Ceres

"Ceres Solver is an open source C++ library for modeling and solving large, complicated optimization problems. It can be used to solve Non-linear Least Squares problems with bounds constraints and general unconstrained optimization problems.

The features include:

Code Quality - Ceres Solver has been used in production at Google for more than four years now. It is clean, extensively tested and well documented code that is actively developed and supported.

Modeling API - It is rarely the case that one starts with the exact and complete formulation of the problem that one is trying to solve. Ceres’s modeling API has been designed so that the user can easily build and modify the objective function, one term at a time. And to do so without worrying about how the solver is going to deal with the resulting changes in the sparsity/structure of the underlying problem.

• Derivatives Supplying derivatives is perhaps the most tedious and error prone part of using an optimization library. Ceres ships with automatic and numeric differentiation. So you never have to compute derivatives by hand (unless you really want to). Not only this, Ceres allows you to mix automatic, numeric and analytical derivatives in any combination that you want.
• Robust Loss Functions Most non-linear least squares problems involve data. If there is data, there will be outliers. Ceres allows the user to shape their residuals using a LossFunction to reduce the influence of outliers.
• Local Parameterization In many cases, some parameters lie on a manifold other than Euclidean space, e.g., rotation matrices. In such cases, the user can specify the geometry of the local tangent space by specifying a LocalParameterization object.

Solver Choice Depending on the size, sparsity structure, time & memory budgets, and solution quality requiremnts, different optimization algorithms will suit different needs. To this end, Ceres Solver comes with a variety of optimization algorithms:

• Trust Region Solvers - Ceres supports Levenberg-Marquardt, Powell’s Dogleg, and Subspace dogleg methods. The key computational cost in all of these methods is the solution of a linear system. To this end Ceres ships with a variety of linear solvers - dense QR and dense Cholesky factorization (using Eigen or LAPACK) for dense problems, sparse Cholesky factorization (SuiteSparse, CXSparse or Eigen) for large sparse problems custom Schur complement based dense, sparse, and iterative linear solvers for bundle adjustment problems.
• Line Search Solvers - When the problem size is so large that storing and factoring the Jacobian is not feasible or a low accuracy solution is required cheaply, Ceres offers a number of line search based algorithms. This includes a number of variants of Non-linear Conjugate Gradients, BFGS and LBFGS.

Speed - Ceres Solver has been extensively optimized, with C++ templating, hand written linear algebra routines and OpenMP based multithreading of the Jacobian evaluation and the linear solvers.

Solution Quality Ceres is the best performing solver on the NIST problem set used by Mondragon and Borchers for benchmarking non-linear least squares solvers.

Covariance estimation - Evaluate the sensitivity/uncertainty of the solution by evaluating all or part of the covariance matrix. Ceres is one of the few solvers that allows you to to do this analysis at scale.

Community Since its release as an open source software, Ceres has developed an active developer community that contributes new features, bug fixes and support."

http://ceres-solver.org/

https://ceres-solver.googlesource.com/ceres-solver

https://github.com/ceres-solver/ceres-solver