On optimizing over lift-and-project closures

The lift-and-project closure is the relaxation obtained by computing all lift-and-project cuts from the initial formulation of a mixed integer linear program or equivalently by computing all mixed integer Gomory cuts read from all tableau's corresponding to feasible and infeasible bases. In this paper, we present an algorithm for approximating the value of the lift-and-project closure. The originality of our method is that it is based on a very simple cut generation linear programming problem which is obtained from the original linear relaxation by simply modifying the bounds on the variables and constraints. This separation LP can also be seen as the dual of the cut generation LP used in disjunctive programming procedures with a particular normalization. We study some properties of this separation LP in particular relating it to the equivalence between lift-and-project cuts and Gomory cuts shown by Balas and Perregaard. Finally, we present some computational experiments and comparisons with recent related works

Lift-and-project inequalities

The lift-and-project technique is a systematic way to generate valid inequalities for a mixed binary program. The technique is interesting both on the theoretical and on the practical point of view. On the theoretical side it allows one to construct the inequality description of the convex hull of all mixed-{0,1} solutions of a binary MIP in n repeated applications of the technique, where n is the number of binary variables. On the practical side, a variant of the method allows one to derive some cutting planes from the simplex tableau rather efficiently

Lattice reformulation cuts

Here we consider the question whether the lattice reformulation of a linear integer program can be used to produce effective cutting planes. In particular, we aim at deriving split cuts that cut off more of the integrality gap than Gomory mixed-integer (GMI) inequalities generated from LP-tableaus, while being less computationally demanding than generating the split closure. We consider integer programs (IPs) in the form max{ Ax=b x =Zn+}, where the reformulation takes the form max\{cx +cQ> -xu u =Zn-m Z n - m\}, where Q is an n (n - m) integer matrix. Working on an optimal LP-tableau in the u -space allows us to generate n - m GMIs in addition to the m GMIs associated with the optimal tableau in the x space. These provide new cuts that can be seen as GMIs associated to n - m nonelementary split directions associated with the reformulation matrix \Q . On the other hand it turns out that the corner polyhedra associated to an LP basis and the GMI or split closures are the same whether working in the x or u spaces. Our theoretical derivations are accompanied by an illustrative computational study. The computations show that the effectiveness of the cuts generated by this approach depends on the quality of the reformulation obtained by the reduced basis algorithm used to generate Q and that it is worthwhile to generate several rounds of such cuts. However, the effectiveness of the cuts deteriorates as the number of constraints is increased

Generating general-purpose cutting planes for mixed-integer programs

Franz WesselmannPaderborn, Univ., Diss., 201

Branching via Cutting Plane Selection: Improving Hybrid Branching

Cutting planes and branching are two of the most important algorithms for solving mixed-integer linear programs. For both algorithms, disjunctions play an important role, being used both as branching candidates and as the foundation for some cutting planes. We relate branching decisions and cutting planes to each other through the underlying disjunctions that they are based on, with a focus on Gomory mixed-integer cuts and their corresponding split disjunctions. We show that selecting branching decisions based on quality measures of Gomory mixed-integer cuts leads to relatively small branch-and-bound trees, and that the result improves when using cuts that more accurately represent the branching decisions. Finally, we show how the history of previously computed Gomory mixed-integer cuts can be used to improve the performance of the state-of-the-art hybrid branching rule of SCIP. Our results show a 4\% decrease in solve time, and an 8\% decrease in number of nodes over affected instances of MIPLIB 2017

Algorithms for Stochastic Integer Programs Using Fenchel Cutting Planes

This dissertation develops theory and methodology based on Fenchel cutting planes for solving stochastic integer programs (SIPs) with binary or general integer variables in the second-stage. The methodology is applied to auto-carrier loading problem under uncertainty. The motivation is that many applications can be modeled as SIPs, but this class of problems is hard to solve. In this dissertation, the underlying parameter distributions are assumed to be discrete so that the original problem can be formulated as a deterministic equivalent mixed-integer program. The developed methods are evaluated based on computational experiments using both real and randomly generated instances from the literature. We begin with studying a methodology using Fenchel cutting planes for SIPs with binary variables and implement an algorithm to improve runtime performance. We then introduce the stochastic auto-carrier loading problem where we present a mathematical model for tactical decision making regarding the number and types of auto-carriers needed based on the uncertainty of availability of vehicles. This involves the auto-carrier loading problem for which actual dimensions of the vehicles, regulations on total height of the auto-carriers and maximum weight of the axles, and safety requirements are considered. The problem is modeled as a two-stage SIP, and computational experiments are performed using test instances based on real data. Next, we develop theory and a methodology for Fenchel cutting planes for mixed integer programs with special structure. Integer programs have to be solved to generate a Fenchel cutting plane and this poses a challenge. Therefore, we propose a new methodology for constructing a reduced set of integer points so that the generation of Fenchel cutting planes is computationally favorable. We then present the computational results based on randomly generated instances from the literature and discuss the limitations of the methodology. We finally extend the methodology to SIPs with general integer variables in the second-stage with special structure, and study different normalizations for Fenchel cut generation and report their computational performance

Topics in exact precision mathematical programming

The focus of this dissertation is the advancement of theory and computation related to exact precision mathematical programming. Optimization software based on floating-point arithmetic can return suboptimal or incorrect resulting because of round-off errors or the use of numerical tolerances. Exact or correct results are necessary for some applications. Implementing software entirely in rational arithmetic can be prohibitively slow. A viable alternative is the use of hybrid methods that use fast numerical computation to obtain approximate results that are then verified or corrected with safe or exact computation. We study fast methods for sparse exact rational linear algebra, which arises as a bottleneck when solving linear programming problems exactly. Output sensitive methods for exact linear algebra are studied. Finally, a new method for computing valid linear programming bounds is introduced and proven effective as a subroutine for solving mixed-integer linear programming problems exactly. Extensive computational results are presented for each topic.Ph.D.Committee Chair: Dr. William J. Cook; Committee Member: Dr. George Nemhauser; Committee Member: Dr. Robin Thomas; Committee Member: Dr. Santanu Dey; Committee Member: Dr. Shabbir Ahmed; Committee Member: Dr. Zonghao G

Combinatorial Optimization

This report summarizes the meeting on Combinatorial Optimization where new and promising developments in the field were discussed. Th