everyone. I want to minimize an objective function \begin{equation} \min_{\mathbf{x},\mathbf{y}} \ \sum_{i=1}^M x_i e^{y_i}, \end{equation} where $\mathbf{x},\mathbf{y}\in \mathbb{R}^M$. As far as I know, the objective function is non-convex, because if $\mathbf{x},\mathbf{y}$ are scalar, we can derive the Hessian as \begin{equation} \mathbf{H} = \begin{bmatrix} 0 & e^y\\ e^y & xe^y \end{bmatrix}, \end{equation} which is negative definite, so the original objective function is concave.
So who can tell me how to minimize my objective function? Here I don't list the constraints. Thanks in advance.
You want your function to be stationary at the minimum that is $$df(x, y)=\nabla_{\bf{x}}f(x, y)\cdot{d\bf{x}}+\nabla_{\bf{y}}f(x, y)\cdot{d\bf{y}}=0$$