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Unformatted text preview: Integer Programming IE418 Lecture 12 Menal Guzelsoy IE418 Lecture 12 1 Reading for This Lecture Duality for MixedInteger Linear Programs by Guzelsoy and Ralphs. IE418 Lecture 12 2 Duality Another method of obtaining a bound is to formulate a dual problem . Let a pure IP be defined by z IP = max { cx  x S} , S = { x Z n +  Ax = b } where c R n , A Q m n , b R m . We refer to this instance as the primal problem . A weak dual problem is an optimization problem of the form z D = min v V f ( v ) , with f : V R , V R k for k N such that z D z IP . A strong dual is a weak dual if z IP is finite and also z D = z IP . IE418 Lecture 12 3 Importance of Duality Note again that if we have a dual to IP , then we can easily obtain bounds on the value of an optimal solution. The advantage of a dual is that we need not solve it to optimality . Any feasible solution to the dual yields a valid bound. The three main categories of duals used most frequently are LP duals Combinatorial duals Lagrangian duals IE418 Lecture 12 4 The Duality Gap In the case of weak duals, there is a gap between the optimal solution to the dual problem and the optimal solution to IP . This gap is known as the duality gap or just the gap . It is typically measured as a percentage of the value of an optimal solution. The size of the gap is a measure of the difficulty of a problem. It can help us estimate how long it will take to solve a given problem by branch and bound. As a rule of thumb, problems with a gap of more than 510% are too difficult to solve in practice. Note that in most cases, we dont know the exact gap because we dont know the exact value of an optimal solution. Usually, the gap is estimated based on the best known solution. IE418 Lecture 12 5 Generalized Dual The previously introduced definition of a dual problem is not at useful, since the dual poroblem is not selected by any measure of goodness. Conceptually, we can improve the situation by choosing the best from a family of dual problem to obtain z D = min f,V min v V f ( v ) , where each pair ( f, V ) is required to comprise a dual problem....
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This note was uploaded on 08/06/2008 for the course IE 418 taught by Professor Ralphs during the Spring '08 term at Lehigh University .
 Spring '08
 Ralphs

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