Algorithm for multidimensional optimization / root-finding / something

Question

I have five values, A, B, C, D and E.

Given the constraint A + B + C + D + E = 1, and five functions F(A), F(B), F(C), F(D), F(E), I need to solve for A through E such that F(A) = F(B) = F(C) = F(D) = F(E).

What's the best algorithm/approach to use for this? I don't care if I have to write it myself, I would just like to know where to look.

EDIT: These are nonlinear functions. Beyond that, they can't be characterized. Some of them may eventually be interpolated from a table of data.

Answer 1

There is no general answer to this question. A solver finding the solution to any equation does not exist. As Lance Roberts already says, you have to know more about the functions. Just a few examples

• If the functions are twice differentiable, and you can compute the first derivative, you might try a variant of Newton-Raphson
• Have a look at the Lagrange Multiplier Method for implementing the constraint.
• If the function F is continuous (which it probably is, if it is an interpolant), you could also try the Bisection Method, which is a lot like binary search.

Before you can solve the problem, you really need to know more about the function you're studying.

Answer 2

As others have already posted, we do need some more information on the functions. However, given that, we can still try to solve the following relaxation with a standard non-linear programming toolbox.

min k
st.
A + B + C + D + E = 1
F1(A) - k = 0
F2(B) - k = 0
F3(C) -k = 0
F4(D) - k = 0
F5(E) -k = 0

Now we can solve this in any manner we wish, such as penalty method

min k + mu*sum(Fi(x_i) - k)^2 st
A+B+C+D+E = 1

or a straightforward SQP or interior-point method.

More details and I can help advise as to a good method.

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