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Secant update version of quasi-Newton PSB with weighted multisecant equations

Nicolas Boutet (UGent) , Robby Haelterman (UGent) and Joris Degroote (UGent)
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Abstract
Quasi-Newton methods are often used in the frame of non-linear optimization. In those methods, the quality and cost of the estimate of the Hessian matrix has a major influence on the efficiency of the optimization algorithm, which has a huge impact for computationally costly problems. One strategy to create a more accurate estimate of the Hessian consists in maximizing the use of available information during this computation. This is done by combining different characteristics. The Powell-Symmetric-Broyden method (PSB) imposes, for example, the satisfaction of the last secant equation, which is called secant update property, and the symmetry of the Hessian (Powell in Nonlinear Programming 31-65, 1970). Imposing the satisfaction of more secant equations should be the next step to include more information into the Hessian. However, Schnabel proved that this is impossible (Schnabel in quasi-Newton methods using multiple secant equations, 1983). Penalized PSB (pPSB), works around the impossibility by giving a symmetric Hessian and penalizing the non-satisfaction of the multiple secant equations by using weight factors (Gratton et al. in Optim Methods Softw 30(4):748-755, 2015). Doing so, he loses the secant update property. In this paper, we combine the properties of PSB and pPSB by adding to pPSB the secant update property. This gives us the secant update penalized PSB (SUpPSB). This new formula that we propose also avoids matrix inversions, which makes it easier to compute. Next to that, SUpPSB also performs globally better compared to pPSB.
Keywords
Control and Optimization, Applied Mathematics, Computational Mathematics, Non-linear optimization, Quasi-Newton formulae, Weighted multiple secant equations, Symmetric gradient, ADJOINT, PERFORMANCE, GRADIENT

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Citation

Please use this url to cite or link to this publication:

MLA
Boutet, Nicolas, et al. “Secant Update Version of Quasi-Newton PSB with Weighted Multisecant Equations.” COMPUTATIONAL OPTIMIZATION AND APPLICATIONS, vol. 75, 2020, pp. 441–66.
APA
Boutet, N., Haelterman, R., & Degroote, J. (2020). Secant update version of quasi-Newton PSB with weighted multisecant equations. COMPUTATIONAL OPTIMIZATION AND APPLICATIONS, 75, 441–466.
Chicago author-date
Boutet, Nicolas, Robby Haelterman, and Joris Degroote. 2020. “Secant Update Version of Quasi-Newton PSB with Weighted Multisecant Equations.” COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 75: 441–66.
Chicago author-date (all authors)
Boutet, Nicolas, Robby Haelterman, and Joris Degroote. 2020. “Secant Update Version of Quasi-Newton PSB with Weighted Multisecant Equations.” COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 75: 441–466.
Vancouver
1.
Boutet N, Haelterman R, Degroote J. Secant update version of quasi-Newton PSB with weighted multisecant equations. COMPUTATIONAL OPTIMIZATION AND APPLICATIONS. 2020;75:441–66.
IEEE
[1]
N. Boutet, R. Haelterman, and J. Degroote, “Secant update version of quasi-Newton PSB with weighted multisecant equations,” COMPUTATIONAL OPTIMIZATION AND APPLICATIONS, vol. 75, pp. 441–466, 2020.
@article{8644687,
  abstract     = {Quasi-Newton methods are often used in the frame of non-linear optimization. In those methods, the quality and cost of the estimate of the Hessian matrix has a major influence on the efficiency of the optimization algorithm, which has a huge impact for computationally costly problems. One strategy to create a more accurate estimate of the Hessian consists in maximizing the use of available information during this computation. This is done by combining different characteristics. The Powell-Symmetric-Broyden method (PSB) imposes, for example, the satisfaction of the last secant equation, which is called secant update property, and the symmetry of the Hessian (Powell in Nonlinear Programming 31-65, 1970). Imposing the satisfaction of more secant equations should be the next step to include more information into the Hessian. However, Schnabel proved that this is impossible (Schnabel in quasi-Newton methods using multiple secant equations, 1983). Penalized PSB (pPSB), works around the impossibility by giving a symmetric Hessian and penalizing the non-satisfaction of the multiple secant equations by using weight factors (Gratton et al. in Optim Methods Softw 30(4):748-755, 2015). Doing so, he loses the secant update property. In this paper, we combine the properties of PSB and pPSB by adding to pPSB the secant update property. This gives us the secant update penalized PSB (SUpPSB). This new formula that we propose also avoids matrix inversions, which makes it easier to compute. Next to that, SUpPSB also performs globally better compared to pPSB.},
  author       = {Boutet, Nicolas and Haelterman, Robby and Degroote, Joris},
  issn         = {0926-6003},
  journal      = {COMPUTATIONAL OPTIMIZATION AND APPLICATIONS},
  keywords     = {Control and Optimization,Applied Mathematics,Computational Mathematics,Non-linear optimization,Quasi-Newton formulae,Weighted multiple secant equations,Symmetric gradient,ADJOINT,PERFORMANCE,GRADIENT},
  language     = {eng},
  pages        = {441--466},
  title        = {Secant update version of quasi-Newton PSB with weighted multisecant equations},
  url          = {http://dx.doi.org/10.1007/s10589-019-00164-z},
  volume       = {75},
  year         = {2020},
}

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