Abstract
We consider the proximal form of a bundle algorithm for minimizing a nonsmooth convex function, assuming that the function and subgradient values are evaluated approximately. We show how these approximations should be controlled in order to satisfy the desired optimality tolerance. For example, this is relevant in the context of Lagrangian relaxation, where obtaining exact information about the function and subgradient values involves solving exactly a certain optimization problem, which can be relatively costly (and as we show, in any case unnecessary). We show that approximation with some finite precision is sufficient in this setting and give an explicit characterization of this precision. Alternatively, our result can be viewed as a stability analysis of standard proximal bundle methods, as it answers the following question: for a given approximation error, what kind of approximate solution can be obtained and how does it depend on the magnitude of the perturbation?
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Kiwiel, K. C., Methods of Descent for Nondifferentiable Optimization. Lecture Notes in Mathematics, Springer Verlag, Berlin, Germany, Vol. 1133, 1985.
Hiriart-Urruty, J. B., and LemarÉchal, C., Convex Analysis and Minimization Algorithms, Springer Verlag, Berlin, Germany, 1993.
Bonnans, J. F., Gilbert, J. C., LemarÉchal, C., and SagastizÁbal, C. A., Optimisation Numérique: Aspects Théoriques et Pratiques, Springer Verlag, Berlin, Germany, 1997.
Bertsekas, D. P., and Tsitsiklis, J. N., Parallel and Distributed Computation, Prentice-Hall, Englewood Cliffs, New Jersey, 1989.
LemarÉchal, C., Lagrangian Decomposition and Nonsmooth Optimization: Bundle Algorithm, Prox Iteration, Augmented Lagrangian, in Nonsmooth Optimization: Methods and Applications, Edited by F. Giannessi, Gordon and Breach, Philadelphia, Pennsylvania, pp. 201-216, 1992.
Bertsekas, D. P., Nonlinear Programming, Athena Scientific, Belmont, Massachusetts, 1995.
Solodov, M. V., Convergence Analysis of Perturbed Feasible Descent Methods, Journal of Optimization Theory and Applications, Vol. 93, pp. 337-353, 1997.
Solodov, M. V., and Zavriev, S. K., Error Stability Properties of Generalized Gradient-Type Algorithms, Journal of Optimization Theory and Applications, Vol. 98, pp. 663-680, 1998.
Kiwiel, K. C., Approximations in Proximal Bundle Methods and Decomposition of Convex Programs, Journal of Optimization Theory and Applications, Vol. 84, pp. 529-548, 1995.
HintermÜller, M., A Proximal Bundle Method Based on Approximate Subgradients, Computational Optimization and Applications, Vol. 20, pp. 245-266, 2001.
Miller, S. A., An Inexact Bundle Method for Solving Large Structured Linear Matrix Inequalities, PhD Thesis, University of California, Santa Barbara, California, 2001.
Kiwiel, K. C., A Method for Solving Certain Quadratic Programming Problems Arising in Nonsmooth Optimization, IMA Journal of Numerical Analysis, Vol. 6, pp. 137-152, 1986.
Kiwiel, K. C., Proximity Control in Bundle Methods for Convex Nondifferentiable Minimization, Mathematical Programming, Vol. 46, pp. 105-122, 1990.
Schramm, H., and Zowe, J., A Version of the Bundle Idea for Minimizing a Nonsmooth Function: Conceptual Idea, Convergence Analysis, Numerical Results, SIAM Journal on Optimization, Vol. 2, pp. 121-152, 1992.
Bonnans, J. F., Gilbert, J. C., LemarÉchal, C., and SagastizÁbal, C., A Family of Variable-Metric Proximal-Point Methods, Mathematical Programming, Vol. 68, pp. 15-47, 1995.
LemarÉchal, C., and SagastizÁbal, C., An Approach to Variable-Metric Bundle Methods, System Modelling and Optimization, Lecture Notes in Control and Information Sciences, Edited by J. Henry, and J. P. Yvon, Springer, Berlin, Germany, Vol. 197, pp. 144-162, 1994.
Mifflin, R., A Quasi-Second-Order Proximal Bundle Algorithm, Mathematical Programming, Vol. 73, pp. 51-72, 1996.
LemarÉchal, C., and SagastizÁbal, C., Variable-Metric Bundle Methods: From Conceptual to Implementable Forms, Mathematical Programming, Vol. 76, pp. 393-410, 1997.
Chen, X., and Fukushima, M., Proximal Quasi-Newton Methods for Nondifferentiable Convex Optimization, Mathematical Programming, Vol. 85, pp. 313-334, 1999.
LukŠan, L., and VlcŠek, J., Globally Convergent Variable-Metric Method for Convex Nonsmooth Unconstrained Optimization, Journal of Optimization Theory and Applications, Vol. 102, pp. 593-613, 1999.
Mangasarian, O. L., Nonlinear Programming, McGraw-Hill, New York, New York, 1969.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Solodov, M.V. On Approximations with Finite Precision in Bundle Methods for Nonsmooth Optimization. Journal of Optimization Theory and Applications 119, 151–165 (2003). https://doi.org/10.1023/B:JOTA.0000005046.70410.02
Issue Date:
DOI: https://doi.org/10.1023/B:JOTA.0000005046.70410.02