The above lecture is brought to you by Skillshare. In a previous post of mine, we introduced weak alternatives. As a small reminder, consider the following two sets
(1)
and
(2)
where
(3)
is the dual function and is the domain of the problem. Since we did not impose any convexity assumption on
‘s neither did we assume that our
‘s are affine, then all we can say about
and
is that they form weak alternatives. In other words,
- If
is feasible, then
is infeasible.
- If
is feasible, then
is infeasible.
In this lecture, we assume the following
are convex
‘s are affine, i.e.
such that
In that case, we write as
(4)
Thanks to the three conditions above, we could strengthen weak alternatives so that they form strong alternatives. That is to say
is feasible
is infeasible.
is feasible
is infeasible.
Indeed, strong alternatives are stronger since (unlike weak alternatives) if we know that one of the sets or
is infeasible, then the other has to be feasible.
In my YouTube lecture, I give two applications relating to linear inequalities and intersection of ellipsoids.
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