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I need to (computationally) solve a system of equations, for the purposes of an interior point method, of the form

$$ \left[\begin{array}{cc}B & A^T \\ A & 0\end{array}\right] \left[\begin{array}{c}x\\y\end{array}\right] = \left[\begin{array}{c}c_x \\ c_y\end{array}\right] $$

where $A$ and $B$ are large and sparse, and $B$ is Hermitian positive definite. By assumption, $A$ has full row rank. My current approach goes:

  1. Compute the Cholesky factor $B = L L^T$.
  2. Solve $L W = A^{T}$ for $W$, producing $W=L^{-1}A^T$ so that $W^T W = A L^{-T} L^{-1} A^T$
  3. Compute the Cholesky factor of $W^T W$
  4. Use this to solve $A L^{-T} L^{-1} A^T y = A L^{-T} L^{-1} c_x - c_y$
  5. Recover $x$ from $L L^T x = c_x - A^T y$

The problem that I'm running into is that $W$, in step 2, may be dense and its sparsity structure indeterminate based on the sparsity of $A$ and $B$. The sparsity of $W$ is dependent on the actual numerical values of $A$ and $L$, which is hard to optimize for.

Is there a way to avoid materializing $W$? I'd like to avoid having to invert $A$, as while $A$ itself usually has a sparse structure, its inverse is not guaranteed to be sparse.

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  • $\begingroup$ Is there any reason that you don’t find $x$ by solving this sparse linear equation: $Ax = c_{y}$ and using any sparse linear solver either direct or iterative and then find $y$ by solving this equation: $A^{T}y = c_{x} - Bx$ again by using any available sparse linear solver instead of using this funny complicated procedure that in my opinion is just try to accomplish something that is already available in any linear algebra library optimized for sparse systems. $\endgroup$ – Alone Programmer Dec 20 '20 at 17:33
  • $\begingroup$ Does $A$ have full column rank? It typically wouldn't in an interior point method. i.e. there are usually more variables than linear constraints on $x$. Have you considered Cholesky factorization of $ABA^{T}$? $\endgroup$ – Brian Borchers Dec 20 '20 at 17:58
  • $\begingroup$ Why don't you simply perform a sparse $LDL^T$ factorization of the complete system? $\endgroup$ – Bill Greene Dec 20 '20 at 19:02
  • $\begingroup$ @BillGreene in interior point methods for LP, the sparse Cholesky factorization of $ABA^{T}$ is generally faster than using a sparse symmetric indefinite factorization of the bigger matrix. For more general nonlinear problems where $B$ can be indefinite, the larger matrix needs to be factorized. $\endgroup$ – Brian Borchers Dec 20 '20 at 19:27
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    $\begingroup$ This is straight forward algebra. $x=B^{-1}(c_{x}-A^{T}y)$. So $Ax=AB^{-1}(c_{x}-A^{T}y)=b$. Thus you can solve $AB^{-1}A^{T}y=AB^{-1}c_{x}-b$ to get $y$ and then solve for $x$. In interior point methods for LP, $B$ is diagonal, so this can be quite efficient. When $B$ isn't diagonal (e.g. in interior point methods for QP) or when $B$ isn't positive definite, then using a general sparse indefinite $PLDLP^{T}$ factorization can be the way to go. $\endgroup$ – Brian Borchers Dec 21 '20 at 3:44
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There are two common ways to deal with systems of equations like this that arise in interior-point methods.

We'll assume that $A$ is of full row rank but not full column rank. That is, $A$ has more columns than rows. Thus we can't solve systems of equations involving $A$ uniquely or find $A^{-1}$.

  1. If $B$ is symmetric and positive definite, we can reduce this system of equations to a smaller system of equations in $y$ by solving for $x$ in terms of $y$,

$x=B^{-1}(c_{x}-A^{T}y)$

and then substituting this into the equation $Ax=b$ to get

$AB^{-1}A^{T}y=AB^{-1}c_{x}-b$.

The matrix in this equation for $y$ is symmetric and positive definite, so we can solve the system by sparse Cholesky factorization.

In interior point methods for linear programming, $B$ is diagonal, so $B^{-1}$ is easy to compute and the the matrix $AB^{-1}A^{T}$ has the same sparsity pattern as as $AA^{T}$. The speed of the Cholesky factorization can be improved by reordering the rows of $A$ (and symmetrically the columns of $A^{T}$) to minimize fill-in. The reordering algorithm only has to be run once since the sparsity pattern of $AB^{-1}A^{T}$ is fixed. Furthermore, there's no need to pivot for numerical stability during the factorization.

  1. In interior point methods for nonlinear programming this approach may not be feasible, either because $B$ is indefinite or because $AB^{-1}A^{T}$ becomes too dense. In these situations, we typically make use of a sparse symmetric indefinite factorization

$\left[ \begin{array}{cc} B & A^{T} \\ A & 0 \\ \end{array} \right] =PLDL^{T}P^{T}$

to solve the larger system of equations. Here, it is generally necessary to pivot for stability.

In your case, $B$ is sparse, symmetric and positive definite. You could compute sparse Cholesky factors of $B$ and use them to compute $B^{-1}A^{T}$ column by column (with a pair of sparse triangular solves for each column) rather than computing $B^{-1}$ and multiplying it times $A^{T}$. Whether this would be faster than a sparse symmetric indefinite factorization is something that you'd want to experiment with.

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  • $\begingroup$ Ah, okay, that makes sense. I'm doing second order cone programming, so in extreme cases (where most of the problem's variables are in a single large second order cone) $B$ is dense. It seems like the sparse symmetric indefinite factorization is the right route for me, then. $\endgroup$ – ckfinite Dec 21 '20 at 6:35
  • $\begingroup$ Of popular packages for SOCP/SDP, SDPT3, SDPA, and SeDuMi all use the smaller reduced system with sparse Cholesky factorization. $\endgroup$ – Brian Borchers Dec 21 '20 at 15:01
  • $\begingroup$ To elaborate a bit- in many SOCP problems the structure isn't one big second order cone but rather a direct product of many small (e.g. 3 dimensional) cones. Sometimes this leads to $AB^{-1}A^{T}$ being sparse or even diagonal. In many other cases, it's fully dense. $\endgroup$ – Brian Borchers Dec 21 '20 at 16:59
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Though digging through all the commentary has muddied (for me, at least) what your exact scenario is, there are sparse-direct algorithms for computing schur complements of the form $\mathbf S = \mathbf A \mathbf B^{-1} \mathbf A^T$, where both the $\mathbf B$ and $\mathbf A$ operands are sparse. I can point you to my own implementation here. The PARDISO package (contained within MKL) has similar functionality.

Their efficiency depends upon the size/shape of $\mathbf A$ relative to $\mathbf B$, the approach only makes sense if the size of $\mathbf S$ is (asymptotically) smaller than $\mathbf B$. The use case that drove me to these methods is substructuring / static condensation, wherein $\mathbf B$ represents a volume-sized discretization of a PDE, while $\mathbf A$ is a surface-sized restriction/sampling over a boundary. (In which case the time/memory required to compute $\mathbf S$ is asymptotically the same as the Cholesky factorization of $\mathbf B$). Unfortunately I can't quite tell if you example falls into this category or not .. basically you need $\mathrm {size}(\mathbf x)$ >> $\mathrm{size}(\mathbf y)$ for this approach to work well.

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