Dynamic constraint and variable aggregation in column generation

Starting from the improved primal simplex (IPS) decomposition, introduced by Elhallaoui et al. (2011) to tackle degeneracy in general linear programs, we introduce and discuss the mathematical foundations of improved column-generation decompositions (ICG) that are better than the standard column-generation decomposition when degeneracy is an issue. We also present an improved dynamic constraint aggregation (IDCA), which is a specialization of ICG to efficiently solve set partitioning problems. We show that IDCA improves the dynamic constraint aggregation (DCA) and the multiphase dynamic constraint aggregation (MPDCA) algorithms (Elhallaoui et al., 2005, 2010) that not only reduce degeneracy but also profit from it to efficiently solve set partitioning problems. IDCA solves at each iteration a complementary problem (CP) to obtain a group of variables that can be aggregated or pivoted into the basis to bypass degenerate pivots and decrease the objective value and to obtain more “central” dual solutions to generate new columns. Our numerical results show reduction factors in the solution times exceeding 10 for IDCA compared to a state-of-the-art standard column-generation solver.