the power set of $N_l$. The solution to FRMP
optimization is attacked with a coordinate ascent method in the dual. The
generation of direction of travel corresponds to solving the restricted master
problem over columns corresponding to the reduced lowest cost column in each
family given specific dual variables based on the incumbent dual, and is easily
generated without resolving complex pricing problems. We apply our algorithm to
the SSCFLP and demonstrate improved performance over two relevant baselines.
