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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
 Linear Programming, Optimization, optimality

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