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

Relaxations of mixed integer sets from lattice-free polyhedra

This paper gives an introduction to a recently established link between the geometry of numbers and mixed integer optimization. The main focus is to provide a review of families of lattice-free polyhedra and their use in a disjunctive programming approach. The use of lattice-free polyhedra in the context of deriving and explaining cutting planes for mixed integer programs is not only mathematically interesting, but it leads to some fundamental new discoveries, such as an understanding under which conditions cutting planes algorithms converge finitel

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

A note on the split rank of intersection cuts

In this note, we present a simple geometric argument to determine a lower bound on the split rank of intersection cuts. As a first step of this argument, a polyhedral subset of the lattice-free convex set that is used to generate the intersection cut is constructed. We call this subset the restricted lattice-free set. It is then shown that ! log 2(l)mixed integer programming, split rank, intersection cuts.

Optimization with mixed-integer, complementarity and bilevel constraints with applications to energy and food markets

In this dissertation, we discuss three classes of nonconvex optimization problems, namely, mixed-integer programming, nonlinear complementarity problems, and mixed-integer bilevel programming. For mixed-integer programming, we identify a class of cutting planes, namely the class of cutting planes derived from lattice-free cross-polytopes, which are proven to provide good approximations to the problem while being efficient to compute. We show that the closure of these cuts gives an approximation that depends only on the ambient dimension and that the cuts can be computed efficiently by explicitly providing an algorithm to compute the cut coefficients in $O(n2^n)$ time, as opposed to solving a nearest lattice-vector problem, which could be much harder. For complementarity problems, we develop a first-order approximation algorithm to efficiently approximate the covariance of the decision in a stochastic complementarity problem. The method can be used to approximate the covariance for large-scale problems by solving a system of linear equations. We also provide bounds to the error incurred in this technique. We then use the technique to analyze policies related to the North American natural gas market. Further, we use this branch of nonconvex problems in the Ethiopian food market to analyze the regional effects of exogenous shocks on the market. We develop a detailed model of the food production, transportation, trade, storage, and consumption in Ethiopia, and test it against exogenous shocks. These shocks are motivated by the prediction that teff, a food grain whose export is banned now, could become a super grain. We present the regional effects of different government policies in response to this shock. For mixed-integer bilevel programming, we develop algorithms that run in polynomial time, provided a subset of the input parameters are fixed. Besides the $\Sigma^p_2$-hardness of the general version of the problem, we show polynomial solvability and $NP$-completeness of certain restricted versions of this problem. Finally, we completely characterize the feasible regions represented by each of these different types of nonconvex optimization problems. We show that the representability of linear complementarity problems, continuous bilevel programs, and polyhedral reverse-convex programs are the same, and they coincide with that of mixed-integer programs if the feasible region is bounded. We also show that the feasible region of any mixed-integer bilevel program is a union of the feasible regions of finitely many mixed-integer programs up to projections and closures

Novel Analysis of the Branch-and-Bound Method for Integer Programming

Mixed-integer linear programming (MILP) has become a pillar of operational decision making and optimization, with large-scale economic and societal impact. MILP solvers drive multi-billion dollar industries and the operation of critical infrastructure, and this ability to use MILPs to effectively make large-scale discrete decisions relies on the ability to solve MILPs efficiently. Despite a half-century of active research on the subject, critical components of these solvers' underlying algorithms remain poorly understood theoretically. This thesis provides novel and fundamental explanations for, and practical insights on, several long-analyzed phenomena in the branch-and-bound method, the workhorse algorithm of all state-of-the-art MILP solvers. In Chapter 1, we give some background on branch-and-bound and related works. These implementations of branch-and-bound typically use variable branching, that is, the child nodes are obtained by fixing some integer constrained variable to one of its possible values. Even though modern MILP solvers are able to solve very large-scale instances efficiently, relatively little attention has been given to understanding why the underlying branch-and-bound algorithm performs so well. In Chapter 2, our goal is to theoretically analyze the performance of the standard variable branching based branch-and-bound algorithm. In order to avoid the exponential worst-case lower bounds, we follow the common idea of considering random instances. More precisely, we consider random integer programs where the entries of the coefficient matrix and the objective function are randomly sampled. Our main result is that with good probability branch-and-bound with variable branching explores only a polynomial number of nodes to solve these instances, for a fixed number of constraints. To the best of our knowledge this is the first known such result for a standard version of branch-and-bound. We believe that this result provides an indication as to why branch-and-bound with variable branching works so well in practice. To understand the difficulties of branch-and-bound, in Chapter 3 we study an algorithm that can be viewed as an abstraction of modern MILP solvers: general branch-and-bound. That is, instances that are challenging for general branch-and-bound are likely to also be challenging for MILP solvers. A general branch-and-bound tree is a branch-and-bound tree which is allowed to use general disjunctions to create child nodes. We construct a packing instance, a set covering instance, and a Traveling Salesman Problem instance, such that any general branch-and-bound tree that solves these instances must be of exponential size. We also verify that an exponential lower bound on the size of general branch-and-bound trees persists even when we add Gaussian noise to the coefficients of the cross-polytope, thus showing that a polynomial-size ``smoothed analysis'' upper bound is not possible. Full strong-branching (henceforth referred to as strong-branching) is a well-known variable selection rule that is known experimentally to produce significantly smaller branch-and-bound trees in comparison to all other known variable selection rules. In Chapter 4, we attempt an analysis of the performance of the strong-branching rule both from a theoretical and a computational perspective. On the positive side for strong-branching we identify vertex cover as a class of instances where this rule provably works well. In particular, for vertex cover we present an upper bound on the size of the branch-and-bound tree using strong-branching as a function of the additive integrality gap, show how the Nemhauser-Trotter property of persistency which can be used as a pre-solve technique for vertex cover is being recursively and consistently used through-out the strong-branching based branch-and-bound tree, and finally provide an example of a vertex cover instance where not using strong-branching leads to a tree that has at least exponentially more nodes than the branch-and-bound tree based on strong-branching. On the negative side for strong-branching, we identify another class of instances where strong-branching based branch-and-bound tree has exponentially larger tree in comparison to another branch-and-bound tree for solving these instances. On the computational side, we conduct experiments on various types of instances like the lot-sizing problem and its variants, packing integer programs (IP), covering IPs, chance constrained IPs, vertex cover, etc., to understand how much larger is the size of the strong-branching based branch-and-bound tree in comparison to the optimal branch-and-bound tree. The main take-away from these experiments is that for all these instances, the size of the strong-branching based branch-and-bound tree is within a factor of two of the size of the optimal branch-and-bound tree. Finally, in Chapter 5 we discuss possible extensions of the work covered in this thesis.Ph.D