Hi, why this statement is TRUE? Isn't the bounded set A can be non-convex? Thanks.
The LMO is simply minimizing a linear function over a set X and the set does not have to be convex. So it makes sense to write \(LMO_A\). The "convexity" of X we see in the slides result from the fact that we are using Frank-Wolfe to solve a optimization problem constrained on a convex set.
But why \( LMO_X (g)\) = \(LMO_A (g) \) ? X is a set larger than A. I didn't get it.
Intuitively \(LMO_X\) minimize a linear function over a convex set X so the minimizers must be located at the "corners" of X. As X=Conv(A), the corners of X must be inside the A. Thus \(LMO_X=LMO_A\).