Augmented Lagrangians quadratic growth and second-order sufficient optimality conditions Damian Fern´andez * * CIEM-CONICET, Facultad de Matem´atica, Astronom´ıa, F´ısica y Computaci´on-Universidad Nacional de C´ordoba (FaMAF-UNC). Medina Allende s/n , Ciudad Universitaria, X5016HUA, C´ordoba, Argentina ARTICLE HISTORY Compiled July 28, 2020 ABSTRACT It is well-know that the primal quadratic growth condition of the classical augmented Lagrangian around a local minimizer can be obtained under the second-order suffi- cient optimality condition. In this paper we show that those conditions are indeed equivalent. Moreover, we prove that the primal quadratic growth condition of the sharp augmented Lagrangian around a local minimizer is in fact equivalent to the weak second-order sufficient optimality condition. In addition, we present some sec- ondary results involving the sharp augmented Lagrangian. KEYWORDS Augmented Lagrangian; sharp augmented Lagrangian; second-order sufficient optimality condition; weak second-order sufficient optimality condition AMS CLASSIFICATION 90C46; 90C30; 65K05 1. Introduction The study of the augmented Lagrangian method is an important topic in the optimiza- tion comunity. Computational implementations such as LANCELOT or ALGENCAN are examples of the state-of-the-art software to solve large nonlinear contrained opti- mization problems. Theoretical aspects such as its global convergence [1] or its local analysis that does not depend on constraints qualifications [2], puts this method in a preferable position. Most of the existing literature deal with the classical augmented Lagrangian, also known as the Powell-Hestenes-Rockafellar augmented Lagrangian, or proximal Lagrangian [3]. In several convergence results using this Lagrangian, the penalty parameter is driving to infinity to achieve a suitable rate of convergence [4– 6]. This behaviour may be related to the quadratic nature of the penalization in the construction of this Lagrangian. The same quadratic nature allows the existence of a duality gap for some problems. It is known that the duality gap can be avoided by using a nondifferentiable penalization, that produces the so-called sharp Lagrangian [3]. CONTACT D. Fern´ andez. Email: dfernandez@famaf.unc.edu.ar