If $\alpha\,\, and\,\, p$ are non-negative scalars, are the two problems given below equivalent? I know that the values of objective functions will be different but I want to know will they result in same values of $\alpha\,\, and\,\, p$. If they are equivalent, how can I prove it mathematically without drawing graphs or any thing like that. I am not a mathematician so I want to know is it possible to have a 7,8 line prof that would satisfy a mathematician that they are equivalent? Also will they be equivalent for $\alpha\geq1$ and in general will they be equivalent for any set of constraints as long as both problems have the same constraints?
\begin{align} Problem 1: \min_{\alpha_{1},p_{1},\alpha_{2},p_{2}}\alpha_{1}p_{1}+\alpha_{2}p_{2},\\ \text{subject to}: \alpha_{1}log(1+p_{1} )\geq5,\\ \alpha_{2}log(1+p_{2})\geq5,\\ \alpha_{1}\,\, and\,\, \alpha_{2}\leq1.\\ Problem 2: \min_{\alpha_{1},p_{1},\alpha_{2},p_{2}}(\alpha_{1}p_{1})^{2}+(\alpha_{2}p_{2})^{2},\\ \text{subject to}: \alpha_{1}log(1+p_{1} )\geq5,\\ \alpha_{2}log(1+p_{2})\geq5,\\\alpha_{1}\,\, and\,\, \alpha_{2}\leq1 .\\ \end{align}
Both problems are separable into two independent problems in $(\alpha_1,p_1)$ and $(\alpha_2,p_2)$, namely $$ \min_{\alpha,p} \alpha p \quad \text{or} \quad \min_{\alpha,p}\,(\alpha p)^2 \\ \text{subject to } \alpha\log(1+p)\ge 5, \alpha \le 1. $$ Since $\alpha p$ is nonnegative, squaring it makes no difference to the location of the minimum.