Variable-Metric Algorithm Employing Linear and Quadratic Penalties

Abstract

A variable-metric algorithm is described which makes use of both linear and quadratic penalty terms for handling nonlinear constraints, and employs both projection and penalty features. Quadratic penalty coefficients are adjusted in a process that attempts to maintain a positive-definite matrix of second partial derivatives of the function, including penalty terms, without generating the large positive eigenvalues that traditionally attend the use of quadratic penalties, which cause zigzagging and slowed convergence. The schemes contemplated make use of inferred second-order properties, not only in terms of the variable metric of DFP (or its relatives), but by estimation of second directional derivatives, by fitting cubics to various functions along directions of search. Some experiments are described with a simple constrained-minimum problem contrived to offer difficulties with methods that use only linear penalties, hence taxing the quadratic-penalty-adjustment procedure.