How to convert Frobenius norm minimization to quadratic optimization problem

Question

I am reading a paper titled "Cross-Calibration for Data Fusion of EO-1/Hyperion and Terra/ASTER" which mentions a particular minimization problem

I am not able to comprehend how this equation is converted (during implementation, in the code) to a quadratic optimization problem specified in the form of and how would the matrices $H$ and $f$ be computed.

I know the question is a bit ambiguous and might require more information. But I'm just looking for some guidance as to how to begin with the conversion between the two forms (Frobenius norm and Quadratic optimization)

Answer 1

Note. For ease of writing, everything is transposed in my answer, but should change anything in the implementation...

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So you have $L_m$ constrained problems of minimization of Frobenius norm. The objective function of the $i$th problem can be written as $ \|y_i - Xr_i\|^2 = \|y_i\|^2 - 2y_i^TXr_i + \|Xr_i\|^2$. Discarding the constant term $\|y_i\|^2$ (it doesn't depend on the optimization variable $r_i$), we can further rewrite the objective as $\frac{1}{2}r_i^T H r_i + f^Tr_i$, where $H := 2X^TX$ and $f_i := -2y_i^TX$.

As for the constraint $\|r_i - r_i^*\|_\infty \le \epsilon r_i^*$, well, as you've been told in (15), they can be rewritten as $Ar_i \le b$, where $A :=\begin{bmatrix}-I\\I\end{bmatrix}$ and $b_i=\begin{bmatrix}-(1-\epsilon)r_i^*\\(1+\epsilon)r^*_i\end{bmatrix}$.

Thus, all in all, your $L_m$ problems are QP problems of the form

$$ \text{minimize}\; \frac{1}{2}r_i^T H r_i + f_i^Tr_i\text{ subject to }Ar_i \le b_i,\; r_i \ge 0, \tag{$P_i$} $$

with the $H$, $f_i$, $A$, and $b_i$ as given above.

Answer 2

For your paper $\tilde{y}_i$ and $r_i$ are row vectors.

We have

\begin{align}\|\tilde{y_i}-r_iX_i\|_F^2 &= (\tilde{y_i}-r_iX_i)(\tilde{y_i}-r_iX_i)^T\\ &=\|\tilde{y_i}\|_F^2-2r_iX_i\tilde{y}_i^T+r_i X_i X_i^Tr_i\end{align}

which is in quadratic form.