Maureen Doyle <doyle@sccm.stanford.edu>
www.stanford.edu/~mdoyle

Barrier Algorithms with Nonlinear Constraints.


The most successful class of algorithms for nonconvex nonlinearly constrained
optimization problems is sequential quadratic programming (SQP) methods. As the
name suggests such methods solve a sequence of quadratic programs (QP). When
second derivatives are available such QPs are  in general nonconvex and this
poses both technical and practical difficulties. For example, the QP may not
have a solution and if it does it may not be unique. To circumvent the
difficulties we proposed removing the inequality constraints using a barrier
function. Such methods were first proposed in the 50s and have recently had a
revival after their success in solving linear problems. They have been
successfully extended to other convex problems although as yet how best to
utilize the ideas for nonconvex nonlinearly constrained problems has still to
be determined.