The things we know, usually minimization of a convex function, unique solution will exist.
My question is, maximization of a strictly convex function, will that give an unique maximum? If so how we can proof that. The objective function looks like below.
max $Ae^{-(d_{0}+d_{1})}+ Be^{-(d_{0}+d_{1}+d_{2})}+C$
A,B and C are constants and $d_{0},d_{1},d_{2}$ are euclidean distances.
In general, the maximum of a strictly convex function over a convex set is not guaranteed to be bounded or unique. A sufficient counter example for boundedness is $f(x)=x^2$, with a counterexample for uniqueness achieved by restricting $x$ to $[-1,1]$.
For this example (assuming $d_i\geq0$), the maximum occurs at $d_0=d_1=d_2=0$.