57 research outputs found
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
Verifying integer programming results
Software for mixed-integer linear programming can return incorrect results for a number of reasons, one being the use of inexact floating-point arithmetic. Even solvers that employ exact arithmetic may suffer from programming or algorithmic errors, motivating the desire for a way to produce independently verifiable certificates of claimed results. Due to the complex nature of state-of-the-art MIP solution algorithms, the ideal form of such a certificate is not entirely clear. This paper proposes such a certificate format designed with simplicity in mind, which is composed of a list of statements that can be sequentially verified using a limited number of inference rules. We present a supplementary verification tool for compressing and checking these certificates independently of how they were created. We report computational results on a selection of MIP instances from the literature. To this end, we have extended the exact rational version of the MIP solver SCIP to produce such certificates
Strengthening Chvátal-Gomory cuts for the stable set problem
The stable set problem is a well-known NP-hard combinatorial optimization problem. As well as being hard to solve (or even approximate) in theory, it is often hard to solve in practice. The main difficulty is that upper bounds based on linear programming (LP) tend to be weak, whereas upper bounds based on semidefinite programming (SDP) take a long time to compute. We propose a new method to strengthen the LP-based upper bounds. The key idea is to take violated Chvátal-Gomory cuts and then strengthen their right-hand sides. Although the strengthening problem is itself NP-hard, it can be solved reasonably quickly in practice. As a result, the overall procedure proves to be capable of yielding competitive upper bounds in reasonable computing times
Branching on multi-aggregated variables
open5siopenGamrath, Gerald; Melchiori, Anna; Berthold, Timo; Gleixner, Ambros M.; Salvagnin, DomenicoGamrath, Gerald; Melchiori, Anna; Berthold, Timo; Gleixner, Ambros M.; Salvagnin, Domenic
Primal Cutting Plane Methods for the Traveling Salesman Problem
Most serious attempts at solving the traveling salesman problem (TSP)
are based on the dual fractional cutting plane approach, which
moves from one lower bound to the next.
This thesis describes methods for implementing a TSP
solver based on a primal cutting plane approach, which moves
from tour to tour with non-degenerate primal simplex pivots and
so-called primal cutting planes. Primal cutting
plane solution of the TSP has received scant attention in the
literature; this thesis seeks to redress this gap, and its findings
are threefold.
Firstly, we develop some theory around the computation of
non-degenerate primal simplex pivots, relevant to general primal
cutting plane computation. This theory guides highly efficient
implementation choices, a sticking point in prior studies.
Secondly, we engage in a practical study of several existing primal separation
algorithms for finding TSP cuts. These algorithms are
all conceptually simpler, easier to implement, or
asymptotically faster than their standard counterparts.
Finally, this thesis may constitute the first
computational study of the work of Fleischer, Letchford, and Lodi
on polynomial-time separation of simple domino parity
inequalities. We discuss exact and heuristic enhancements, including a
shrinking-style heuristic which makes the algorithm more suitable for
application on large-scale instances.
The theoretical developments of this thesis are integrated into a
branch-cut-price TSP solver which is used to obtain computational
results on a variety of test instances
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
Combining Precision Boosting with LP Iterative Refinement for Exact Linear Optimization
This article studies a combination of the two state-of-the-art algorithms for
the exact solution of linear programs (LPs) over the rational numbers, i.e.,
without any roundoff errors or numerical tolerances. By integrating the method
of precision boosting inside an LP iterative refinement loop, the combined
algorithm is able to leverage the strengths of both methods: the speed of LP
iterative refinement, in particular in the majority of cases when a
double-precision floating-point solver is able to compute approximate solutions
with small errors, and the robustness of precision boosting whenever extended
levels of precision become necessary. We compare the practical performance of
the resulting algorithm with both puremethods on a large set of LPs and
mixed-integer programs (MIPs). The results show that the combined algorithm
solves more instances than a pure LP iterative refinement approach, while being
faster than pure precision boosting. When embedded in an exact branch-and-cut
framework for MIPs, the combined algorithm is able to reduce the number of
failed calls to the exact LP solver to zero, while maintaining the speed of the
pure LP iterative refinement approach
Mixed Integer Conic Optimization and its Applications
In this dissertation, we present our work on the theory and applications of Mixed Integer Linear Optimization (MILO) and Mixed Integer Second Order Cone Optimization (MISOCO). The dissertation is separated in three parts.In the first part, we focus on th
Almost Symmetries and the Unit Commitment Problem
This thesis explores two main topics. The first is almost symmetry detection on graphs. The presence of symmetry in combinatorial optimization problems has long been considered an anathema, but in the past decade considerable progress has been made. Modern integer and constraint programming solvers have automatic symmetry detection built-in to either exploit or avoid symmetric regions of the search space. Automatic symmetry detection generally works by converting the input problem to a graph which is in exact correspondence with the problem formulation. Symmetry can then be detected on this graph using one of the excellent existing algorithms; these are also the symmetries of the problem formulation.The motivation for detecting almost symmetries on graphs is that almost symmetries in an integer program can force the solver to explore nearly symmetric regions of the search space. Because of the known correspondence between integer programming formulations and graphs, this is a first step toward detecting almost symmetries in integer programming formulations. Though we are only able to compute almost symmetries for graphs of modest size, the results indicate that almost symmetry is definitely present in some real-world combinatorial structures, and likely warrants further investigation.The second topic explored in this thesis is integer programming formulations for the unit commitment problem. The unit commitment problem involves scheduling power generators to meet anticipated energy demand while minimizing total system operation cost. Today, practitioners usually formulate and solve unit commitment as a large-scale mixed integer linear program.The original intent of this project was to bring the analysis of almost symmetries to the unit commitment problem. Two power generators are almost symmetric in the unit commitment problem if they have almost identical parameters. Along the way, however, new formulations for power generators were discovered that warranted a thorough investigation of their own. Chapters 4 and 5 are a result of this research.Thus this work makes three contributions to the unit commitment problem: a convex hull description for a power generator accommodating many types of constraints, an improved formulation for time-dependent start-up costs, and an exact symmetry reduction technique via reformulation
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