An exploratory computational analysis of dual degeneracy in mixed-integer programming

Gerald Gamrath,Timo Berthold,Domenico Salvagnin

doi:10.1007/s13675-020-00130-z

Gerald Gamrath, Timo Berthold + Show 1 more

Open Access

https://doi.org/10.1007/s13675-020-00130-z

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Abstract

Dual degeneracy, i.e., the presence of multiple optimal bases to a linear programming (LP) problem, heavily affects the solution process of mixed integer programming (MIP) solvers. Different optimal bases lead to different cuts being generated, different branching decisions being taken and different solutions being found by primal heuristics. Nevertheless, only a few methods have been published that either avoid or exploit dual degeneracy. The aim of the present paper is to conduct a thorough computational study on the presence of dual degeneracy for the instances of well-known public MIP instance collections. How many instances are affected by dual degeneracy? How degenerate are the affected models? How does branching affect degeneracy: Does it increase or decrease by fixing variables? Can we identify different types of degenerate MIPs? As a tool to answer these questions, we introduce a new measure for dual degeneracy: the variable–constraint ratio of the optimal face. It provides an estimate for the likelihood that a basic variable can be pivoted out of the basis. Furthermore, we study how the so-called cloud intervals—the projections of the optimal face of the LP relaxations onto the individual variables—evolve during tree search and the implications for reducing the set of branching candidates.

Highlights

We analyze dual degeneracy emerging in linear programming (LP) relaxations solved during the solution process of generic mixed-integer programs (MIPs) in standard form: min{cT x : Ax = b, x ≥ 0, x j ∈ Z ∀ j ∈ J }
We performed a computational analysis of dual degeneracy in mixed integer programming (MIP) instances and demonstrated that it is very common in practical instances from standard MIP problem collections
We introduced a new metric, the variable–constraint ratio, and combined it with the share of degenerate non-basic variables to obtain an improved measure of dual degeneracy

Summary

Introduction

The y-coordinate of a point represents the average difference between the degeneracy of that instance at the corresponding depth level to its degeneracy rate at the root node. The variable–constraint ratio is larger than 1.2 for more than half of the instances up to depth level 17 Together, these numbers indicate that many variables can change their status from basic to non-basic or vice versa in alternative optimal LP solutions. We are interested in the case that the cloud interval contains an integer value, because this implies that there is at least one optimal solution for which that variable is integral This might have implications for solver components, e.g., branching on such variables should typically be avoided since the dual bound will not improve for at least one child node. Variable-constraint ratio variables with non-trivial cloud interval (%)

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Findings

Conclusions