How can I minimize $\|A*w\|$ for all w by minimizing $\|A\|$? (dimension of $A: n\times n$, dimension of $w: n\times1$)

Question

Question: Could I use problem 2 instead of problem 1? In other words, do problem 1 and problem 2 result in a similar optimal value for matrix K?

Problem 1: minimizing norm-2 of $(A+B*K)*(w)$ for all possible $w$; decision variable: $K$ Problem 2: minimizing matrix norm of $(A+B*K)$; decision variable: $K$

Symbols: There are two matrices $A$ and $B$ with known real values; dimension of $A$: $n$ by $n$, the dimension of B: n by 1 $K$ is a matrix which is not known and in fact, it is a variable; dimension of $K$: $1$ by $n$ w is a vector with real values which belongs to the compact and closed set W; dimension of $w$: n by $1$

My sense: Could I benefit from the following property to prove the equality of problem 1 and problem 2? $$\|(A+B*K)*(w)\|\le\|A+B*K\|*\|w\|$$

Answer 1

I think some constraints should be added, such as $\|w\|=1$, otherwise, the optimal solution must be $w=0$, and $\|(A+BK)w\|=0$.

$\min\limits_{\|w\|=1}\|(A+BK)w\|^2=\min\limits_{\|w\|=1} w^T(A+BK)^T(A+BK)w$

So, the minimal is $\lambda_{\min}(A+BK)^T(A+BK)$, where $\lambda_{\min}$ represents the minimal eigenvalue.

Indeed, problem 1 is equivalent to minimizing the smallest eigenvalue, which is different from minimizing the norm.