This repository contains an C++ implementation of FBstab: The proximally regularized Fischer-Burmeister method for convex quadratic programming. FBstab solves quadratic programs of the form,

min.  1/2 z'Hz + f'z
s.t.      Gz  = h
          Az <= b

and its dual

min.  1/2 z'Hz + b'l + h'v
s.t.      Hz + f + G'l + A'v = 0
          v >= 0.

It assumes that H is positive semidefinite and can detect primal and/or dual infeasibility. FBstab requires no regularity assumptions on the G or A matrices. A mathematical description of FBstab can be found in the following research article.

FBstab was originally designed for solving Model Predictive Control (MPC) problems, it can exploit sparsity and be easily warmstarted. However, the algorithm is applicable to general QPs and the core algorithm is written so as to be easily extensible. We currently provide specializations to dense QPs and linear-quadratic optimal control problems, i.e., MPC problems, and are in the process of writing a version for general sparse QPs. FBstab uses Eigen for both backend linear algebra calculations and its interface. We currently provide two versions of FBstab:

FBstabMpc solves linear-quadratic optimal control problems of the following form:

min.  \sum_{i=0}^N 1/2 [x(i)]' * [Q(i) S(i)'] [x(i)] + [q(i)]'*[x(i)]
                       [u(i)]    [S(i) R(i) ] [u(i)]   [r(i)]  [u(i)]

s.t.  x(i+1) = A(i)*x(i) + B(i) u(i) + c(i), i = 0 ... N-1
      x(0) = x0
      E(i)*x(i) + L(i)*u(i) + d(i) <= 0,     i = 0 ... N
      
Where
      [ Q(i),S(i)']
      [ S(i),R(i) ]
 
is positive semidefinite for all i \in [0,N].

FBstabDense solves dense QPs of the following form:

min.    1/2  z'Hz + f'z
s.t.         Gz  = h
             Az <= b

A third version 'FBstabSparse' is currently being developed.

FBstab is written in C++ 11 and uses Bazel as its primary build system. FBstab has the following dependencies:

1. Eigen
2. googletest (for testing only)

Once you've cloned the repo and installed bazel you can check if everything is working by nagivating to the project root and running bazel test //... in your terminal. Make sure you have at least Bazel 1.1.0 installed, this is a common issue.

If your project uses Bazel then FBstab's targets can be made directly available. We also provide basic cmake files so that FBstab can be included in cmake projects using the add_subdirectory command.

By default, Bazel will automatically download appropriate versions of all dependencies. If you wish to use a local "installation" of Eigen instead, e.g., if other parts of your project rely on it and you want to enforce consistent versioning, we provide the option to do so, see the comments in the WORKSPACE file.

We currently have the following interfaces planned/available:

1. C++: See fbstab/tests/ for usage examples. Interfaces are exposed in the header files:
   • fbstab/fbstab_mpc.h
   • fbstab/fbstab_dense.h
2. MATLAB (In progress)
3. Python (Planned)

FBstab is still under active development so, while the mathematical problem definitions are more or less stable, the interfaces will likely change as the project matures.

FBstab is primarily developed on macOS but is highly portable, as long as a compliant C++ 11 compiler is available you should be able to build it. Versions of FBstab have been tested on Xenial and Bionic with gcc and clang compilers in the past and we plan to support MacOS, Linux, and Windows. We hope to get CI up and running soon.

If you find FBstab useful and would like to cite it in your academic publications, we recommend the following BibTeX citation:

@article{liao2020fbstab,
  title={Fbstab: A proximally stabilized semismooth algorithm for convex quadratic programming},
  author={Liao-McPherson, Dominic and Kolmanovsky, Ilya},
  journal={Automatica},
  volume={113},
  pages={108801},
  year={2020},
  publisher={Elsevier}
}

The FBstab algorithm was originally developed with support from the National Science Foundation through the award CMMI 1562209. This C++ implementation was developed jointly by the Toyota Research Institute (TRI) and by the University of Michigan (UM) through a TRI/UM collaborative project.