What it does

QPgen is a MATLAB toolbox that generates C code that solve convex parametric problems of the form:

minimize \(\tfrac{1}{2}x^THx+g^Tx+h(Cx)\)
subject to \(Ax=b\),


  • \(x\in\mathbf{R}^n\) is the variable to be decided

  • \(H\in\mathbf{R}^{n\times n}\) is a positive semi definite real data matrix

  • \(C\in\mathbf{R}^{p\times n}\) and \(A\in\mathbf{R}^{m\times n}\) are fixed data matrices

  • \(h : \mathbf{R}^p\to\mathbf{R}\cup\{\infty\}\) is a sum of piecewise linear functions which models, e.g.,

    • Upper and lower bounds on \(Cx\)

    • Soft upper and lower bounds on \(Cx\)

    • The 1-norm

The generated C code takes \(g\in\mathbf{R}^n\) and/or \(b\in\mathbf{R}^m\) as inputs and solves the corresponding problem instance.

How it does it

The generated C code uses one of the following three algorithms to solve the optimization problem instances:

  • The alternating direction method of multipliers (ADMM)

  • Fast dual proximal gradient method (FGMdual)

  • Fast primal proximal gradient method (FGMprimal)

Features of the generated C code:

  • It usually has very good performance (up to 10 000 times faster than standard software, see examples)

  • It can be made library free

  • It can be made division free for fixed-point implementations

Keys points that enable good performance:

  • Optimal preconditioning and parameter selection

  • Efficient handling of soft constraints (without introducing additional variables)

  • Solver tailoring to specific parametric program by code generation

  • Offline matrix factorization

  • Limited online data checks