Lecture #5 clarification

Kevin asked in class: in the “geometric” proof of duality we talked about, did we need the fact that existed a BFS, i.e., that there were {n} tight constraints? (Recall we were considering the LP {\max\{ c^\top x \mid Ax \leq b\}}; since we were not talking about equational form LPs, the optimum could be achieved at a non-BFS.) He was right: we don’t. Here’s a proof.

Pick an optimal feasible point for the LP, call this point {x^*}. Let {a_i^\top x \leq b_i} for {i \in I} be all the constraints tight at {x^*}—there may be {n} linearly independent constraints in there, or there may be fewer, we don’t care. Now the claim is that the objective function vector {c} is contained in the cone {K = \{ x \mid x = \sum_{i \in I} \lambda_i a_i, \lambda_i \geq 0 \}} generated by these vectors {\{a_i\}_{i \in I}}. If we prove this claim, we’re done–the rest of the argument follows that in lecture.

(Indeed, this was precisely where we were being hand-wavy earlier, even when we assumed {x^*} was a BFS. So this also answers Jamie’s request.)

Ok, back to business at hand. Suppose {c} does not lie in the cone {K} generated by the vectors {\{a_i\}_{i \in I}}. Then there must be a separating hyperplane between {c} and {K}: i.e., there exists a vector {d \in {\mathbb R}^n} such that {a_i^\top d \leq 0} for all {i \in I}, but {c^\top d > 0}. So now consider the point {z = x^* + \epsilon d} for some tiny {\epsilon > 0}. Note the following:

  • We claim that for small enough {\epsilon > 0}, the point {z} satisfies the constraints {Ax \leq b}. Consider {a_j^\top x \leq b} for {j \not \in I}: since this constraint was not tight, we won’t violate it if we choose {\epsilon} small enough. And for {a_j^\top x \leq b} with {j \in I}? See, {a_j^\top z = a_j^\top x + \epsilon\, a_j^\top d = b + \epsilon\, a_j^\top d \leq b}, since {\epsilon > 0} and {a_j^\top d \leq 0}.
  • What about the objective function value? {c^\top z = c^\top x^* + \epsilon\, c^\top d > c^\top x^*}.

But this contradicts the fact that {x^*} was optimal.

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