Systems Optimization Laboratory
Stanford, CA 943054121 USA

Freely Available Optimization Software
The following software packages are provided by SOL under the terms of
The MIT License (MIT).
The software may alternatively be used under the terms of a
BSD License
(BSDlicense.txt).
 SYMMLQ:
Fortran, MATLAB, and Python software for sparse symmetric linear equations \(Ax = b\),
where \(A\) is definite or indefinite.
(Iterative method. Allows a positivedefinite preconditioner.)
 MINRES:
Fortran, MATLAB, and Python software for sparse symmetric linear equations \(Ax = b\),
where \(A\) is definite or indefinite, possibly singular.
(Iterative method. Allows a positivedefinite preconditioner.)
 MINRESQLP:
MATLAB software for sparse symmetric linear equations \(Ax = b\)
or leastsquares problems \( \min \Ax  b\ \),
where \(A\) is definite or indefinite, possibly singular.
(Iterative method. Allows a positivedefinite preconditioner.
For singular systems, computes the minimumnorm solution.)
 cgLanczos:
MATLAB software for sparse symmetric positivedefinite linear equations \(Ax = b\).
(Iterative method; Lanczosbased implementation of CG, the conjugategradient method.)
Special feature: Returns an estimate of the diagonals of
\( A^{1} \).
 LSQR:
MATLAB, Fortran, C, C++, .NET, and Python software for sparse linear equations
and sparse least squares. (Iterative method; more stable
than symmetric conjugategradient method on normal equations.
Allows positive "damping".)
Special feature: Returns an estimate of the diagonals of
\( (A^T A)^{1} \).
 LSMR:
MATLAB, Python, and Fortran 90 software for sparse linear equations
and sparse least squares. (Iterative method; more stable
than MINRES on normal equations.
Allows positive "damping".)
Special feature: Both \( \r\ \) and \( \A^T r\ \) decrease monotonically.
For LSQR, only \( \r\ \) is monotonic.
 CGLS:
MATLAB software for sparse linear equations
and sparse least squares. (Iterative method; more stable
than symmetric conjugategradient method on normal equations.)
Special feature: Allows positive or negative "damping"
(although negative is potentially unstable).
 CRAIG:
MATLAB software for underdetermined sparse linear equations \( Ax=b \).
Returns the minimumlength solution.
 covLSQR:
MATLAB software for sparse linear equations
and sparse least squares, derived from LSQR.
Special feature: Can estimate any principal submatrix
of the covariance matrix \( (A^T A)^{1} \).
 LSRN:
Python and C++ (with MATLAB interface) software for strongly overdetermined
or underdetermined, possibly rankdeficient, linear least squares.
 LUSOL:
Fortran and MATLAB software for maintaining a sparse square or rectangular
factorization \(A = LU\). (Suitable as a basis factorization package
for sparse simplex method.)
 lusolZ:
MATLAB software for computing a nullspace operator \(Z\) of the
transpose of a sparse matrix \(S\) (so that \(S^T Z = 0\))
using sparse QR factors of either \(S\) or \(S^T\) computed by
SuiteSparseQR,
or sparse LU factors of either \(S\) or \(S^T\) computed by
LUSOL.
MATLAB routines {spqrZ, spqrZv, spqrZt} and {lusolZ, lusolZv, lusolZt}
are provided to factorize \(S\) or \(S^T\)
and to compute products of the form \(w = Zv\) and \(s = Z^T t\)
for given vectors \(v\) and \(t\).
 LUMOD:
Fortran software for updating a dense square factorization \(LC = U\)
when rows and columns of \(C\) are added, deleted or replaced.
(Suitable as basis factorization package for dense simplex method,
or for updating sparse factorizations via the Schurcomplement
(BlockLU) method.)
 ASP:
MATLAB software implementing an activeset method for sparse optimization:
minimize \(\lambda \x\_1 + \frac12 \Axb\_2^2\).
This includes basis pursuit (BP), basis pursuit denoising (BPDN),
and NNLS (nonnegative least squares).
For certain matching values of \(\lambda\) and \(\tau\),
BPDN is equivalent to the Lasso problem:
minimize \(\frac12 \Axb\_2^2\) subject to \(\x\_1 \le \tau\).
Special feature: \(A\) may be a dense or sparse matrix
or a function for computing \(Ax\) and \(A^Ty\).
 PDCO:
MATLAB software implementing a primaldual interior method
for sparse linear programming, least squares,
or convex optimization subject to linear constraints \(Ax=b\), \(l \le x \le u\).
This includes NNLS (nonnegative least squares).
Special feature: \(A\) may be a dense or sparse matrix
or a function for computing \(Ax\) and \(A^Ty\).
 PDSCO:
Superseded by PDCO.
 PNOPT:
MATLAB solver that uses proximal Newtontype methods to minimize
composite functions (the sum of a smooth and a nonsmooth function).
 qdotdd:
Fortran 90 quadprecision dotproduct of doubleprecision vectors.
