We consider two minimization problems of exponential functions. Using numerical analysis we find that the two problems, when properly parameterized, have equivalent solutions. Can the equivalence of the problems be proved?
The problems are parameterized by positive scalars $s,s_1$ and $s_2$ and with decision variables that are vectors represented by $\boldsymbol{w}$. Let $\boldsymbol{x}$ be a vector conformable with $\boldsymbol{w}$ and with elements that are random variables. The first problem is
\begin{equation} \boldsymbol{w_1} = \underset{\boldsymbol{w}}{\mathrm{argmin}} \text{ } E[\exp{(-s_1 \boldsymbol{x'w})}]+ E[\exp{(-s_2 \boldsymbol{x'w})}] \label{mixed} \end{equation}
and the second problem is
\begin{equation} \boldsymbol{w_2} = \underset{\boldsymbol{w}}{\mathrm{argmin}} \text{ } E[\exp{(-s \boldsymbol{x'w})}] \label{single} \end{equation}
Using numerical analysis we find there is always an $s$, $s_1 \leq s \leq s_2$, such that $\boldsymbol{w_1} = \boldsymbol{w_2}$.
Can this be proven analytically? Any pointers will be appreciated.