Back QL

Quadratic Programming

Version 3.13 (2012)


QL solves quadratic programming problems with a positive definite objective function matrix and linear equality and inequality constraints.

Numerical Method

The algorithm is an implementation of the dual method of Goldfarb and Idnani and a modification of the original implementation of Powell. Initially, the algorithm computes a solution of the unconstrained problem by performing a Cholesky decomposition and by solving the triangular system. In an iterative way, violated constraints are added to a working set and a minimum with respect to the new subsystem with one additional constraint is calculated. Whenever necessary, a constraint is dropped from the working set. The internal matrix transformations are performed in numerically stable way.

Program Organization

QL is a FORTRAN subroutine where all data are passed by subroutine arguments.

Special Features

  • separate handling of upper and lower bounds
  • initially given Cholesky decomposition exploited
  • full documentation
  • FORTRAN source code (close to F77, conversion to C by f2c possible)


As an essential part of the nonlinear programming routine NLPQLP, QL solves the internal quadratic programming subproblem of the SQP-method and has therefore the same domain of application as NLPQLP.