Combinatorial Optimization

K. Subramani, LCSEE, West Virginia University January 28, 2013

1. Weak duality theorem.

2. Strong duality theorem.

3. Slack and Surplus variables.

4. Weak complementary slackness theorem.

5. Strong complementary slackness theorem.

6. Complementary slackness for canonical form. Let s = b − A · x and t = y · A − c. If x^{∗} and y^{∗} are the primal and dual optimal, then, x_{j}∗ · t_{j}^{∗} = 0, ∀j = 1, 2, … n and s_{i}∗ · u_{i}^{∗} = 0, ∀i = 1, 2, … m. Proof:

c · x^{∗} = (y^{∗} · A − t^{∗}) · x^{∗}
= y^{∗} · A · x^{∗} − t^{∗} · x^{∗}
= y^{∗} · (b − s^{∗}) − t^{∗} · x^{∗}
= y^{∗} · b − y^{∗} · s^{∗} − t^{∗} · y^{∗}
⇒ y^{∗} · s^{∗} + t^{∗} · y^{∗} = 0

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7. Graphical Solution.

8. The Simplex method (Samir’s notes).