I want to optimize the following equation in order to get the optimal $\beta$, where $U \in R^{m \times n}, V \in R^{m \times n},\beta \in R^{k}$.
Are three optimization equations are equal to obtain $\beta$?
First One:
It is NOT squared Frobenius norm in the 1st term.
$$\text{arg} \min_{\beta}\left\|U- \sum_{i=1}^{k} \beta_i V_i \right\|_F + \lambda\|\beta\|_2^2 $$
Second One:
It is squared Frobenius norm in 1st term compared First One.
$$\text{arg} \min_{\beta}\left\|U- \sum_{i=1}^{k} \beta_i V_i \right\|_F^{2} + \lambda\|\beta\|_2^2$$
Third One:
There is a coefficient $\frac{1}{2 \sigma n}$ which is also constant in the 1st term compared to First One.
$$\text{arg} \min_{\beta} \frac{1}{2\sigma n} \|U- \sum_{i=1}^{k} \beta_i V_i \|_F + \lambda\|\beta\|_2^2$$
I want to know if these three optimization are equal to obtain the optimal $\beta$? That is to say: the optimal $\beta$ should be same for the above three objective functions?
I understand the Second One can be equal to standard form ridge regression optimization problem as suggested by optimize weight coefficient: $\text{arg} \min_{\beta}\|U- \sum_{i=1}^{k} \beta_i V_i \|_F + \lambda\|\beta\|_2^2$
When optimizing, can I arbitrarily add or substract constant coefficients for 1st or 2nd terms, (like First one and Third one)?
Thanks.