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5.1.7 (Duality Gap of the Knapsack Problem) www

5.1.7 (Duality Gap of the Knapsack Problem) www

Practical Management Science
Practical Management Science
6th Edition
ISBN:9781337406659
Author:WINSTON, Wayne L.
Publisher:WINSTON, Wayne L.
Chapter2: Introduction To Spreadsheet Modeling
Section: Chapter Questions
Problem 20P: Julie James is opening a lemonade stand. She believes the fixed cost per week of running the stand...
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5.1.7 (Duality Gap of the Knapsack Problem) www Given objects i = 1,... , n with positive weights w; and values vi, we want to assemble a subset of the objects so that the sum of the weights of the subset does not exceed a given A > 0, and the sum of the values of the subset is maximized. %3D (a) Show that this problem can be written as maximize > i=1 subject to > Wit; < A, Σ xi E {0,1}, i = 1,..., n. i=1 (b) Use a graphical procedure to solve the dual problem where a Lagrange multiplier is assigned to the constraint , w;X; < A. (c) Let f* and q* be the optimal values of the primal and dual problems, respectively. Show that 0 < q* – f* < max vi. | i=1,...,n (d) Consider the problem where A is multiplied by a positive integer k and each object is replaced by k replicas of itself, while the object weights and values stay the same. Let f* (k) and q*(k) be the corresponding optimal primal and dual values. Show that q*(k) - f*(k) f* (k) 1 maxi=1,...,n Vi f* k so that the relative value of the duality gap tends to 0 as k o.
Transcribed Image Text:5.1.7 (Duality Gap of the Knapsack Problem) www Given objects i = 1,... , n with positive weights w; and values vi, we want to assemble a subset of the objects so that the sum of the weights of the subset does not exceed a given A > 0, and the sum of the values of the subset is maximized. %3D (a) Show that this problem can be written as maximize > i=1 subject to > Wit; < A, Σ xi E {0,1}, i = 1,..., n. i=1 (b) Use a graphical procedure to solve the dual problem where a Lagrange multiplier is assigned to the constraint , w;X; < A. (c) Let f* and q* be the optimal values of the primal and dual problems, respectively. Show that 0 < q* – f* < max vi. | i=1,...,n (d) Consider the problem where A is multiplied by a positive integer k and each object is replaced by k replicas of itself, while the object weights and values stay the same. Let f* (k) and q*(k) be the corresponding optimal primal and dual values. Show that q*(k) - f*(k) f* (k) 1 maxi=1,...,n Vi f* k so that the relative value of the duality gap tends to 0 as k o.
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