Computing Second-Order Points Under Equality Constraints: Revisiting Fletcher’s Augmented Lagrangian

Florentin Goyens (), Armin Eftekhari and Nicolas Boumal
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Florentin Goyens: LAMSADE, Université Paris Dauphine-PSL
Nicolas Boumal: Ecole Polytechnique Fédérale de Lausanne (EPFL)

Journal of Optimization Theory and Applications, 2024, vol. 201, issue 3, No 9, 1198-1228

Abstract: Abstract We address the problem of minimizing a smooth function under smooth equality constraints. Under regularity assumptions on these constraints, we propose a notion of approximate first- and second-order critical point which relies on the geometric formalism of Riemannian optimization. Using a smooth exact penalty function known as Fletcher’s augmented Lagrangian, we propose an algorithm to minimize the penalized cost function which reaches $$\varepsilon $$ ε -approximate second-order critical points of the original optimization problem in at most $${\mathcal {O}}(\varepsilon ^{-3})$$ O ( ε - 3 ) iterations. This improves on current best theoretical bounds. Along the way, we show new properties of Fletcher’s augmented Lagrangian, which may be of independent interest.

Keywords: Nonconvex optimization; Constrained optimization; Augmented Lagrangian; Complexity; Riemannian optimization (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s10957-024-02421-6

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