Maintaining the Positive Definiteness of the Matrices in Reduced Secant Methods for Equality Constrained Optimization

Gilbert, J.C. (1987). Maintaining the Positive Definiteness of the Matrices in Reduced Secant Methods for Equality Constrained Optimization. IIASA Working Paper. IIASA, Laxenburg, Austria: WP-87-123

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Abstract

This paper proposes an algorithm for minimizing a function f on R^n in the presence of m equality constraints c that locally is a reduced secant method. The local method is globalized using a nondifferentiable augmented Lagrangian whose decrease is obtained by both a longitudinal search that decreases mainly f and a transversal search that decreases mainly ||c||.

The main objective of the paper is to show that the longitudinal path can be designed in order to maintain the positive definiteness of the reduced matrices by means of the positivity of gamma_{k}^{T}, where gamma_{k} is the change in the reduced gradient and bk is the reduced longitudinal displacement.

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