Abstract
In this paper the sensitivity analysis is discussed for the parameter-dependent optimization and constraint optimization. The sensitivity of the optimality value function with respect to the change in parameters plays a significant role in the inverse problems and the optimization theory, including economics, finance, the Hamilton–Jacobi theory, the inf-sup duality and the topological design and the bi-level optimization. We develop the calculus for the value function and present its applications in the variational calculus, the bi-level optimization and the optimal control and optimal design, shape calculus and inverse problems.
Keywords
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
G. Allaire, F. Jouve, A.M. Toader, Structural optimization using sensitivity analysis and a level-set method. J. Comput. Phys. 194, 363–393 (2004)
D.P. Bertsekas, Nonlinear Programming Belmont (Athena Scientific, Belmont, 1999)
J.F. Bonnans, J.C. Gilbert, C. Lemarechal, C.A. Sagastizabal, Numerical Optimization-Theoretical and Practical Aspects (Springer, Berlin, 2006)
P. Bernhard, A. Rapaport, On a theorem of Danskin with an application to a theorem of Von Neumann-Sion. Nonlinear Anal.: Theory Methods Appl. 24, 1163–1181 (1995)
E. Castillo, A. Conejo, C. Castillo, R. Minguez, D. Ortigosa, A perturbation approach to sensitivity. J. Optim. Theory Appl. 128, 49–74 (2006)
M.C. Delfour, J.-P. Zolesio, Shapes and Geometries: Metrics, Analysis, Differential Calculus, and Optimization, 2nd edn. (SIAM, Philadelphia, 2011)
A.V. Fiacco, Introduction to Sensitivity and Stability Analysis in Nonlinear Programming (Academic, New York, 1983)
T.H. Lee, J.S. Arora, A computational method for design sensitivity analysis of elastoplastic structures. Comput. Methods Appl. Mech. Eng. 122, 27–50 (1995)
K. Ito, B. Jin, Inverse Problems: Tikhonov Theory and Algorithms (World Scientific, Singapore, 2014)
K. Ito, B. Jin, T. Takeuchi, A regularization parameter for nonsmooth Tikhonov regularization. SIAM J. Sci. Comput. 33, 1415–1438 (2011)
K. Ito, K. Kunisch, On the choice of the regularization parameter in nonlinear inverse problems, (with K. Kunisch). SIAM J. Optim. 2, 376–404 (1992)
K. Ito, K. Kunisch, Optimal control of elliptic variational inequalities. Appl. Math. Optim. 41, 343–364 (2000)
K. Ito, K. Kunisch, Lagrange Multiplier Approach to Variational Problems and Applications. SIAM Advances in Design and Control (2008)
K. Ito, K. Kunisch, G. Peich, Variational approach to shape derivatives. ESAIM: Control Optim. Calc. Var. 14, 517–539 (2008)
M. Hintermuller, K. Kunisch, Path-following methods for a class of constrained minimization problems in function space, SIAM J. Optim. 17, 159–187 (2006)
A. Rothwell, Optimization Methods in Structural Design. Solid Mechanics and Its Applications (Springer, Berlin, 2017)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Singapore Pte Ltd.
About this paper
Cite this paper
Ito, K. (2020). Value Function Calculus and Applications. In: Cheng, J., Lu, S., Yamamoto, M. (eds) Inverse Problems and Related Topics. ICIP2 2018. Springer Proceedings in Mathematics & Statistics, vol 310. Springer, Singapore. https://doi.org/10.1007/978-981-15-1592-7_13
Download citation
DOI: https://doi.org/10.1007/978-981-15-1592-7_13
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-15-1591-0
Online ISBN: 978-981-15-1592-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)