hw3solns - 15-750 — Graduate Algorithms — Spring...

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Unformatted text preview: 15-750 — Graduate Algorithms — Spring 2009Miller and Dinitz and TangwongsanAssignment 3 Solutions1Duality Theory(25 pts.)Given a linear program min{c>x:Ax≥b,x≥}and a solutionx, how can one decide whetherxis an optimal solution? More generally, how can one calculate agoodlower bound on such linearprograms? Duality is a key concept in linear programming that can help answer these questions.Since we did not have time in class to cover such an important topic, you will learn about dualitytheory first-hand in this problem.More concretely, suppose we are given a linear program (which we will call the primal LP)minimizec>x,subject toAx≥bandx≥.The following LP is called theduallinear program:maximizey>b,subject toA>y≤candy≥.Several useful properties about the dual program can be derived. For our discussion, defineP={x∈Rd:Ax≥bandx≥}andD={A>y≤candy≥}.(a)Weak duality.Show that ifx∈Pandy∈D, theny>b≤c>x.Note that the dual program therefore gives a lower bound for a minimization LP. In fact,what you have just proven here implies the following theorem, known asweak duality.Theorem:(Weak duality) If the primal is a minimization linear program with op-timum valuez, then it has a dual, which is a maximization problem with optimumvaluewandw≤z.Solution:We will be working with two types of inequalities. To tell them apart, we will write≤cwfor the component-wise version and≤for the scalar one. Letx∈Pandy∈D. Webegin by observing that since≤cwyandb≤cwAx(becausexis feasble), we know thaty>b≤y>(Ax). By a similar reasoning, we have(A>y)>x≤c>x. Therefore, we establishy>b≤y>(Ax) = (A>y)>x≤c>x,where the equality follows from the matrix identity(AB)>=B>A>. This concludes the proof.(b)Strong duality.The strong duality theorem states that if both the primal and the dual arefeasible (i.e.,P6=∅), thenmin{c>x:x∈P}= max{y>b:y∈D}.In this part, you will prove the strong duality theorem. For simplicity, assume thatPandDare bounded. You can use the following lemma without providing a proof:15-750HW 3 Solutions2Farkas’s Lemma:LetA∈Rm×n, andb∈Rm. Exactly one of the followingholds:1. There existsx∈Rnsuch thatAx≤b.2. There existsy∈Rm≥such thatA>y=andb>y<0.Solution:Letz= min{c>x:x∈P}andw= max{y>b:y∈D}. Assume without loss ofgenerality thatAx≤balready contains the constraintx≥cw(otherwise, construct a new LPwith this constraint added in). We know from weak duality thatw≤z. To show thatw=z, itremains to show thatw≥z. Indeed, we are looking for a vectorysuch thatA>y≤cwcandb>y≥z. We will show that such a vector exists.Suppose for a contradiction that no suchyexists. Use Farkas’s lemma withA=A>-b>andb=c-z....
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hw3solns - 15-750 — Graduate Algorithms — Spring...

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