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

Question

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?

Answer 1

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.