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Unformatted text preview: Introduction to Mathematical Programming IE496 Quiz 2 Review Dr. Ted Ralphs IE496 Quiz 2 Review 1 Reading for The Quiz Material covered in detail in lecture Bertsimas 4.14.5, 4.8, 5.15.5, 6.16.3 Material covered briefly in lecture Bertsimas 4.6, 4.9 IE496 Quiz 2 Review 2 Deriving the Dual Problem Consider a standard form LP min { c T x : Ax = b,x } . To derive the dual problem , we use Lagrangian relaxation and consider the function g ( p ) = min x c T x + p T ( b Ax ) / in which infeasibility is penalized by a vector of dual prices p . For every vector p , g ( p ) is a lower bound on the optimal value of the original LP. To achieve the best bound, we considered maximizing g ( p ) , which is equivalent to maximize p T b s.t. p T A c This LP is the dual to the original one. IE496 Quiz 2 Review 3 From the Primal to the Dual We can dualize general LPs as follows PRIMAL minimize maximize DUAL b i constraints b i variables = b i free c j variables c j constraints free = c j IE496 Quiz 2 Review 4 Relationship of the Primal and the Dual The following are the possible relationships between the primal and the dual: Finite Optimum Unbounded Infeasible Finite Optimum Possible Impossible Impossible Unbounded Impossible Impossible Possible Infeasible Impossible Possible Possible IE496 Quiz 2 Review 5 Strong Duality and Complementary Slackness Theorem 1. ( Strong Duality ) If a linear programming problem has an optimal solution, so does its dual, and the respective optimal costs are equal. Theorem 2. If x and p are feasible primal and dual solutions, then x and p are optimal if and only if p T ( Ax b ) = 0 , ( c T p T A ) x = 0 . From complementary slackness, we can derive a number of alternative optimality conditions . The simplex algorithm always maintains complementary slackness IE496 Quiz 2 Review 6 LPs with General Upper and Lower Bounds In many problems, the variables have explicit nonzero upper or lower bounds. These upper and lower bounds can be dealt with implicitly instead of being included as constraints. In this more general framework, all nonbasic variables are fixed at either their upper or lower bounds . For minimization, variables eligible to enter the basis are either Variables at their lower bounds with negative reduced costs , or Variables at their upper bounds with positive reduced cost . When no such variables exist, we are at optimality . For maximization, we can just reverse the signs. IE496 Quiz 2 Review 7 Economic Interpretation The dual variables tell us the marginal change in the objective function per unit change in the righthand side of a constraint....
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 Spring '08
 Ralphs

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