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On the exact separation of mixed integer knapsack cuts
 Proceedings of the 2007 Integer Programming and Combinatorial Optimization conference
, 2007
"... During the last decades, much research has been conducted deriving classes of valid inequalities for singlerow mixed integer programming polyhedrons. However, no such class has had as much practical success as the MIR inequality when used in cutting plane algorithms for general mixed integer progra ..."
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During the last decades, much research has been conducted deriving classes of valid inequalities for singlerow mixed integer programming polyhedrons. However, no such class has had as much practical success as the MIR inequality when used in cutting plane algorithms for general mixed integer programming problems. In this work we analyze this empirical observation by developing an algorithm which takes as input a point and a mixed integer knapsack polyhedron, and either proves the point is in the convex hull of said polyhedron, or finds a separating hyperplane, or knapsack cut. The main feature of this algorithm is a specialized subroutine for solving the Mixed Integer Knapsack Problem which exploits dominance relationships. To our knowledge, this is the first algorithm proposed for this problem. Exactly separating over the closure of mixed integer knapsack sets allows us to establish natural benchmarks by which to evaluate specific classes of knapsack cuts. Using these benchmarks on Miplib 3.0 and Miplib 2003 instances we analyze the performance of MIR inequalities. Our computations, which are performed in exact arithmetic, are surprising: Averaging over the 78 instances in which knapsack cuts afford bound improvements, MIR cuts alone achieve 95 % of the observed gain. 1
Existence and Welfare Properties of Equilibrium in an Exchange Economy with Multiple Divisible, Indivisible Commodities and Linear Production Technologies
"... In this paper we consider a class of economies with a nite number of divisible commodities, linear production technologies, and indivisible goods, and a finite number of agents. This class contains several wellknown economies with indivisible goods and money as special cases. It is shown that if th ..."
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In this paper we consider a class of economies with a nite number of divisible commodities, linear production technologies, and indivisible goods, and a finite number of agents. This class contains several wellknown economies with indivisible goods and money as special cases. It is shown that if the utility functions are continuous on the divisible commodities and are weakly monotonic both on one of the divisible commodities and on all the indivisible commodities, if each agent initially owns a sufficient amount of one of the divisible commodities, and if a "noproductionwithoutinput"like assumption on production sector holds, then there exists a competitive equilibrium for any economy in this class. The usual convexity assumption is not needed here. Furthermore, by imposing strong monotonicity on one of the divisible commodities we show that any competitive equilibrium is in the core of the economy and therefore the rst theorem of welfare also holds. We further obtain a second welfare theorem stating that under some condtions a Pareto efficient allocation can be sustained by a competitive equilibrium allocation for some wellchosen redistribution of the total initial endowments.
A New Constructive Proof to the Existence of an Integer Zero Point of a Mapping with the Direction Preserving Property
, 2009
"... Let f: Z n → R n be a mapping satisfying the direction preserving property that fi(x)> 0 implies fi(y) ≥ 0 for any integer points x and y with ∥x − y∥ ∞ ≤ 1. We assume that there is an integer point x 0 with c ≤ x 0 ≤ d satisfying that max 1≤i≤n (xi − x 0 i) fi(x)> 0 for any integer point ..."
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Let f: Z n → R n be a mapping satisfying the direction preserving property that fi(x)> 0 implies fi(y) ≥ 0 for any integer points x and y with ∥x − y∥ ∞ ≤ 1. We assume that there is an integer point x 0 with c ≤ x 0 ≤ d satisfying that max 1≤i≤n (xi − x 0 i) fi(x)> 0 for any integer point x with f (x) ∕ = 0 on the boundary of H = {x ∈ R n ∣ c − e ≤ x ≤ d + e}, where c and d are two finite integer points with c ≤ d and e = (1,1,⋅⋅ ⋅,1) ⊤ ∈ R n. This assumption is implied by one of two different conditions for the existence of an integer zero point of the mapping in van der Laan et al. (2004). Under the assumption, there is an integer point x ∗ ∈ H such that f (x ∗ ) = 0. A constructive proof of the existence is derived from an application of the wellknown (n + 1)ray algorithm for computing a fixed point. The existence result has applications in general equilibrium models with indivisible commodities.
ILIN: An Implementation of the Integer Labeling Algorithm for Integer Programming
"... this paper we discuss a practical implementation of the algorithm and present a computer program (ILIN) for solving integer programming using integer labeling algorithm. We also report on the solution of a number of tested examples with up to 500 integer variables. Numerical results indicate that th ..."
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this paper we discuss a practical implementation of the algorithm and present a computer program (ILIN) for solving integer programming using integer labeling algorithm. We also report on the solution of a number of tested examples with up to 500 integer variables. Numerical results indicate that the algorithm is computationally simple, flexible, efficient and stable.
Computing an Integer Point of a Class of Polytopes with an Arbitrary Starting Variable Dimension Algorithm
"... Abstract An arbitrary starting variable dimension algorithm is developed for computing an integer point of a polytope, P = {x  Ax ≤ b}, which satisfies that each row of A has at most one positive entry. The algorithm is derived from an integer labelling rule and a triangulation of the space. It con ..."
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Abstract An arbitrary starting variable dimension algorithm is developed for computing an integer point of a polytope, P = {x  Ax ≤ b}, which satisfies that each row of A has at most one positive entry. The algorithm is derived from an integer labelling rule and a triangulation of the space. It consists of two phases, one of which forms a variable dimension algorithm and the other a fulldimensional pivoting procedure. Starting at an arbitrary integer point, the algorithm interchanges from one phase to the other, if necessary, and follows a finite simplicial path that either leads to an integer point of the polytope or proves that no such point exists.
A SIMPLICIAL ALGORITHM FOR TESTING THE INTEGRAL PROPERTY OF A POLYTOPE
, 1994
"... Link to publication Citation for published version (APA): Yang, Z. F. (1994). A simplicial algorithm for testing the integral property of a polytope. (CentER Discussion Paper; Vol. 199475). CentER, Center for Economic Research. General rights Copyright and moral rights for the publications made acc ..."
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Link to publication Citation for published version (APA): Yang, Z. F. (1994). A simplicial algorithm for testing the integral property of a polytope. (CentER Discussion Paper; Vol. 199475). CentER, Center for Economic Research. General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.? Users may download and print one copy of any publication from the public portal for the purpose of private study or research? You may not further distribute the material or use it for any profitmaking activity or commercial gain? You may freely distribute the URL identifying the publication in the public portal Take down policy If you believe that this document breaches copyright, please contact us providing details, and we will remove access to the work immediately and investigate your claim. Download date: 12. mei. 2016 ~„ ~ ks ~ Discussion I l~lllllllhlul llllllll! II P INU~l ll~''III
Commodities and Linear Production Technologies 1
, 2000
"... Existence and welfare properties of equilibrium in an exchange economy with multiple divisible and indivisible commodities and linear production van der Laan, G.; Talman, Dolf; Yang, Z.F. ..."
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Existence and welfare properties of equilibrium in an exchange economy with multiple divisible and indivisible commodities and linear production van der Laan, G.; Talman, Dolf; Yang, Z.F.