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Primal-dual methods for nonlinear optimization


Nonlinearly constrained optimization problems may be solved by minimizing a sequence of simpler subproblems based on the properties of a so-called merit function that balances the (usually) conflicting aims of reducing the objective function and satisfying the constraints. Sometimes this merit function is minimized directly as an unconstrained function, in which case convergence is achieved by adjusting the relative weighting of the objective and constraints between subproblems. Alternatively, some model of the merit function is minimized subject to simple bounds and/or linearizations of the constraints. In this case, the merit function drives the algorithm by assessing the "quality" of points generated by the subproblem. generated by the subproblem that may be minimized with respect to both the primal and dual variables. A benefit of this approach is that each subproblem may be regularized by imposing explicit bounds on the dual variables. Two primal-dual variants of classical primal methods are given: a primal-dual bound constrained Lagrangian (pdBCL) method and a primal-dual ℓ₁ linearly constrained Lagrangian (pdℓ₁-LCL) method

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