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Reformulations in Mathematical Programming: A Computational Approach
"... Mathematical programming is a language for describing optimization problems; it is based on parameters, decision variables, objective function(s) subject to various types of constraints. The present treatment is concerned with the case when objective(s) and constraints are algebraic mathematical ex ..."
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Mathematical programming is a language for describing optimization problems; it is based on parameters, decision variables, objective function(s) subject to various types of constraints. The present treatment is concerned with the case when objective(s) and constraints are algebraic mathematical expressions of the parameters and decision variables, and therefore excludes optimization of blackbox functions. A reformulation of a mathematical program P is a mathematical program Q obtained from P via symbolic transformations applied to the sets of variables, objectives and constraints. We present a survey of existing reformulations interpreted along these lines, some example applications, and describe the implementation of a software framework for reformulation and optimization.
Incremental Lower Bounds for Additive Cost Planning Problems
"... We present a novel method for computing increasing lower bounds on the cost of solving planning problems, based on repeatedly solving and strengthening the delete relaxation of the problem. Strengthening is done by compiling select conjunctions into new atoms, similar to theP m ⋆ construction. Becau ..."
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Cited by 15 (7 self)
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We present a novel method for computing increasing lower bounds on the cost of solving planning problems, based on repeatedly solving and strengthening the delete relaxation of the problem. Strengthening is done by compiling select conjunctions into new atoms, similar to theP m ⋆ construction. Because it does not rely on search in the state space, this method does not suffer some of the weaknesses of admissible search algorithms and therefore is able to prove higher lower bounds for many problems that are too hard for optimal planners to solve, thus narrowing the gap between lower bound and cost of the best known plan, providing better assurances of plan quality.
An investigation into Mathematical Programming for Finite Horizon Decentralized POMDPs, in "The
 Journal of Artificial Intelligence Research
"... Decentralized planning in uncertain environments is a complex task generally dealt with by using a decisiontheoretic approach, mainly through the framework of Decentralized Partially Observable Markov Decision Processes (DECPOMDPs). Although DECPOMDPS are a general and powerful modeling tool, sol ..."
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Cited by 10 (0 self)
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Decentralized planning in uncertain environments is a complex task generally dealt with by using a decisiontheoretic approach, mainly through the framework of Decentralized Partially Observable Markov Decision Processes (DECPOMDPs). Although DECPOMDPS are a general and powerful modeling tool, solving them is a task with an overwhelming complexity that can be doubly exponential. In this paper, we study an alternate formulation of DECPOMDPs relying on a sequenceform representation of policies. From this formulation, we show how to derive Mixed Integer Linear Programming (MILP) problems that, once solved, give exact optimal solutions to the DECPOMDPs. We show that these MILPs can be derived either by using some combinatorial characteristics of the optimal solutions of the DECPOMDPs or by using concepts borrowed from game theory. Through an experimental validation on classical test problems from the DECPOMDP literature, we compare our approach to existing algorithms. Results show that mathematical programming outperforms dynamic programming but is less efficient than forward search, except for some particular problems. The main contributions of this work are the use of mathematical programming for DECPOMDPs and a better understanding of DECPOMDPs and of their solutions. Besides, we argue that our alternate representation of DECPOMDPs could be helpful for designing novel algorithms looking for approximate solutions to DECPOMDPs. 1
Compact formulations as a union of polyhedra
 MATHEMATICAL PROGRAMMING
, 2006
"... We explore one method for finding the convex hull of certain mixed integer sets. The approach is to break up the original set into a small number of subsets, find a compact polyhedral description of the convex hull of each subset, and then take the convex hull of the union of these polyhedra. The re ..."
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Cited by 8 (7 self)
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We explore one method for finding the convex hull of certain mixed integer sets. The approach is to break up the original set into a small number of subsets, find a compact polyhedral description of the convex hull of each subset, and then take the convex hull of the union of these polyhedra. The resulting extended formulation is then compact, its projection is the convex hull of the original set, and optimization over the mixed integer set is reduced to solving a linear program over the extended formulation. The approach is demonstrated on three different sets: a continuous mixing set with an upper bound and a mixing set with two divisible capacities both arising in lotsizing, and a single node flow model with divisible capacities that arises as a subproblem in network design.
A Branchandcut Algorithm for Integer Bilevel Linear Programs
, 2008
"... We describe a rudimentary branchandcut algorithm for solving integer bilevel linear programs that extends existing techniques for standard integer linear programs to this very challenging computational setting. The algorithm improves on the branchandbound algorithm of Moore and Bard in that it u ..."
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Cited by 8 (0 self)
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We describe a rudimentary branchandcut algorithm for solving integer bilevel linear programs that extends existing techniques for standard integer linear programs to this very challenging computational setting. The algorithm improves on the branchandbound algorithm of Moore and Bard in that it uses cutting plane techniques to produce improved bounds, does not require specialized branching strategies, and can be implemented in a straightforward way using only linear solvers. An implementation built using software components available in the COINOR software repository is described and preliminary computational results presented.
