Is the Frobenius norm defining a Trust-Region for a matrix?

I am minimizing a non convex function $$f$$ defined over the positive semidefinite cone $$S_+^n$$ through linearization methods (sequential linear/convex approximation) and I would like to constraint the problem in such a way that at every iteration, the new matrix at step $$k+1$$, i.e. $$S^{(k+1)}$$ is "not so far" from the matrix at the previous iteration $$S^{(k)}$$ (This goes under the name sequential convex\linear approximation methods).

Formally:

$$\min_S f(S) \\ \text{s.t. } S \in S_+^n \\ \qquad dist(S,S^{(k)})< \epsilon$$

where $$dist(S,S^{(k)})$$ encodes a notion of trust-region.

Which is a good trust region in such a way the two matrix are similar? In other words, which metric between the two function can be defined in such a way the cost function evaluated at the two matrices does not differ so much?

What about $$\|S-S^{(k)}\|_{F}^2< \epsilon$$? Which other metrics are suitable for matrices?

Frobenius norm is an obvious one to consider. As appropriate, you could consider a weighted Frobenius norm, where elements are weighted according to their importance or perhaps (if known) sensitivity of the objective function to the particular elements. A rationale for use of Frobenius norm is that it is the length of the matrix step corresponding to the inner product distance of the matrix step (difference of the matrices), i.e., $$\sqrt{\text{trace}((S^{(k)-S)^T*(S^{(k)-S))} = \text{Frobenius norm}$$.