How to solve an objective function with inequality constraint?

Question

I have an objective function as follows,

$\displaystyle\min_{\mathbf{A},\mathbf{B}}\lVert \mathbf{X}-\mathbf{ABZ} \rVert_F^2$,

where $\mathbf{X},\mathbf{Z}\in \mathbb{R}^{p\times n}$, $\mathbf{A}\in \mathbb{R}^{p\times r}$, and $\mathbf{B}\in \mathbb{R}^{r\times p}$. The objective function can be solved by setting the derivatives to zero with respect to $\mathbf{A}$ and $\mathbf{B}$ as,

$\mathbf{A} = \mathbf{X}\mathbf{Z}^{\top}\mathbf{B}^{\top}{(\mathbf{BZZ}^{\top}\mathbf{B}^{\top})}^{-1}$,

$\mathbf{B}={(\mathbf{A}^{\top}\mathbf{A})}^{-1}\mathbf{A}^{\top}\mathbf{XZ}^{\top}{(\mathbf{ZZ}^{\top})}^{-1}$.

Now, I want to add an inequality constraint like this,

$\displaystyle\min_{\mathbf{A},\mathbf{B}}\lVert \mathbf{X}-\mathbf{ABZ} \rVert_F^2$
$s.t. \lVert \mathbf{AB} \rVert_F^2 \leq 1$.

How can I solve the new objective function with respect to the inequality constraint? If I should add the constraint to the main objective function, how to achieve a smooth new objective function?

Answer 1

You can use the multiplier method by introducing the Lagrange multiplier for the constraint, then forming the Lagrangian.

You then have to write and solve the KKT conditions. Give it a shot and let us know :)

Edit: if you know that the inequality constraint is active at the solution of the constrained problem, you can directly write the problem with an equality constraint. It makes things easier.