When Lift-and-Project Cuts are Different

In this paper, we present a method to determine if a lift-and-project cut for a mixed-integer linear program is irregular, in which case the cut is not equivalent to any intersection cut from the bases of the linear relaxation. This is an important question due to the intense research activity for the past decade on cuts from multiple rows of simplex tableau as well as on lift-and-project cuts from non-split disjunctions. While it is known since Balas and Perregaard (2003) that lift-and-project cuts from split disjunctions are always equivalent to intersection cuts and consequently to such multi-row cuts, Balas and Kis (2016) have recently shown that there is a necessary and sufficient condition in the case of arbitrary disjunctions: a lift-and-project cut is regular if, and only if, it corresponds to a regular basic solution of the Cut Generating Linear Program (CGLP). This paper has four contributions. First, we state a result that simplifies the verification of regularity for basic CGLP solutions from Balas and Kis (2016). Second, we provide a mixed-integer formulation that checks whether there is a regular CGLP solution for a given cut that is regular in a broader sense, which also encompasses irregular cuts that are implied by the regular cut closure. Third, we describe a numerical procedure based on such formulation that identifies irregular lift-and-project cuts. Finally, we use this method to evaluate how often lift-and-project cuts from simple $t$-branch split disjunctions are irregular, and thus not equivalent to multi-row cuts, on 74 instances of the MIPLIB benchmarks.Comment: INFORMS Journal on Computing (to appear

Tight Bounds for Gomory-Hu-like Cut Counting

By a classical result of Gomory and Hu (1961), in every edge-weighted graph $G=(V,E,w)$, the minimum $st$-cut values, when ranging over all $s,t\in V$, take at most $|V|-1$ distinct values. That is, these $\binom{|V|}{2}$ instances exhibit redundancy factor $\Omega(|V|)$. They further showed how to construct from $G$ a tree $(V,E',w')$ that stores all minimum $st$-cut values. Motivated by this result, we obtain tight bounds for the redundancy factor of several generalizations of the minimum $st$-cut problem. 1. Group-Cut: Consider the minimum $(A,B)$-cut, ranging over all subsets $A,B\subseteq V$ of given sizes $|A|=\alpha$ and $|B|=\beta$. The redundancy factor is $\Omega_{\alpha,\beta}(|V|)$. 2. Multiway-Cut: Consider the minimum cut separating every two vertices of $S\subseteq V$, ranging over all subsets of a given size $|S|=k$. The redundancy factor is $\Omega_{k}(|V|)$. 3. Multicut: Consider the minimum cut separating every demand-pair in $D\subseteq V\times V$, ranging over collections of $|D|=k$ demand pairs. The redundancy factor is $\Omega_{k}(|V|^k)$. This result is a bit surprising, as the redundancy factor is much larger than in the first two problems. A natural application of these bounds is to construct small data structures that stores all relevant cut values, like the Gomory-Hu tree. We initiate this direction by giving some upper and lower bounds.Comment: This version contains additional references to previous work (which have some overlap with our results), see Bibliographic Update 1.

A Finite-Time Cutting Plane Algorithm for Distributed Mixed Integer Linear Programming

Many problems of interest for cyber-physical network systems can be formulated as Mixed Integer Linear Programs in which the constraints are distributed among the agents. In this paper we propose a distributed algorithm to solve this class of optimization problems in a peer-to-peer network with no coordinator and with limited computation and communication capabilities. In the proposed algorithm, at each communication round, agents solve locally a small LP, generate suitable cutting planes, namely intersection cuts and cost-based cuts, and communicate a fixed number of active constraints, i.e., a candidate optimal basis. We prove that, if the cost is integer, the algorithm converges to the lexicographically minimal optimal solution in a finite number of communication rounds. Finally, through numerical computations, we analyze the algorithm convergence as a function of the network size.Comment: 6 pages, 3 figure

A parallelizable algorithmic framework for solving large scale multi-stage stochastic mixed 0-1 problems under uncertainty

Preprint submitted to Computers & Operations Researchmulti-stage stochastic mixed 0-1 optimization, nonsymmetric scenario trees, implicit and explicit nonanticipativity constraints, splitting variable and compact representations, scenario cluster partitioning

Computational Experiments with Cross and Crooked Cross Cuts

In this paper, we study whether cuts obtained from two simplex tableau rows at a time can strengthen the bounds obtained by Gomory mixed-integer (GMI) cuts based on single tableau rows. We also study whether cross and crooked cross cuts, which generalize split cuts, can be separated in an effective manner for practical mixed-integer programs (MIPs) and can yield a nontrivial improvement over the bounds obtained by split cuts. We give positive answers to both these questions for MIPLIB 3.0 problems. Cross cuts are a special case of the t-branch split cuts studied by Li and Richard [Li Y, Richard J-PP (2008) Cook, Kannan and Schrijvers's example revisited. Discrete Optim. 5:724–734]. Split cuts are 1-branch split cuts, and cross cuts are 2-branch split cuts. Crooked cross cuts were introduced by Dash, Günlük, and Lodi [Dash S, Günlük O, Lodi A (2010) MIR closures of polyhedral sets. Math Programming 121:33–60] and were shown to dominate cross cuts by Dash, Günlük, and Molinaro [Dash S, Günlük O, Molinaro M (2012b) On the relative strength of different generalizations of split cuts. IBM Technical Report RC25326, IBM, Yorktown Heights, NY].United States. Office of Naval Research (Grant N000141110724

Safe and Verified Gomory Mixed Integer Cuts in a Rational MIP Framework

This paper is concerned with the exact solution of mixed-integer programs (MIPs) over the rational numbers, i.e., without any roundoff errors and error tolerances. Here, one computational bottleneck that should be avoided whenever possible is to employ large-scale symbolic computations. Instead it is often possible to use safe directed rounding methods, e.g., to generate provably correct dual bounds. In this work, we continue to leverage this paradigm and extend an exact branch-and-bound framework by separation routines for safe cutting planes, based on the approach first introduced by Cook, Dash, Fukasawa, and Goycoolea in 2009. Constraints are aggregated safely using approximate dual multipliers from an LP solve, followed by mixed-integer rounding to generate provably valid, although slightly weaker inequalities. We generalize this approach to problem data that is not representable in floating-point arithmetic, add routines for controlling the encoding length of the resulting cutting planes, and show how these cutting planes can be verified according to the VIPR certificate standard. Furthermore, we analyze the performance impact of these cutting planes in the context of an exact MIP framework, showing that we can solve 21.5% more instances and reduce solving times by 26.8% on the MIPLIB 2017 benchmark test set