Minimizing $\| x A - B\|_F^2$ With a Constraint

Question

I have previously asked an optimization question Here. I will reiterate the question and simply add a constraint to it:

I have 2 known grayscale images (256×256 matrices) $A$ and $B$ and want to find the unknown scalar variable $x$ so that:

$$\text{Minimize} \quad \|xA-B\|_F^2$$ $$\text{So That All Of The Final Image Values (256x256) Remain Between} \quad 0 \leq xA-B \leq 256 $$

The previous solution works just fine but I want to know how to optimize $x$ considering the constraint. Is this considered a Linear Programming problem?

Thanks.

You'll get simpler answers if your constraint is that $0\le xA-B\le256$ componentwise. — kimchi lover, May 17 '20 at 12:45
@kimchilover Thanks. I edited the question to what you suggested. — Cypher, May 17 '20 at 13:10

score 1 · Answer 1 · answered May 17 '20 at 13:40

1

This can be solved as a linearly-constrained linear least squares problem, for which there are many available numerical solvers, accessible from a variety of computer languages and packages. Alternatively, by not squaring the Frobenius norm, it can be solved as a Second Order Cone Problem (SOCP), again for which there are a number of solvers.

Here is how it can be formulated in CVX (under MATLAB), which will transform it to an SOCP, and call a solver to solve it.

cvx_begin
variable x
minimize(norm(x*A-B,'fro'))
subject to
0 <= x*A - B <= 256
cvx_end

answered May 17 '20 at 13:40

Mark L. Stone

4,496

Thanks for the informed details of your answer. Is there a Python solution available too? – Cypher May 17 '20 at 13:53
A similar formulation be be done in CVXPY, under Python; ths syntax is a little different due to Python vs. MATLAB. There are other options as well for Python. – Mark L. Stone May 17 '20 at 13:58

Minimizing $\| x A - B\|_F^2$ With a Constraint

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