233 research outputs found
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
Strong Bounds for Resource Constrained Project Scheduling: Preprocessing and Cutting Planes
Resource Constrained Project Scheduling Problems (RCPSPs) without preemption
are well-known NP-hard combinatorial optimization problems. A feasible RCPSP
solution consists of a time-ordered schedule of jobs with corresponding
execution modes, respecting precedence and resources constraints. In this
paper, we propose a cutting plane algorithm to separate five different cut
families, as well as a new preprocessing routine to strengthen resource-related
constraints. New lifted versions of the well-known precedence and cover
inequalities are employed. At each iteration, a dense conflict graph is built
considering feasibility and optimality conditions to separate cliques,
odd-holes and strengthened Chv\'atal-Gomory cuts. The proposed strategies
considerably improve the linear relaxation bounds, allowing a state-of-the-art
mixed-integer linear programming solver to find provably optimal solutions for
754 previously open instances of different variants of the RCPSPs, which was
not possible using the original linear programming formulations.Comment:
Light on the Infinite Group Relaxation
This is a survey on the infinite group problem, an infinite-dimensional
relaxation of integer linear optimization problems introduced by Ralph Gomory
and Ellis Johnson in their groundbreaking papers titled "Some continuous
functions related to corner polyhedra I, II" [Math. Programming 3 (1972),
23-85, 359-389]. The survey presents the infinite group problem in the modern
context of cut generating functions. It focuses on the recent developments,
such as algorithms for testing extremality and breakthroughs for the k-row
problem for general k >= 1 that extend previous work on the single-row and
two-row problems. The survey also includes some previously unpublished results;
among other things, it unveils piecewise linear extreme functions with more
than four different slopes. An interactive companion program, implemented in
the open-source computer algebra package Sage, provides an updated compendium
of known extreme functions.Comment: 45 page
Iterated Chvátal--Gomory cuts and the geometry of numbers
Chvátal--Gomory cutting planes (CG-cuts for short) are a fundamental tool in integer programming. Given any single CG-cut, one can derive an entire family of CG-cuts, by “iterating' its multiplier vector modulo one. This leads naturally to two questions: first, which iterates correspond to the strongest cuts, and, second, can we find such strong cuts efficiently? We answer the first question empirically, by showing that one specific approach for selecting the iterate tends to perform much better than several others. The approach essentially consists of solving a nonlinear optimization problem over a special lattice associated with the CG-cut. We then provide a partial answer to the second question, by presenting a polynomial-time algorithm that yields an iterate that is strong in a certain well-defined sense. The algorithm is based on results from the algorithmic geometry of numbers
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