# How can I show that these two problems have the same optimal solution?

How can I show that these two problems have the same optimal solution:

$$\inf \{ x^TAx + b^Tx : 1-x^Tx \ge 0,\ x \in \mathbb R^n\}$$

$$\inf \{ x^TAx + b^Tx : 1-x^Tx = 0,\ x \in \mathbb R^n\}$$

when $A$ is not positive semi-definite?

Start with $\|x^{\text{opt}}\|_2 < 1$ as a hypothetical optimal solution for $(1)$ and improve the value of the objective function by scaling $x^{\text{opt}}$. The infinmum exists because $\overline{B_1(0)}$ is compact in $\mathbb R^n$.
This gives a contradiction and shows that the infimum is attained on $\|x^{\text{opt}}\|_2 = 1$ so the problems are equivalent.
• Thank you! I am unfamiliar with the notation; what does $B_1(0)$ mean? – user192348 Dec 4 '14 at 22:27
• @user2340818 The (in this case closed) unit ball. Usually it refers to the open ball: $$B_r(x) = \{ y | \|x-y\|_2 < r \}$$ I changed the notation to use the open ball, wich is more standard. – AlexR Dec 4 '14 at 22:35
• @user2340818 We assumed $x^{\text{opt}}$ to be an interior point on wich the infimum is attained and we can see that $\frac1{\|x^{\text{opt}}\|_2} x^{\text{opt}}$ is a point with smaller objective function. – AlexR Dec 4 '14 at 22:38