Minimum point of Constraint set

Question

Let $ c \in R^n $ be non-zero, and consider the problem of minimizing the function $f(x)=c^Tx $ on some constraint set $ S$. Show that a minimum point of this problem cannot lie in the interior of the set $S$.

score 1 · Answer 1 · answered Jul 23 '13 at 17:20

1

An interior point of extremum must be a critical point. The gradient of $f$ is $c$ (identically), which is a nonzero vector. Hence, $f$ has no interior points of extremum.

answered Jul 23 '13 at 17:20

40 votes

9,736

pitchounet · Answer 2 · 2013-07-23T17:16:58.787

0

If the constraint set $S$ in bounded, one answer I can think of (there is probably a more simple answer) is : $f$ is continuous, non-constant on $\mathbb{R}^{n}$ and harmonic. Then (as a consequence of the maximum principle for harmonic functions) a minimum point of $f$ lies on the boundary of $S$.

edited Jul 23 '13 at 17:16

answered Jul 23 '13 at 15:16

pitchounet

6,576

Yes, there is a simpler answer. – 40 votes Jul 23 '13 at 17:21

Minimum point of Constraint set

2 Answers2