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Title: A STRUCTURED QUASI-NEWTON ALGORITHM FOR OPTIMIZING WITH INCOMPLETE HESSIAN INFORMATION

Abstract

We present a structured quasi-Newton algorithm for unconstrained optimization problems that have unavailable second-order derivatives or Hessian terms. We provide a formal derivation of the well-known Broyden-Fletcher-Goldfarb-Shanno (BFGS) secant update formula that approximates only the missing Hessian terms, and we propose a linesearch quasi-Newton algorithm based on a modification of Wolfe conditions that converges to first-order optimality conditions. We also analyze the local convergence properties of the structured BFGS algorithm and show that it achieves superlinear convergence under the standard assumptions used by quasi-Newton methods using secant updates. We provide a thorough study of the practical performance of the algorithm on the CUTEr suite of test problems and show that our structured BFGS-based quasi-Newton algorithm outperforms the unstructured counterpart(s).

Authors:
; ;
Publication Date:
Research Org.:
Argonne National Lab. (ANL), Argonne, IL (United States)
Sponsoring Org.:
USDOE Office of Science (SC); National Science Foundation (NSF)
OSTI Identifier:
1573037
DOE Contract Number:  
AC02-06CH11357
Resource Type:
Journal Article
Journal Name:
SIAM Journal on Optimization
Additional Journal Information:
Journal Volume: 29; Journal Issue: 2
Country of Publication:
United States
Language:
English

Citation Formats

Petra, Cosmin G., Chiang, Nai-Yuan, and Anitescu, Mihai. A STRUCTURED QUASI-NEWTON ALGORITHM FOR OPTIMIZING WITH INCOMPLETE HESSIAN INFORMATION. United States: N. p., 2019. Web. doi:10.1137/18M1167942.
Petra, Cosmin G., Chiang, Nai-Yuan, & Anitescu, Mihai. A STRUCTURED QUASI-NEWTON ALGORITHM FOR OPTIMIZING WITH INCOMPLETE HESSIAN INFORMATION. United States. doi:10.1137/18M1167942.
Petra, Cosmin G., Chiang, Nai-Yuan, and Anitescu, Mihai. Tue . "A STRUCTURED QUASI-NEWTON ALGORITHM FOR OPTIMIZING WITH INCOMPLETE HESSIAN INFORMATION". United States. doi:10.1137/18M1167942.
@article{osti_1573037,
title = {A STRUCTURED QUASI-NEWTON ALGORITHM FOR OPTIMIZING WITH INCOMPLETE HESSIAN INFORMATION},
author = {Petra, Cosmin G. and Chiang, Nai-Yuan and Anitescu, Mihai},
abstractNote = {We present a structured quasi-Newton algorithm for unconstrained optimization problems that have unavailable second-order derivatives or Hessian terms. We provide a formal derivation of the well-known Broyden-Fletcher-Goldfarb-Shanno (BFGS) secant update formula that approximates only the missing Hessian terms, and we propose a linesearch quasi-Newton algorithm based on a modification of Wolfe conditions that converges to first-order optimality conditions. We also analyze the local convergence properties of the structured BFGS algorithm and show that it achieves superlinear convergence under the standard assumptions used by quasi-Newton methods using secant updates. We provide a thorough study of the practical performance of the algorithm on the CUTEr suite of test problems and show that our structured BFGS-based quasi-Newton algorithm outperforms the unstructured counterpart(s).},
doi = {10.1137/18M1167942},
journal = {SIAM Journal on Optimization},
number = 2,
volume = 29,
place = {United States},
year = {2019},
month = {1}
}