Mathematics Colloquia and Seminars

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Many methods in constrained optimization involve reformulating the problem as an unconstrained problem by passing the constraint(s) into the objective function. We will first discuss two of the most common current methods: the augmented Lagrangian method and the proximal point method. In the process, we will motivate the construction of both methods and highlight parallels between both methods. These two methods will lead us to a discussion of a recent active set method with a convergence rate superior to all other methods for least squares problems with box or l1-regularization constraints. This method has been constructed as the Newton system of both a particular augmented Lagrangian and a modified Moreau envelope. We will proceed by showing the equivalence of these value functions under first orthant constraints (x ≥ 0). Finally, time allowing, we will close with a proposal to use duality theory, and in particular conjugate functions in the augmented Lagrangian, to create a generalized augmented Lagrangian with first order conditions that we conjecture will result in a single variable value function that is identical to the modified Moreau envelope.