Relations between facets of low- and high-dimensional group problems

One-dimensional infinite group problems have been extensively studied and have yielded strong cutting planes for mixed integer programs. Although numerical and theoretical studies suggest that group cuts can be significantly improved by considering higher-dimensional groups, there are no known facets for infinite group problems whose dimension is larger than two. In this paper, we introduce an operation that we call sequential-merge. We prove that the sequential-merge operator creates a very large family of facet-defining inequalities for high-dimensional infinite group problems using facet-defining inequalities of lower-dimensional group problems. Further, they exhibit two properties that reflect the benefits of using facets of high-dimensional group problems: they have continuous variables' coefficients that are not dominated by those of the constituent low-dimensional cuts and they can produce cutting planes that do not belong to the first split closure of MIPs. Further, we introduce a general scheme for generating valid inequalities for lower-dimensional group problems using valid inequalities of higher-dimensional group problems. We present conditions under which this construction generates facet-defining inequalities when applied to sequential-merge inequalities. We show that this procedure yields some two-step MIR inequalities of Dash and Günlük.