I have a non-linear programming problem, in which all the inequality is linear and only the optimization goal is in a non-linear form. The problem is as following. $x_j$ is the variables and $a_{k,j}$ is constant with value being either $0$ or $1$. $y_k$ and $\sigma_j$ are also constant.
\begin{eqnarray} y_0 & \leq & a_{0,0}\cdot{}x_0 + a_{0,1}\cdot{}x_1 + \dots{} + a_{0,n-1}\cdot{}x_{n-1} \\ \nonumber y_1 & \leq & a_{1,0}\cdot{}x_0 + a_{1,1}\cdot{}x_1 + \dots{} + a_{1,n-1}\cdot{}x_{n-1} \\ \nonumber & \dots & \\ \nonumber y_{l-1} & \leq & a_{l-1,0}\cdot{}x_0 + a_{l-1,1}\cdot{}x_1 + \dots{} + a_{l-1,n-1}\cdot{}x_{n-1} \\ \nonumber \text{minimize} && \sigma_0\cdot{}e^{x_0} + \sigma_1\cdot{}e^{x_1} + \dots + \sigma_{n-1}\cdot{}e^{x_{n-1}} \end{eqnarray}
I would like to know there is any "easy" or programmable way to solve problem of this form. My current thinking is to transform this problem to a linear programming problem. But I have problems in two aspects:
1) is there a way to transfer this problem to a linear form?
2) is there any software (like solver) that can directly solve this problem? If possible, C or Java library are preferred.
PS: Not sure if useful, but in my problem setting, the coefficient matrix $[a_{k,j}]$ is very sparse, meaning that a lot of $a$ is of $0$ value.