In this paper a bundle method for nonconvex nonsmooth optimization in infinite-dimensional Hilbert spaces is developed and analyzed. The algorithm requires only inexact function value and subgradient information. Global convergence to approximately stationary points is proved, where the final accuracy depends on the error level in the function and subgradient data. The method is then applied to an optimal control problem governed by the obstacle problem. For adaptively controlling the inexactness, implementable conditions are developed, first on a general level and then for the concrete case of a FEM discretization for optimal control of an obstacle problem. Numerical results are presented.


  1. nonsmooth optimization
  2. nonconvex bundle method
  3. inexact function values
  4. inexact subgradients
  5. optimal control of obstacle problem
  6. error estimates

MSC codes

  1. 65K05
  2. 90C56
  3. 49M37
  4. 49J52
  5. 49K20
  6. 90C26

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Information & Authors


Published In

cover image SIAM Journal on Control and Optimization
SIAM Journal on Control and Optimization
Pages: 3137 - 3165
ISSN (online): 1095-7138


Submitted: 22 October 2018
Accepted: 3 July 2019
Published online: 17 September 2019


  1. nonsmooth optimization
  2. nonconvex bundle method
  3. inexact function values
  4. inexact subgradients
  5. optimal control of obstacle problem
  6. error estimates

MSC codes

  1. 65K05
  2. 90C56
  3. 49M37
  4. 49J52
  5. 49K20
  6. 90C26



Funding Information

Deutsche Forschungsgemeinschaft https://doi.org/10.13039/501100001659 : 1962

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