Nonlinearly constrained optimization via sequential regularized linear programming

Abstract

This thesis proposes a new active-set method for large-scale nonlinearly constrained optimization. The method solves a sequence of linear programs to generate search directions. The typical approach for globalization is based on damping the search directions with a trust-region constraint; our proposed approach is instead based on using a 2-norm regularization term in the objective. Numerical evidence is presented which demonstrates scaling inefficiencies in current sequential linear programming algorithms that use a trust-region constraint. Specifically, we show that the trust-region constraints in the trustregion subproblems significantly reduce the warm-start efficiency of these subproblem solves, and also unnecessarily induce infeasible subproblems. We also show that the use of a regularized linear programming (RLP) step largely eliminates these inefficiencies and, additionally, that the dual problem to RLP is a bound-constrained least-squares problem, which may allow for very efficient subproblem solves using gradient-projection-type algorithms. Two new algorithms were implemented and are presented in this thesis, based on solving sequences of RLPs and trust-region constrained LPs. These algorithms are used to demonstrate the effectiveness of each type of subproblem, which we extrapolate onto the effectiveness of an RLP-based algorithm with the addition of Newton-like steps. All of the source code needed to reproduce the figures and tables presented in this thesis is available online at http://www.cs.ubc.ca/labs/scl/thesis/10crowe/