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

We propose an algorithm for minimizing a functionf on ℝn in the presence ofm equality constraintsc 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 mainlyf and a transversal search that decreases mainly ∥c∥. Our 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 ofγkTδk, whereγk is the change in the reduced gradient and δk is the reduced longitudinal displacement