710 research outputs found
Reformulation and decomposition of integer programs
In this survey we examine ways to reformulate integer and mixed integer programs. Typically, but not exclusively, one reformulates so as to obtain stronger linear programming relaxations, and hence better bounds for use in a branch-and-bound based algorithm. First we cover in detail reformulations based on decomposition, such as Lagrangean relaxation, Dantzig-Wolfe column generation and the resulting branch-and-price algorithms. This is followed by an examination of Benders’ type algorithms based on projection. Finally we discuss in detail extended formulations involving additional variables that are based on problem structure. These can often be used to provide strengthened a priori formulations. Reformulations obtained by adding cutting planes in the original variables are not treated here.Integer program, Lagrangean relaxation, column generation, branch-and-price, extended formulation, Benders' algorithm
Facets of a mixed-integer bilinear covering set with bounds on variables
We derive a closed form description of the convex hull of mixed-integer
bilinear covering set with bounds on the integer variables. This convex hull
description is determined by considering some orthogonal disjunctive sets
defined in a certain way. This description does not introduce any new
variables, but consists of exponentially many inequalities. An extended
formulation with a few extra variables and much smaller number of constraints
is presented. We also derive a linear time separation algorithm for finding the
facet defining inequalities of this convex hull. We study the effectiveness of
the new inequalities and the extended formulation using some examples
Separation and extension of cover inequalities for second-order conic knapsack constraints with GUBs
On the integration of Dantzig-Wolfe and Fenchel decompositions via directional normalizations
The strengthening of linear relaxations and bounds of mixed integer linear
programs has been an active research topic for decades. Enumeration-based
methods for integer programming like linear programming-based branch-and-bound
exploit strong dual bounds to fathom unpromising regions of the feasible space.
In this paper, we consider the strengthening of linear programs via a composite
of Dantzig-Wolfe and Fenchel decompositions. We provide geometric
interpretations of these two classical methods. Motivated by these geometric
interpretations, we introduce a novel approach for solving Fenchel sub-problems
and introduce a novel decomposition combining Dantzig-Wolfe and Fenchel
decompositions in an original manner. We carry out an extensive computational
campaign assessing the performance of the novel decomposition on the
unsplittable flow problem. Very promising results are obtained when the new
approach is compared to classical decomposition methods
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Zero-one IP problems: Polyhedral descriptions & cutting plane procedures
A systematic way for tightening an IP formulation is by employing classes of linear inequalities that define facets of the convex hull of the feasible integer points of the respective problems. Describing as well as identifying these inequalities will help in the efficiency of the LP-based cutting plane methods. In this report, we review classes of inequalities that partially described zero-one poly topes such as the 0-1 knapsack polytope, the set packing polytope and the travelling salesman polytope. Facets or valid inequalities derived from the 0-1 knapsack and the set packing polytopes are algorithmically identifie
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