My optimization problem looks like this: $$ \underset{\boldsymbol{x}}{\max}\quad f\left( \boldsymbol{x} \right) +\underset{y\in Y}{\max}g\left( \boldsymbol{x},y \right) \\ \mathrm{s.t.} 0\le x_i \le x_{max}, \forall x_i \in \boldsymbol{x} $$ where $\boldsymbol{x}\in\mathbb{C}^{N\times1}$ is a vector and $y$ is an integer. $f(\boldsymbol{x})$ is concave with respect to $\boldsymbol{x}$, and $g\left( \boldsymbol{x},y \right)$ is concave with respect to $\boldsymbol{x}$ with given $y$.
By applying alternating optimization (AO) method, I can get the optimal $y^{\star}$ with given $\boldsymbol{\tilde{x}}\in X$ and get the optimal $\boldsymbol{x}^{\star}$ with given $\tilde{y}\in Y$. I want to solve this optimization problem without using AO method to get a better performance, but it is difficult for me to deal with the term $\underset{y\in Y}{\max}g\left( \boldsymbol{x},y \right) $. Is there any other way?
Just maximize $f(x)+g(x,y)$ subject to \begin{align} 0 \le x_i &\le x_\max &&\text{for all $i$} \\ y &\in Y \end{align}