@article{iiasa14158,
          volume = {50},
          number = {1-3},
           title = {Maintaining the positive definiteness of the matrices in reduced secant methods for equality constrained optimization},
       publisher = {Springer},
         journal = {Mathematical Programming},
             doi = {10.1007/BF01594922},
           pages = {1--28},
            year = {1991},
        keywords = {Augmented Lagrangian; constrained optimization; exact penalty function; global convergence; optimization algorithm; reduced secant method; superlinear convergence; Wolfe's step-size selection},
             url = {https://pure.iiasa.ac.at/id/eprint/14158/},
            issn = {0025-5610},
        abstract = {We propose an algorithm for minimizing a functionf on {$R$}n 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 {$\parallel$}c{$\parallel$}. 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 of{\ensuremath{\gamma}}kT{\ensuremath{\delta}}k, where{\ensuremath{\gamma}}k is the change in the reduced gradient and {\ensuremath{\delta}}k is the reduced longitudinal displacement.},
          author = {Gilbert, J. C.}
}