how do I solve the equation with double sum?

Question

How do I solve the following optimization problem?

$$\min_{c}\ \sum_{i=1}^{P} \sum_{n=1}^{N} \left[ (q_{n}^{i} - c_n)^2 - (g^i)^2 \right]^2.$$

what are the different options available to achieve the solution?

Answer 1

So first of all, notice that the objective function splits into $N$ terms involving disjoint optimization variables. It therefore suffices to minimize each term individually: $$\min_{c_n}\ \sum_{i=1}^P \left[(q^i_n-c_n)^2-(g^i)^2\right]^2.$$ This objective function is a quartic polynomial in one variable $(c_n)$ that clearly attains its global minimum. Its derivative is $$\sum_{i=1}^P -4\left[(q^i_n-c_n)^2-(g^i)^2\right](q_n^i-c_n)$$ which is cubic. Use your favorite polynomial root-finding method (e.g. Jenkins-Traub) to solve for the real roots of the cubic (there will be one or three of them), and plug them into the above objective function to test for which one is the global minimum.