Lecture5

# Lecture5 - C&O 355 Mathematical Programming Fall 2010...

This preview shows pages 1–6. Sign up to view the full content.

Mathematical Programming Fall 2010 Lecture 5 N. Harvey

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Review of our Theorems Fundamental Theorem of LP: Every LP is either Infeasible, Unbounded, or has an Optimal Solution. Not Yet Proven! Weak Duality Theorem: If x feasible for primal and y feasible for dual then c T x · b T y. Strong LP Duality Theorem: 9 x optimal for primal ) 9 y optimal for dual. Furthermore, c T x=b T y. Since the dual of the dual is the primal, we also get: 9 y optimal for dual ) 9 x optimal for primal. Primal LP: Dual LP:
Variant of Strong Duality Fundamental Theorem of LP: Every LP is either Infeasible, Unbounded, or has an Optimal Solution. Variant of Strong LP Duality Theorem: If primal is feasible and dual is feasible, then 9 x optimal for primal and 9 y optimal for dual. Furthermore, c T x=b T y. Proof: Since primal and dual are both feasible, primal cannot be unbounded. (by Weak Duality) By FTLP, primal has an optimal solution x. By Strong Duality, dual has optimal solution y and c T x=b T y. ¥ Primal LP: Dual LP:

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Proof of Variant from Farkas’ Lemma Theorem: If primal is feasible and dual is feasible, then 9 x optimal for primal and 9 y optimal for dual. Furthermore, c T x=b T y. Primal LP: Dual LP: Existence of optimal solutions is equivalent to solvability of { Ax · b, A T y = c, y ¸ 0, c T x ¸ b T y } We can write this as: Suppose this is unsolvable. Farkas’ Lemma: If Mp · d has no solution, then 9 q ¸ 0 such that q T M=0 and q T d < 0.
Farkas’ Lemma: If Mp · d has no solution, then 9 q ¸ 0 such that q T M=0 and q T d < 0. So if this is unsolvable, then there exists [ u, v

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 19

Lecture5 - C&O 355 Mathematical Programming Fall 2010...

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online