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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.

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    The objective function is the inner product of two vectors: $$ and $$ and $i=1,\cdots ,M$. if you set $y_i=0$ and $x_i \rightarrow -\infty $ then the minimum of the function is clearly unbounded. – 2017-02-20
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    Actually, $x_i, \forall i$ are Boolean, and I have a constraint on $y_i$ as $\sum_{i=1}^My_i \leq y_{\max}$. So my idea to optimize the objective function is that, first fix $y$ as initialization and use convex programming to solve $x$; then based on the obtained $x$ solve $y$. Does this idea make sense? Can it obtain global optimum? Or does anyone know some optimization techniques for solving this kind of problems? – 2017-02-20

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You want your function to be stationary at the minimum that is $$df(x, y)=\nabla_{\bf{x}}f(x, y)\cdot{d\bf{x}}+\nabla_{\bf{y}}f(x, y)\cdot{d\bf{y}}=0$$

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    Actually, $x_i, \forall i$ are Boolean, and I have a constraint on $y_i$ as $\sum_{i=1}^My_i \leq y_{\max}$. So my idea to optimize the objective function is that, first fix $y$ as initialization and use convex programming to solve $x$; then based on the obtained $x$ solve $y$. Does this idea make sense? Can it obtain global optimum? Or does anyone know some optimization techniques for solving this kind of problems? – 2017-02-20