I am facing a challenging maximization problem involving a function with cosine terms and scalar parameters, and I would appreciate some insights or guidance on how to approach it effectively.
$$\max_{\phi_{1},\phi_{2},...,\phi_{N}} \sum_{m} \sum_{l=m+1} \cos(\phi_{m} - \phi_{l} + \delta_{m,l}) $$
I'm afraid that if I use the normal gradient descent or newton method will stuck on one of the local minimum of the cos function.
My idea is to approximate the cosinus function inside by the taylor serie expansion at x=0. Why choosing 0 because at x = 0, cos function attain its local max which is also a global max.
Second idea is to approximate it with pade approximations..
Please, Let me know if I'm going on the right way. Especially, the first idea.