Abstract
This paper deals with stochastic semidefinite chance constrained problems. Semidefinite optimization generalizes linear programs, and generally solves deterministic optimization. We propose a new sampling method to solve chance constrained semidefinite optimization problems. Numerical results are given in order to compare the performances of our approach to the state-of-the-art.
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
Similar content being viewed by others
References
Charnes, A., Cooper, W.W., Symonds, G.H.: Cost horizons and certainty equivalents: an approach to stochastic programming of heating oil. Manag. Sci. 4, 235–263 (1958)
Prékopa, A.: On probabilistic constrained programming. In: Proceedings of the Princeton symposium on mathematical programming, pp. 113–138. Princeton University Press, Princeton (1970)
Dentcheva, D., Prékopa, A., Ruszczynski, A.: Concavity and efficient points of discrete distributions in probabilistic programming. Math. Program. 89, 55–77 (2000)
Prékopa, A.: Probabilistic programming. In: Handbooks in Operations Research and Management Science, vol. 10, pp. 267–351 (2003)
Henrion, R., Strugarek, C.: Convexity of chance constraints with independent random variables. Comput. Optim. Appl. 41, 263–276 (2008)
Nemirovski, A., Shapiro, A.: Convex approximations of chance constrained programs. SIAM J. Optim. 17, 969–996 (2006)
Nemirovski, A.: On safe tractable approximations of chance constraints. Eur. J. Oper. Res. 219, 707–718 (2012)
Calafiore, G., Campi, M.C.: Uncertain convex programs: randomized solutions and confidence levels. Math. Program. 102, 25–46 (2005)
Calafiore, G.C., Campi, M.C.: The scenario approach to robust control design. IEEE Trans. Autom. Control 51, 742–753 (2006)
Nemirovski, A., Shapiro, A.: Scenario approximations of chance constraints. In: Calafiore, G., Dabbene, F. (eds.) Probabilistic and Randomized Methods for Design Under Uncertainty, pp. 3–47. Springer, London (2006)
Campi, M.C., Garatti, S.: A sampling-and-discarding approach to chance-constrained optimization: feasibility and optimality. J. Optim. Theory Appl. 148, 257–280 (2011)
Pagnoncelli, B.K., Reich, D., Campi, M.C.: Risk-return trade-off with the scenario approach in practice: a case study in portfolio selection. J. Optim. Theory Appl. 155, 707–722 (2012)
Garatti, S., Campi, M.C.: Modulating robustness in control design: principles and algorithms. IEEE Control Syst. 33, 36–51 (2013)
Cheung, S.S., Man-Cho So, A., Wang, K.: Linear matrix inequalities with stochastically dependent perturbations and applications to chance-constrained semidefinite optimization. SIAM J. Optim. 22, 1394–1430 (2012)
Ariyawansa, K., Zhu, Y.: Chance-Constrained Semidefinite Programming. Technical report, DTIC Document (2000)
Yao, D.D., Zhang, S., Zhou, X.Y.: LQ Control without Riccati Equations: Stochastic Systems. FEW-Econometrie en besliskunde (1999)
Zhu, Y.: Semidefinite Programming Under Uncertainty. Ph.D. thesis, Washington State University (2006)
Luedtke, J., Ahmed, S.: A sample approximation approach for optimization with probabilistic constraints. SIAM J. Optim. 19, 674–699 (2008)
Nemirovski, A.: Lectures on modern convex optimization. In: Society for Industrial and Applied Mathematics (SIAM), Citeseer (2001)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Xu, C., Cheng, J., Lisser, A. (2015). Stochastic Semidefinite Optimization Using Sampling Methods. In: de Werra, D., Parlier, G., Vitoriano, B. (eds) Operations Research and Enterprise Systems. ICORES 2015. Communications in Computer and Information Science, vol 577. Springer, Cham. https://doi.org/10.1007/978-3-319-27680-9_6
Download citation
DOI: https://doi.org/10.1007/978-3-319-27680-9_6
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-27679-3
Online ISBN: 978-3-319-27680-9
eBook Packages: Computer ScienceComputer Science (R0)