As an example, if I have a minimzation problem where my objective function is represented by a sphere in n dimensions (one dimension per decision variable), and all my constraints are linear, then does a unique solution always exist? If not, what conditions are required on the constraints for this to be the case?
I am particularly interested in situations where all the constraints are linear equalities.
Best Answer
Writing $f(x,y,z)=x^2+y^2+z^2$ (since it turned out to be what you meant) would be a lot more clear.
Since the function $f$ is strictly convex, it has unique point of minimum on a any convex set; in particular, on a set defined by linear constraints. (As Rahul Narain already said in comments).