Please let me know whether there are closed-form optimal results (or upper bound) for the following optimization problem:$$\max (\prod_{1\leq i\leq n}(x_i)^{y_i}-\prod_{1\leq i\leq n}(x_i-\alpha)^{y_i})$$with the following  constraints:$$x_i\geq \alpha; \sum_{1\leq i\leq n} x_i=1$$$$y_i\in\{0,1,\dots,m\}; \sum_{1\leq i\leq n} y_i=m$$Only $x_i,y_i$ are variables. $\alpha$ is known and  $0\leq \alpha \leq1/n$.Thank you very much.
Upper bound of an optimization problem
upper bounds;convex optimization