Valid inequalities based on the interpolation procedure

Abstract

We study the interpolation procedure of , which generates cutting planes for general integer programs from facets of cyclic group polyhedra. This idea has recently been re-considered by Evans (2002) and Gomory, . We compare inequalities generated by this procedure with mixed-integer rounding (MIR) based inequalities discussed in Dash and Gunluk (2003). We first analyze and extend the shooting experiment described in Gomory, Johnson and Evans. We show that MIR based inequalities dominate inequalities generated by the interpolation procedure in some important cases. We also show that the Gomory mixed-integer cut is likely to dominate any inequality generated by the interpolation procedure in a certain probabilistic sense. We also generalize a result of Cornuéjols, Li and Vandenbussche (2003) on comparing the strength of the Gomory mixed-integer cut with related inequalities.

Mixed-Integer Cuts from Cyclic Groups

“…The outcome is that Gomory mixed-integer cuts play a very special role among subadditive cuts, in that they typically produce, alone, a lower bound increase which is comparable to that obtained when the whole family of cuts is considered. This result confirms the theoretical findings of Dash and Gunluk [7], who showed that interpolated subadditive cuts are dominated by Gomory mixed-integer cuts in a probabilistic sense, as well as the computational experience of Cornuejols, Li and Vandenbussche [5] on the subfamily of k-cuts. Some interesting directions of work are finally addressed in Section 6.…”

“…A key observation at this point is that, being k ideal, the actual value of g(·) outside the sample points i/k is immaterial, since g(·) only needs to be evaluated on these sample points when computing the coefficients in (7). Therefore, the interpolation procedure does not actually restrict the space of the possible subadditive functions-as it would be the case for a different choice of k. As a consequence, we can exactly rephrase g-SEP as the following LP:…”

“…where r := k φ(β), t * i := j {x * j : φ(α j ) = i/k} for i ∈ {1, · · · , k − 1} \ {r}, and t * r := j {x * j : φ(α j ) = r/k} − 1 so as to take into account the role of the right hand side g(β) = g r in (7). With these definitions, the objective function k−1 i=1 t * i g i is precisely the opposite of the violation of a cut of the form (7), hence a violated such cut exists if and only if the optimal value of g − SEP k is strictly negative.…”

“…To our knowledge, however, the practical benefit that can be obtained by embedding the whole family of cyclic-group cuts in a cutting plane algorithm was not investigated computationally by previous authors. As a matter of fact, a number of recent papers [15,16,6,7,8,3] deals only implicitly with cyclic-group separation, as they address the so-called Gomory's shooting experiment. Roughly speaking, in this experiment the point t * ∈ R k−1 to be separated is generated at random (hence corresponding to a random "shooting direction" in the T -space), and statistics on the frequency of the most-violated facets of T (k, r) are collected.…”

“…This paper is organized as follows. In Section 2 we present the theory of Gomory and Johnson [12,13,14] on interpolated subadditive functions (called template functions in [7]) and their role in generating valid inequalities for P I . We also introduce an exact separation procedure for interpolated subadditive cuts based on an LP model taken from the Gomory-Johnson characterization of the master cyclic group polyhedron.…”