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I am trying to solve a problem of the form:

minimize $ \quad f_0(X), \quad X\in\mathbb{R_{+}^{m \times n}}$

subject to $ \quad lb \leq x_{ij} \quad \forall i,j$

where:

$ f_0(X) = ||AX-B||_F, \quad A \in \mathbb{R_{+}^{k \times m}} ,B\in\mathbb{R_{+}^{k \times n}}$

$A$ and $C$ are both matrices of given values and this problem looks like a constrained least square problem except for the fact that X is a matrix instead of a vector.

Since most of the solvers accepts $f_0$ as a user defined function, I could pass $x$ as a vector and then reshape $x$ to $X$ during the evaluation of $f_0$, nevertheless I am not sure if this will invalidate the assumptions on which the solver relies (I am using a Trust-Region Constrained Algorithm). Thank you in advance for any help. I found the question Minimize $\|A-XB\|_F$ subject to $Xv=0$ that is closely related to mine, but my focus is more on which solver (if any) allows to treat the problem in the current formulation instead of how to convert it to a proper constrained least squares.

Royi
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Andrea Citrolo
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1 Answers1

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Using Kronecker Product identity you may transform your problem into a vector form:

$$ \arg \min_{\boldsymbol{x}} \frac{1}{2} {\left\| \boldsymbol{C} \boldsymbol{x} - \boldsymbol{b} \right\|}_{2}^{2}, \; \text{ subject to } {x}_{i} \geq l $$

Where $\boldsymbol{x} = \operatorname{vec} \left( \boldsymbol{X} \right)$, $\boldsymbol{b} = \operatorname{vec} \left( \boldsymbol{B} \right)$ and $\boldsymbol{C} = \boldsymbol{I} \otimes \boldsymbol{A}$.

This problem is Bounded Linear Least Squares / Box Constrained Linear Least Squares.
It has many optimized solvers. See Solving Box Constrained Linear Least Squares.

Royi
  • 10,050