Can anyone please help me with solving the constrained minimization problem below?
$$\mathbf{x}^* = \arg\min \sum_{i=1}^m q_i e^{-2x_i} $$ $$s.t.$$ $$\sum_{i=1}^m x_i = c$$ $$x_i\geq0, i = 1,\cdots,m$$
where $\mathbf{x}=[x_1,\cdots,x_m]^T$ is the unknown vector and $\mathbf{q}=[q_1,\cdots,q_m]^T$ and $c$ are given constants.
Do I need to consider the KKT conditions for this?
Or, can this problem be solved as a linear program by considering the exponential part as a sort of kernel?
Any reason not to solve directly as an NLP (nonlinear programming problem)? It seems convex and well-behaved. I had no problem with: