How to minimize a two nexted summations?

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my problem is: I want to multiply a bidimensional array of values A by a vector of weights W (A*transpose(W)) but I do not know W but I know the result of the multiplication R, so, I want to estimate the weights W which multiplying A give R as a result.

I suppose it is minimization function by using a langragarian approach The function is

$ \sum_{i=1}^{l}(\sum_{k=1}^{R}a_{ijk}*w_k -v_i) $

with constraints

$\sum_{k=1}^{R}w_k = 1$

and

$w_k \geq 0$, k =0,1,2,.., R

but I do not know how to continue.

Thank you