Sorry in advance if this question is very simple for some. I need to minimize the sum of positive 2-D functions: $\sum_{i=1}^{N} f_i(k_0,k_1)$, and say that the values that optimize each of these functions are known, i.e., $k_{0,i}, k_{1,i}$. Is there a way to prove that the values $K_0, K_1$ that optimize the sum of the functions above satisfy the following
$\underset{i}{\mathrm{min}} \{k_{1,i}\} \leq K_1 \leq \underset{i}{\mathrm{max}} \{k_{1,i}\}$,
$\underset{i}{\mathrm{min}} \{k_{0,i}\} \leq K_0 \leq \underset{i}{\mathrm{max}} \{k_{0,i}\}$?
Best Answer
Already in one dimension we can construct functions which are counterexamples.
Here are three $k=\{0,1,2\}$:
$$f_k(x) = \exp\left(-\frac{|x-k|^{1.6}}{6}\right)\cdot (1-\text{sinc}(x-k)^2)^6$$ As we can see they do have local minima in between the individual minima but they have pairwise maximas exactly on each others minima. The global minimum for the sum is $0$ ( at $x = \pm \infty$ ):
For two dimensions we can for example create separable functions with similar properties, global minimum in the middle but local maxima overlapping each others minimum. For example a separable one: $\text{sinc}(x)\text{sinc}(y)$ or a radial one: $\text{sinc}(\sqrt{x^2+y^2})$, which looks like this: