I want to construct a family o continuous functions but not Lipschitzians, defined on $[0,1]^{n}$, with a prescribed number of global minimums and "many" local minimuns.
For instance, consider
$$ F:=\{\sum_{i=1}^{n}|x_{i}-a_{i}|^{r_{i}}:a_{i},r_{i}\in (0,1)\} ,$$ then given any $f(x_{1},\ldots,x_{n})\in F$, is clear that $f$ is not Lipschitzian, and has one global minimum at $(a_{1},\ldots,a_{n})$.
How can I define a family "like" $F$ but with a prefixed number of global minimums (say, 1,2,..., m)? Some idea?
Also if I want to add "many" local minimums to such function, Maybe adding some trigonometric function?
Many thanks in advance for your comments.
Take $k$ point $a_1,...,a_k \in (0,1)^n$. Then define
$$F:[0,1]^n \rightarrow \mathbb{R}$$
$$F(x):=\min_{i\in\{1,...,n\}}||x-a_i||^\alpha, \quad 0 <\alpha <1$$
If we want add $a_{n+1}$ as a local minimum in $a_{n+1}$ consider $$G(x):=\min \{F(x), ||x-a_{n+1}||+\frac{1}{2}\}$$