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Unformatted text preview: etermine that no duality gap exists. Toward that
end, we must first define the Lagrangian function as Saddle Point Optimality Theorem  SPOT (Theorem 6.2.5): If there exists an ̅
and
̅
such that
̅
̅̅
̅ for all
and
, then ̅ solves problem
P and ̅ solves problem D with no duality gap; that is, ̅
̅.
Conceptually: ̅ ̅ is a high point with respect to all the u’s but a low point with
respect to all x’s. Proof:
We are given that ̅ ̅̅
̅ ̅
̅ For this to hold true for any ̅ ̅ ̅ ̅
, we must have
218 ̅ , so ̅ is feasible to P. Since we have
implies that ̅ ̅
̅ But we had ̅ ̅ ̅ , it must also be true for , which .
and ̅ , so that must mean that we have ̅ ̅ . Returning to what we were given in the theorem, the second half of the inequality
says that
̅̅ ̅ ̅ ̅ Since we just showed that ̅
̅ ̅ ̅
, we have ̅ From weak duality, we had ̅ ̅ ̅ ̅ ̅ . Therefore, we must have ̅ ̅ ̅. Relationship between SPOT and KKT Conditions If ̅ is a KKT point and ,
If ̅ ̅ satisfy SPOT, ̅ are convex for
̅ . Then ̅ ̅ satisfy SPOT.
, and , are differentiable, then ̅ is a KKT point. Conceptually: For convex problems, the Lagrange multipliers in the KKT system
serve as the multipliers in SPOT.
Summary The Lagrangian dual problem can be shown to have an optimal objective value that
is equal to that of problem P under certain conditions. was shown to be a concave function, which implies that a local maximum is a
global maximum. As a result, maximizing
is “relatively” easy, so it could be
more appealing than solving problem P. However, it is not always easy to optimize
, since a closed form solution for it is
frequently unavailable. Usually
is obtained by evaluating at a point , which
requires the minimization of a separate problem. Another issue is that
is not always differentiable, so we need to use algorithms
that will work on nondifferentiable functions. We have now had a brief introduction to the important aspects of Lagrangian Duality. In
the next topic, we will address some techniques for solving the Lagrangian Dual. 219...
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This document was uploaded on 11/28/2013.
 Fall '13

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