# Toeplitz equality-constrained least-squares problem

What is the fastest known algorithm for least-squares optimization with a linear equality constraint?

$$\begin{array}{ll} \text{minimize} & \|K x - y\|^2 + \mu \|x\|^2\\ \text{subject to} & Q x = v\end{array}$$

where $Q$ is Toeplitz circular, not necessarily symmetric sparse matrix, $x$ is unknown vector, and matrices $Q$, $K$, vectors $y$, $v$ and scalar $\mu$ are given?

## 1 Answer

I doubt there is a specialized algorithm for this. This is just an equality-constrained least-squares problem, readily solved using the Lagrange multiplier approach. I don't even think you can exploit the structure of $Q$.

The Lagrangian is $$L(x,\lambda) = \|Kx-y\|^2+\mu\|x\|^2-\lambda^T(Qx-v)$$ The optimality conditions are $$2K^T(Kx-y)+2\mu x - Q^T\lambda=0 \quad Qx=v$$ Solving the first equation for $x$ we get $$x=\tfrac{1}{2}(K^TK+\mu I)^{-1} Q^T \lambda$$ Substituting this into the second equation we get $$Q(K^TK+\mu I)^{-1}Q^T \lambda = 2v$$ Assuming that Q has full row rank, you simply solve this equation for $\lambda$ and substitute the result into the previous equation for $x$. A typical algorithm would proceed as follows:

1. Compute the Cholesky factorization $R_1^TR_1=K^TK+\mu I$. For a more expensive but more accurate approach, compute the QR factorization $Q_1R_1=\begin{bmatrix} K \\ \mu^{1/2} I \end{bmatrix}$.
2. Compute $\tilde{Q}=QR_1^{-1}$, and then compute the Cholesky factorization $R_2^TR_2=\tilde{Q}\tilde{Q}^T$. Similarly, you can compute the QR factorization $Q_2R_2=\tilde{Q}^T$.
3. Compute $\lambda = 2R_2^{-1}R_2^{-T}v$.
4. Compute $x=\tfrac{1}{2}R_1^{-1}R_1^{-T}Q^T\lambda=\tfrac{1}{2}R_1^{-1}\tilde{Q}^T\lambda$.

If you're going to exploit the structure of $Q$, it will have to be in step 2, but I'm not entirely sure how you will accomplish that. Perhaps you fully invert $R_1$ and exploit the sparsity of $Q$, but I don't see how the Toeplitz structure in particular will be helpful.

A projected conjugate gradient method might allow you to exploit the full structure of $Q$, but I'm still not certain that would be a significant driver of performance.

• Why did $\mathrm y$ disappear when solving the first equation for $\mathrm x$? – Rodrigo de Azevedo Aug 4 '16 at 14:16