daniel suriya 2022-07-19
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Find a vector x such that: min c T x, subject to A x = b and x 0.

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I am trying to solve the linear program for Wasserstein Distance between two discrete distributions. In the standard case, b represents the marginals for each datapoint. I know the marginals for the target distribution but the marginals from my source distribution are unknown. I am wondering if there is an efficient way to optimize the marginals for my source distribution such that the Wasserstein distance is minimized.
asked 2022-06-01
Find x 1 , x 2 such that min x 1 , x 2 x 1 2 + 2 x 1 x 2 where x 1 , x 2 are subject to constraint x 1 2 x 2 10.
I have changed the constraint into the equality x 1 2 x 2 10 s 2 = 0 and attained the gradients which result in 4 equations and 4 unknowns:
x 1 2 x 2 10 s 2 = 0 2 x 1 + 2 x 2 = λ ( 2 x 1 x 2 ) 2 x 1 = λ x 1 2 0 = λ ( 2 s )
But I am unsure of how to proceed from here. Additionally, I am struggling to find the dual problem.
asked 2022-06-24
Suppose that p 1 , , p n are nonnegative real numbers such that p 1 + + p n = 1; denote the corresponding set of vectors by Δ n .

I am interested in the following function, f : Δ n R + , given by
f ( p ) = k = 1 n p k j = k n p j .
We always have f ( p ) n, using the lower bound j k p j p k . However I feel this must be a loose bound on the quantity
sup p Δ n f ( p ) ,
since it requires that j > k p j = 0 for all k to be met with equality. Hence, I am wondering what the largest f ( p ) can be when evaluated over the simplex?

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