Helo, every one. May I ask for help about how to solve this problem.$\begin{align} &  \text{max}_{x_i} \quad |\sum_{i=1}^{4} a_i x_i | \\&  s.t. \quad \sum_{i=1}^4 x_i^2=1\end{align}$The goal is to find the optimal $x_i$, the $a_i$ is known.Thank you very much. Anyone can give me some tips?Best
How to solve this optimization problem with abs object function?
optimization;nonlinear programming
It looks like you are optimizing over the surface of a ball. That constraint is non-convex, but I think with your objective the solution will always be on the surface, so you can relax it to be $ \leq 1 $ and now its convex - thats a good start. So lets make that change.Now if you ignore the absolute sign, you are maximizing a linear objective over a ball. Thats easy. So solve two optimization problems$\begin{align} &  \text{max}_{x_i} \quad \sum_{i=1}^{4} a_i x_i  \\&  s.t. \quad \sum_{i=1}^4 x_i^2 \leq 1\end{align}$and$\begin{align} &  \text{max}_{x_i} \quad -\sum_{i=1}^{4} a_i x_i  \\&  s.t. \quad \sum_{i=1}^4 x_i^2 \leq 1\end{align}$and take the best of the two. Both these problems have a closed form solution - you can use the Karush-Kuhn-Tucker conditions to find it, or just throw it into a solver like Gurobi/CPLEX/Mosek/IPOPT/whatever you have a available that can handle quadratic constraints.