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Unformatted text preview: CS 435 : Linear Optimization Fall 2008 Lecture 16: Duality: Bounding the Primal Solution from Below Lecturer: Sundar Vishwanathan Computer Science & Engineering Indian Institute of Technology, Bombay In this lecture we will see another motivation for duality. However, before that, a quick recap. Consider the following primal-dual pair. P : max c T x D : min y T b Ax b A T y = c y The duality theorem in its complete avatar states the following. If P is feasible and has a nite maximum then D is feasible and the two optimum values coincide. If P is infeasible and D is feasible then D is unbounded. If P feasible and unbounded then D is infeasible. Exercise: Prove the theorem using the discussion in the last lecture and the duality theorem. 1 Duality from Lower Bounds Consider the following LP. max 14 x 1 + 7 x 2 + 22 x 3 + 10 x 4 s.t. 10 x 1 + 3 x 2 + 10 x 3 + 7 x 4 20 (1) 3 x 1 12 x 2 13 x 3 + 14 x 4 35 (2) 4 x 1 + 4 x 2 + 12 x 3 + 3 x 4 4 (3) Looking at the above equations closely we can see that an upper bound on the objective is 24. This is because if we add (1) and (3), we get 14 x 1 + 7 x 2 + 22 x 3 + 10 x 4 24. We could have multiplied the above equations by any non-negative factors (if we multiply by negative factors, the direction of the...
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This note was uploaded on 01/04/2010 for the course CSE CS435 taught by Professor Profsundar during the Summer '09 term at IIT Bombay.
- Summer '09
- Computer Science