Maintaining the positive definiteness of the matrices in reduced secant methods for equality constrained optimization

The paper proposes an algorithm for minimizing a function f on ℝⁿ 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 . The present main objective is to show that the longitudinal path can be designed to maintain the positive definiteness of the reduced matrices by means of the positivity of , where is the change in the reduced gradient and is the reduced longitudinal displacement.