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Unformatted text preview: CS 435 : Linear Optimization Fall 2008 Lecture 15: Complementary Slackness Lecturer: Sundar Vishwanathan Computer Science & Engineering Indian Institute of Technology, Bombay 1 Complementary Slackness Primal Dual max c T x min y T b Ax b A T y = c y We proved this in the last lecture. Theorem 1 If the primal is feasible and the cost is bounded, then the dual is feasible and its cost is also bounded. Moreover, their optimum values coincide. The following theorem follows from the above theorem. Theorem 2 Consider an x and y , feasible in the primal and dual respectively. Then both are optimum i c T x = y T b . The proof is an exercise. One way is to use the hint given below. Note that c T x f = y T f Ax y T f b for any feasible x f and y f . Theorem 3 (Complementary Slackness) Consider an x and y , feasible in the primal and dual respectively. That is, Ax b and A T y = c ; y . Then c T x = y T b if and only if ( y ) i > ) A i x = b i ....
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