So I have a collection of systems of equations, basically $n$ systems of equations, each composed of $k$ equations:$$\frac{a_1x_{1j}}{a_1x_{1j} + \cdots + a_kx_{kj}} + \log x_{1j} + 1 - B_{1j} = 0$$$$\frac{a_2x_{2j}}{a_1x_{1j} + \cdots + a_kx_{kj}} + \log x_{2j} + 1 - B_{2j} = 0$$$$\vdots$$$$\frac{a_kx_{kj}}{a_1x_{1j} + \cdots + a_kx_{kj}} + \log x_{kj} + 1 - B_{kj} = 0$$for $j = 1,...,n$, with the restriction that each $(x_{1j},...,x_{kj})$ lies on the simplex.As you can see the systems are all very similar to each other, the only difference being the $B_{ij}$ constant at the end.  Also if it helps all the $a_i$ and $B_{ij}$ are positive. I'm wondering first if these systems are actually simple enough that they have analytic solutions, and if not, then upon finding the solution to one of them (probably by using some sort of constraint-modified Newton's method), whether I can basically use this solution to immediately find the solution to the others.
Finding quick solution to a collection of systems of fairly simple but nonlinear equations
optimization;nonlinear equations