Suppose I want to minimize the following matrix norm: $$\min_{\alpha\in\mathbb{R}, \beta\in\mathbb{R}^{n\times 1}} ||A-\alpha B-c*\beta'||_2, $$ where $A, B \in \mathbb{R}^{m\times n}$ and $c \in \mathbb{R}^{m\times1}$.
Are there any standard approaches to solving such a problem? Note, that in my case $B$ is a constant matrix consisting solely of ones.
$ \def\a{\alpha}\def\b{\beta} \def\qiq{\quad\implies\quad} $The problem can be broken into two subproblems, each of which has a closed-form for its least-squares solution $$\eqalign{ \a B &= A-c\b^T &\qiq \a = \frac{B:(A-c\b^T)}{B:B} \\ c\b^T &= A-\a B &\qiq \b^T = \frac{c^T(A-\a B)}{c^Tc} \\ }$$ where $(:)$ denotes the matrix inner product, which is a concise notation for the trace $$\eqalign{ A:B &= \sum_{i=1}^m\sum_{j=1}^n A_{ij}B_{ij} \;=\; {\rm trace}(A^TB) \\ A:A &= \|A\|^2_F \\ }$$ Alternate between solving for $\a$ then for $\b$ until convergence is obtained,
i.e. an Alternating Least Squares (ALS) approach.