- #1
smehdi
- 16
- 0
Given an ##N## dimensional binary vector ##\mathbf{v}## whose conversion to decimal is equal to ##j##, is there a way to linearly map the vector ##\mathbf{v}## to an ##{2^N}## dimensional binary vector ##\mathbf{e}## whose ##(j+1)##-th element is equal to ##1## (assuming the index starts from zero)?
For instance if ##\mathbf{v} = [1 \quad 0 \quad1]## or ##\mathbf{v} = [0 \quad 1 \quad1]## (that represent 5 and 3 in decimal, respectively), how ##\mathbf{v}## can be linearly mapped (with a unique mapper of course) to ##\mathbf{e}^\mathrm{T} = [0\quad 0\quad 0\quad 0\quad 0\quad 1 \quad 0 \quad 0]## and ##\mathbf{e}^\mathrm{T} = [0\quad 0\quad 0\quad 1 \quad 0\quad 0 \quad 0 \quad 0]##, respectively.
This is a part of an objective function of a MILP. I guess such a mapper does not exist but hopefully, I am wrong!
For instance if ##\mathbf{v} = [1 \quad 0 \quad1]## or ##\mathbf{v} = [0 \quad 1 \quad1]## (that represent 5 and 3 in decimal, respectively), how ##\mathbf{v}## can be linearly mapped (with a unique mapper of course) to ##\mathbf{e}^\mathrm{T} = [0\quad 0\quad 0\quad 0\quad 0\quad 1 \quad 0 \quad 0]## and ##\mathbf{e}^\mathrm{T} = [0\quad 0\quad 0\quad 1 \quad 0\quad 0 \quad 0 \quad 0]##, respectively.
This is a part of an objective function of a MILP. I guess such a mapper does not exist but hopefully, I am wrong!
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