This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 primaldual 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 nonnegative factors (if we multiply by negative factors, the direction of the...
View
Full
Document
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
 ProfSundar
 Computer Science

Click to edit the document details