Abstract
In this paper, we provide an exact reformulation of Nonsmooth Constrained optimization Problems (NCP) using the Moreau-Yosida regularization. This reformulation allows the transformation of (NCP) to a sequence of convex programs of which solutions are feasible for (NCP). This sequence of solutions of auxiliary programs converges to a local solution of (NCP). Assuming Slater constraint qualification and basing on an exact penalization, our reformulation combined with a nonconvex proximal bundle method provides a local solution of (NCP). Our bundle method allows a strong update of the level set, may reduce significantly the number of null-steps and gives a new stopping criterion. Finally, numerical simulations are carried out.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Kelley Jr., J.E.: The cutting-plane method for solving convex programs. J. Soc. Ind. Appl. Math. 8, 703–712 (1960)
Kiwiel, K.C.: An exact penalty function algorithm for non-smooth convex constrained minimization problems. IMA J. Numer. Anal. 5(1), 111–119 (1985). https://academic.oup.com/imajna/article-lookup/doi/10.1093/imanum/5.1.111
Yang, Y., Pang, L., Ma, X., Shen, J.: Constrained nonconvex nonsmooth optimization via proximal bundle method. J. Optim. Theory Appl. 163(3), 900–925 (2014). https://doi.org/10.1007/s10957-014-0523-9
Ouorou, A.: A proximal cutting plane method using Chebychev center for nonsmooth convex optimization. Math. Program. 119(2), 239–271 (2009). https://doi.org/10.1007/s10107-008-0209-x
De Oliveira, W., Solodov, M.: A doubly stabilized bundle method for nonsmooth convex optimization. Math. Program. 156(1–2), 125–159 (2016). https://doi.org/10.1007/s10107-015-0873-6
Nemirovski, A.: Efficient Methods in Convex Programming (2007). https://www.semanticscholar.org/paper/ Efficient-Methods-in-Convex-Programming-Nemirovski/a2e65acd1dc3642ffd91aadc9c420573b611694d
Couenne, a solver for non-convex MINLP problems. https://www.coin-or.org/ Couenne
us-en\_analytics\_SP\_cplex-optimizer — IBM Analytics. https://www.ibm.com/analytics/cplex-optimizer
Welcome to Python.org. https://www.python.org/
Pyomo. http://www.pyomo.org/
Fooladivanda, D., Al Daoud, A., Rosenberg, C.: Joint channel allocation and user association for heterogeneous wireless cellular networks. In: 2011 IEEE 22nd International Symposium on Personal, Indoor and Mobile Radio Communications, pp. 384–390. IEEE (2011). http://ieeexplore.ieee.org/document/6139988/
Acknowledgement
The first author was supported by the German Academic Exchange Service (DAAD). Gratitute is expressed to the projects CEA-SMA and NLAGA for the support of this work. The authors thank the anonymous referees for useful comments and suggestions.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this paper
Cite this paper
Dembélé, A., Ndiaye, B.M., Ouorou, A., Degla, G. (2020). A Triple Stabilized Bundle Method for Constrained Nonconvex Nonsmooth Optimization. In: Le Thi, H., Le, H., Pham Dinh, T., Nguyen, N. (eds) Advanced Computational Methods for Knowledge Engineering. ICCSAMA 2019. Advances in Intelligent Systems and Computing, vol 1121. Springer, Cham. https://doi.org/10.1007/978-3-030-38364-0_7
Download citation
DOI: https://doi.org/10.1007/978-3-030-38364-0_7
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-38363-3
Online ISBN: 978-3-030-38364-0
eBook Packages: Intelligent Technologies and RoboticsIntelligent Technologies and Robotics (R0)