An optimization problem involving perceptron What I do not understand is that is there only one input for each weights or there is a vector of inputs from 1 to n?

I also dont get the notation for inputs and outputs? Is that a notation or a power operation?
The perceptron receives a $n$ number of $m$-dimensional vectors, so, $x^{i} \in \ R^{m}\  \forall\ i \in [1,n]$.

To iterate, the dataset has $n$ different examples and each of which has $m$ dimensions. It's not a power notation, it just denotes the $i$th example.