On the Complexity of Selecting Disjunctions in Integer Programming
 SIAM Journal on Optimization
"... The imposition of general disjunctions of the form “πx ≤ π0 ∨ πx ≥ π0 + 1”, where π,π0 are integer valued, is a fundamental operation in both the branchandbound and cuttingplane algorithms for solving mixed integer linear programs. Such disjunctions can be used for branching at each iteration of t ..."
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Cited by 7 (3 self)
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The imposition of general disjunctions of the form “πx ≤ π0 ∨ πx ≥ π0 + 1”, where π,π0 are integer valued, is a fundamental operation in both the branchandbound and cuttingplane algorithms for solving mixed integer linear programs. Such disjunctions can be used for branching at each iteration of the branchandbound algorithm or to generate split inequalities for the cuttingplane algorithm. We first consider the problem of selecting a general disjunction and show that the problem of selecting an optimal such disjunction, according to specific criteria described herein, is N Phard. We further show that the problem remains N Phard even for binary programs or when considering certain restricted classes of disjunctions. We observe that the problem of deciding whether a given inequality is a split inequality can be reduced to one of the above problems, which leads to a proof that the problem is N Pcomplete. 1
On the connection of the SheraliAdams Closure and Border
 Bases, 2009, Working Paper, Technische Universität Darmstadt / Massachusetts Institute of Technology
"... ABSTRACT. The SheraliAdams liftandproject hierarchy is a fundamental construct in integer programming, which provides successively tighter linear programming relaxations of the integer hull of a polytope. We initiate a new approach to understanding the SheraliAdams procedure by relating it to me ..."
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Cited by 6 (4 self)
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ABSTRACT. The SheraliAdams liftandproject hierarchy is a fundamental construct in integer programming, which provides successively tighter linear programming relaxations of the integer hull of a polytope. We initiate a new approach to understanding the SheraliAdams procedure by relating it to methods from computational algebraic geometry. Our two main results are the equivalence of the SheraliAdams procedure to the computation of a border basis, and a refinement of the SheraliAdams procedure that arises from this new connection. We present a modified version of the border basis algorithm to generate a hierarchy of linear programming relaxations that are tighter than those of Sherali and Adams, and over which one can still optimize in polynomial time (for a fixed number of rounds in the hierarchy). In contrast to the wellknown Gröbner bases approach to integer programming, our procedure does not create primal solutions, but constitutes a novel approach of using computeralgebraic methods to produce dual bounds. 1.
Intersection cuts for nonlinear integer programming: Convexification techniques for structured sets
 Mathematical Programming
"... We study the generalization of split and intersection cuts from Mixed Integer Linear Programming to the realm of Mixed Integer Nonlinear Programming. Constructing such cuts requires calculating the convex hull of the difference of two convex sets with specific geometric structures. We introduce two ..."
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We study the generalization of split and intersection cuts from Mixed Integer Linear Programming to the realm of Mixed Integer Nonlinear Programming. Constructing such cuts requires calculating the convex hull of the difference of two convex sets with specific geometric structures. We introduce two techniques to give precise characterizations of such convex hulls and use them to construct split and intersection cuts for several classes of sets. In particular, we give simple formulas for split cuts for essentially all convex sets described by a single quadratic inequality and for more general intersection cuts for a wide variety of convex quadratic sets.
A RelaxandCut Framework for Gomory MixedInteger Cuts
"... Gomory MixedInteger Cuts (GMICs) are widely used in modern branchandcut codes for the solution of MixedInteger Programs. Typically, GMICs are iteratively generated from the optimal basis of the current Linear Programming (LP) relaxation, and immediately added to the LP before the next round of ..."
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Cited by 5 (2 self)
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Gomory MixedInteger Cuts (GMICs) are widely used in modern branchandcut codes for the solution of MixedInteger Programs. Typically, GMICs are iteratively generated from the optimal basis of the current Linear Programming (LP) relaxation, and immediately added to the LP before the next round of cuts is generated. Unfortunately, this approach is prone to instability. In this paper we analyze a different scheme for the generation of rank1 GMICs read from a basis of the original LP—the one before the addition of any cut. We adopt a relaxandcut approach where the generated GMICs are not added to the current LP, but immediately relaxed in a Lagrangian fashion. Various elaborations of the basic idea are presented, that lead to very fast— yet accurate—variants of the basic scheme. Very encouraging computational results are presented, with a comparison with alternative techniques from the literature also aimed at improving the GMIC quality. We also show how our method can be integrated with other cut generators, and successfully used in a cutandbranch enumerative framework.