I`m trying to solve a maximization problem:

$\underset{\mathbf{p}}{max}w\text{ln}{\textstyle (}\sum _{i=1}^{I}{p}_{i}{a}_{i}{\textstyle )}-\sum _{i=1}^{I}{p}_{i}{d}_{i},$

where ${p}_{i}\in \{0,1\}$ is binary variable and $\sum _{i=1}^{I}{p}_{i}=1$. I need to find optimal ${p}_{i}^{\ast}$, where

${\mathbf{p}}^{\ast}=\mathrm{arg}\underset{\mathbf{p}}{max}w\text{ln}{\textstyle (}\sum _{i=1}^{I}{p}_{i}{a}_{i}{\textstyle )}-\sum _{i=1}^{I}{p}_{i}{d}_{i}$

Is there any way to obtain the closed-form of ${p}_{i}^{\ast}$ as the function of $w$, ${a}_{i}$, and ${d}_{i}$?

$\underset{\mathbf{p}}{max}w\text{ln}{\textstyle (}\sum _{i=1}^{I}{p}_{i}{a}_{i}{\textstyle )}-\sum _{i=1}^{I}{p}_{i}{d}_{i},$

where ${p}_{i}\in \{0,1\}$ is binary variable and $\sum _{i=1}^{I}{p}_{i}=1$. I need to find optimal ${p}_{i}^{\ast}$, where

${\mathbf{p}}^{\ast}=\mathrm{arg}\underset{\mathbf{p}}{max}w\text{ln}{\textstyle (}\sum _{i=1}^{I}{p}_{i}{a}_{i}{\textstyle )}-\sum _{i=1}^{I}{p}_{i}{d}_{i}$

Is there any way to obtain the closed-form of ${p}_{i}^{\ast}$ as the function of $w$, ${a}_{i}$, and ${d}_{i}$?