On Hessian- and Jacobian-Free SQP Methods - a Total Quasi-Newton Scheme with Compact Storage

Summary

In this paper we describe several modifications to reduce the memory requirement of the total quasi-Newton method proposed by Andreas Griewank et al.

The idea is based on application of the compact representation formulae for the wellknown BFGS and SR1 update for unconstrained optimization. It is shown how these definitions can be extended to a total quasi-Newton approach for the constrained case.

A brief introduction to the limited-memory approach is described in the present paper using an updated null-space factorization for the KKT system as well as an efficient numerical implementation of the null-space method in which the null-space representation is not stored directly. It can be proven that the number of operations per iteration is bounded by a bilinear order $\mathcal{O}(n \cdot max(m,l))$ instead of a cubic order $\mathcal{O}(m \cdot {n^2})$ for standard SQP methods. Here ndenotes the number of variables, m the maximal number of active constraints, and lthe user-selected number of stored update vectors.

^† Supported by the DFG Research Center MATHEON “Mathematics for Key technologies” , Straße des 17. Juni 136, 10623 Berlin, Germany, www.matheon.de