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Unformatted text preview: CS 435 : Linear Optimization Fall 2008 Lecture 7: Maximising c T x Over a Convex Set Lecturer: Sundar Vishwanathan Computer Science & Engineering Indian Institute of Technology, Bombay 1 Convex Sets Recall the de nitions of a convex set from the previous lecture. Theorem 1 If S 1 and S 2 are two convex sets, then S 1 \ S 2 is a convex set. Proof: Let x 1 ; x 2 2 S 1 \ S 2 . Now since x 1 and x 2 belong to S 1 (which is convex), any convex combination of them lies in S 1 . Similarly we can say that this convex combination of x 1 and x 2 lies in S 2 . Thus the convex combination lies in S 1 \ S 2 . Thus S 1 \ S 2 is convex. £ We can use the previous theorem to give a proof of the fact that f x : Ax b g is a convex set. It is, in a sense, the same as the previous proof but easier to visualise. Consider A 1 x b 1 . All the points satisfying this inequality lie on one side of the hyperplane A 1 x = b 1 . This set is convex. If we take two points satisfying this inequality, it can be easily checked that so does every convex combination. Similarly the solution sets of the other inequalities A j x b j are also convex. Thus, Ax b , which is an intersection of all of these convex regions, is convex. 2 Maximize c T x Recall our quest. How do we maximize c T x over the set of all x satisfying Ax b ? We now know that Ax b is a convex set. In subsequent lectures we will have more to say about what it looks like. It is instructive to see how the value of the function...
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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