An implementation of Newton-like methods on nonlinearly constrained networks

Abstract

The minimization of a nonlinear function with linear and nonlinear constraints and simple bounds can be performed by minimizing an augmented Lagrangian function, including only the nonlinear constraints. This procedure is particularly interesting when the linear constraints are flow conservation equations, as there exist efficient techniques for solving nonlinear network problems. It is then necessary to estimate their multipliers, and variable reduction techniques can be used to carry out the successive minimizations. This work analyzes the possibility of estimating the multipliers of the nonlinear constraints using Newton-like methods. Also, an algorithm is designed to solve nonlinear network problems with nonlinear inequality side constraints, which combines first and superlinear-order multiplier methods. The computational performance of this method is compared with that of MINOS 5.5. Scope and purpose Real problems with the structure of a nonlinear network flow problem with nonlinear side constraints exist and have a high dimensionality (e.g. the short-term hydrothermal coordination of electric power generation). They often need to be solved and it is important to find the procedure that will solve them with the highest efficiency. In previous works this author has analyzed the viability of estimating the multipliers of the nonlinear constraints from the Kuhn–Tucker system in order to improve the inexpensive Hestenes–Powell update. However, the conclusion was that there is no significant advantage for multipliers estimated through the Kuhn–Tucker system. The purpose of this paper is mainly to use the convergence properties of the Newton-like methods in order to develop a new estimator of the nonlinear constraint multipliers that improves the efficiency and the robustness of this augmented Lagrangian method.