Implementing proximal point methods for linear programming
Journal of Optimization Theory and Applications
Lagrange multipliers and optimality
SIAM Review
Augmented Lagrangian algorithms for linear programming
Journal of Optimization Theory and Applications
On second-order directional derivatives
Nonlinear Analysis: Theory, Methods & Applications
Second-order global optimality conditions for convex composite optimization
Mathematical Programming: Series A and B
An Augmented Lagrangian Function with Improved Exactness Properties
SIAM Journal on Optimization
A Nonlinear Lagrangian Approach to Constrained Optimization Problems
SIAM Journal on Optimization
The Zero Duality Gap Property and Lower Semicontinuity of the Perturbation Function
Mathematics of Operations Research
A Unified Augmented Lagrangian Approach to Duality and Exact Penalization
Mathematics of Operations Research
Partially Strictly Monotone and Nonlinear Penalty Functions for Constrained Mathematical Programs
Computational Optimization and Applications
Unified theory of augmented Lagrangian methods for constrained global optimization
Journal of Global Optimization
A primal dual modified subgradient algorithm with sharp Lagrangian
Journal of Global Optimization
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In this paper, we present a necessary and sufficient condition for a zero duality gap between a primal optimization problem and its generalized augmented Lagrangian dual problems. The condition is mainly expressed in the form of the lower semicontinuity of a perturbation function at the origin. For a constrained optimization problem, a general equivalence is established for zero duality gap properties defined by a general nonlinear Lagrangian dual problem and a generalized augmented Lagrangian dual problem, respectively. For a constrained optimization problem with both equality and inequality constraints, we prove that first-order and second-order necessary optimality conditions of the augmented Lagrangian problems with a convex quadratic augmenting function converge to that of the original constrained program. For a mathematical program with only equality constraints, we show that the second-order necessary conditions of general augmented Lagrangian problems with a convex augmenting function converge to that of the original constrained program.