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Newton's method is usually preferred when solving optimization problems due to its superior convergence properties compared to gradient-based or derivative-free optimization algorithms. However, deriving and computing second-order derivatives needed by Newton's method often is not trivial and, in some cases, not possible. In such cases quasi-Newton algorithms are a great alternative. In this paper, we provide a new derivation of well-known quasi-Newton formulas in an infinite-dimensional Hilbert space setting. It is known that quasi-Newton update formulas are solutions to certain variational problems over the space of symmetric matrices. In this paper, we formulate similar variational problems over the space of bounded symmetric operators in Hilbert spaces. By changing the constraints of the variational problem we obtain updates (for the Hessian and Hessian inverse) not only for the Broyden-Fletcher-Goldfarb-Shanno (BFGS) quasi-Newton method but also for Davidon--Fletcher--Powell (DFP), Symmetric Rank One (SR1), and Powell-Symmetric-Broyden (PSB). In addition, for an inverse problem governed by a partial differential equation (PDE), we derive DFP and BFGS ``structured" secant formulas that explicitly use the derivative of the regularization and only approximates the second derivative of the misfit term. We show numerical results that demonstrate the desired mesh-independence property and superior performance of the resulting quasi-Newton methods.
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Second Order Adjoints in Optimization
Second order, Newton-like algorithms exhibit convergence properties superior to gradient-based or derivative-free optimization algorithms. However, deriving and computing second order derivatives--needed for the Hessian-vector product in a Krylov iteration for the Newton step--often is not trivial. Second order adjoints provide a systematic and efficient tool to derive second derivative infor- mation. In this paper, we consider equality constrained optimization problems in an infinite-dimensional setting. We phrase the optimization problem in a general Banach space framework and derive second order sensitivities and second order adjoints in a rigorous and general way. We apply the developed framework to a partial differential equation-constrained optimization problem.
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- Award ID(s):
- 1654311
- PAR ID:
- 10352319
- Editor(s):
- Al-Baali, Mehiddin; Purnama, Anton; Grandinetti, Lucio
- Date Published:
- Journal Name:
- Numerical Analysis and Optimization
- Volume:
- 354
- Page Range / eLocation ID:
- 209-230
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1007/978-3-030-72040-7_10
