The definition of a linear program is following: Find a vector x such that: <mo movab

Kyla Ayers

Kyla Ayers

Answered question


The definition of a linear program is following:

Find a vector x such that: min c T x, subject to A x = b and x 0.

Generally, b is assumed to be a fixed constant. However is it possible to construct a program where values of b are part of the optimization? Could I included b in the optimization by changing A x = b to A x b = 0. If so, would I also be able to place constraints upon b like b = 1 and 1 > b > 0? Finally, would such a program be possible to solve efficiently?

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.

Answer & Explanation



Beginner2022-06-15Added 30 answers

If you want b to be variable, yes, you can move it to the LHS and change the RHS to 0. An optimization modeling language would perform such transformations on your behalf. If you are instead directly using a solver that requires one constraint matrix and constant vectors for the objective and RHS, you will need to explicitly augment the matrix and vectors to accommodate the new variables.

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