Complexity of a linearized augmented Lagrangian method for nonconvex minimization with nonlinear equality constraints
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| Format: | Preprint |
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2023
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| author | Bourkhissi, Lahcen El Necoara, Ion |
| author_facet | Bourkhissi, Lahcen El Necoara, Ion |
| contents | In this paper, we consider a nonconvex optimization problem with nonlinear equality constraints. We assume that both, the objective function and the functional constraints are locally smooth. For solving this problem, we propose a linearized augmented Lagrangian method, i.e., we linearize the objective function and the functional constraints in a Gauss-Newton fashion at the current iterate within the augmented Lagrangian function and add a quadratic regularization, yielding a subproblem that is easy to solve, and whose solution is the next primal iterate. The update of the dual multipliers is also based on the linearization of functional constraints. Under a novel dynamic regularization parameter choice, we prove boundedness and global asymptotic convergence of the iterates to a first-order solution of the problem. We also derive convergence guarantees for the iterates of our method to an $ε$-first-order solution in $\mathcal{O}(\sqrtρ ε^{-2})$ Jacobian evaluations, where $ρ$ is the penalty parameter. Moreover, when the problem exhibits a benign nonconvex property, we derive improved convergence results to an $ε$-second-order solution. Finally, we validate the performance of the proposed algorithm by numerically comparing it with the existing methods and software from the literature. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2301_08345 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Complexity of a linearized augmented Lagrangian method for nonconvex minimization with nonlinear equality constraints Bourkhissi, Lahcen El Necoara, Ion Optimization and Control In this paper, we consider a nonconvex optimization problem with nonlinear equality constraints. We assume that both, the objective function and the functional constraints are locally smooth. For solving this problem, we propose a linearized augmented Lagrangian method, i.e., we linearize the objective function and the functional constraints in a Gauss-Newton fashion at the current iterate within the augmented Lagrangian function and add a quadratic regularization, yielding a subproblem that is easy to solve, and whose solution is the next primal iterate. The update of the dual multipliers is also based on the linearization of functional constraints. Under a novel dynamic regularization parameter choice, we prove boundedness and global asymptotic convergence of the iterates to a first-order solution of the problem. We also derive convergence guarantees for the iterates of our method to an $ε$-first-order solution in $\mathcal{O}(\sqrtρ ε^{-2})$ Jacobian evaluations, where $ρ$ is the penalty parameter. Moreover, when the problem exhibits a benign nonconvex property, we derive improved convergence results to an $ε$-second-order solution. Finally, we validate the performance of the proposed algorithm by numerically comparing it with the existing methods and software from the literature. |
| title | Complexity of a linearized augmented Lagrangian method for nonconvex minimization with nonlinear equality constraints |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2301.08345 |