Suppose that we are given a set of vectors {x(1),...,x(p)}, x(i) n, i = 1,...,p. Find the vector x n such that the average squared distance (norm) between and x(1),..., x(p),
is minimized. Use the SOSC to prove that the vector found above is a strict local minimizer. How is related to the centroid (or center of gravity) of the given set of points {x(1),...,x(p)}?"
"Prove the following generalization of the second-order sufficient condition:
Theorem: Let Ω be a convex subset of n, f 2 a real-valued function on Ω, and x* a point in Ω. Suppose that there exists c , c > 0, such that for all feasible directions d at x* (d ≠ 0), the following hold:
1. d∇f(x*) ≥ 0.
2. dF(x*)d ≥ c||d||2.
Then, x* is a strict local minimizer of f."