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- Augmented_Lagrangian_method abstract "Augmented Lagrangian methods are a certain class of algorithms for solving constrained optimization problems. They have similarities to penalty methods in that they replace a constrained optimization problem by a series of unconstrained problems; the difference is that the augmented Lagrangian method adds an additional term to the unconstrained objective. This additional term isdesigned to mimic a Lagrange multiplier. The augmented Lagrangian is not the same as the method of Lagrange multipliers.Viewed differently, the unconstrained objective is the Lagrangian of the constrained problem, with an additional penalty term (the augmentation).The method was originally known as the method of multipliers, and was studied much in the 1970 and 1980s as a good alternative to penalty methods.It was first discussed by Magnus Hestenes in 1969and by Powell in 1969The method was studied by R. Tyrrell Rockafellar in relation to Fenchel duality, particularly in relation to proximal-point methods, Moreau–Yosida regularization, and maximal monotone operators: These methods were used in structural optimization. The method was also studied and implemented by Dimitri Bertsekas, notably in his 1982 book, and with respect to entropic regularization (which accelerate the rate of convergence for his "exponential method of multipliers").Since the 1970s, sequential quadratic programming (SQP) and interior point methods (IPM) have had increasing attention, in part because they more easily use sparse matrix subroutines from numerical software libraries, and in part because IPMs have proven complexity results via the theory of self-concordant functions. The augmented Lagrangian method was rejuvenated by the optimization systems LANCELOT and AMPL, which allowed sparse matrix techniques to be used on seemingly dense but "partially separable" problems. The method is still useful for some problems. As of around 2007, there has been a resurgence of Augmented Lagrangian methods (and ADMM in particular) in fields such as total-variation denoising and compressed sensing; for example, the SALSA package was proposed in 2009.A variant of the standard Augmented Lagrangian method that uses partial updates (similar to the Gauss-Seidel method for solving linear equations) is known as the alternating direction method of multipliers or ADMM.".
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- Augmented_Lagrangian_method subject Category:Optimization_algorithms_and_methods.
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- Augmented_Lagrangian_method comment "Augmented Lagrangian methods are a certain class of algorithms for solving constrained optimization problems. They have similarities to penalty methods in that they replace a constrained optimization problem by a series of unconstrained problems; the difference is that the augmented Lagrangian method adds an additional term to the unconstrained objective. This additional term isdesigned to mimic a Lagrange multiplier.".
- Augmented_Lagrangian_method label "Augmented Lagrangian method".
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