I don't think this question is well posed.
One problem is that I don't know what you mean by linear function. Are you looking for the intersection of two hyper planes? If yes, this should be a set of dimension N-1, not a simple point.
Let's assume instead f is an arbitrary function, it could have numerous local minima in the region of space where all coordinate values are less than zero. Which minimum should we then choose?
There is a similar problem, solved by Lagrange Multipliers. This optimization problem is well stated, so maybe you can use it as an example to reformulate your question. Until then I don't think we will see a satisfactory answer.