On Distributionally Robust Chance-Constrained Linear Programs1

In this paper, we discuss linear programs in which the data that specify the constraints are subject to random uncertainty. A usual approach in this setting is to enforce the constraints up to a given level of probability. We show that, for a wide class of probability distributions (namely, radial distributions) on the data, the probability constraints can be converted explicitly into convex second-order cone constraints; hence, the probability-constrained linear program can be solved exactly with great efficiency. Next, we analyze the situation where the probability distribution of the data is not completely specified, but is only known to belong to a given class of distributions. In this case, we provide explicit convex conditions that guarantee the satisfaction of the probability constraints for any possible distribution belonging to the given class.

[1]  Giuseppe Carlo Calafiore,et al.  Distributionally Robust Chance-Constrained Linear Programs with Applications , 2006 .

[2]  Ioana Popescu,et al.  Optimal Inequalities in Probability Theory: A Convex Optimization Approach , 2005, SIAM J. Optim..

[3]  Laurent El Ghaoui,et al.  Worst-Case Value-At-Risk and Robust Portfolio Optimization: A Conic Programming Approach , 2003, Oper. Res..

[4]  N. Cristianini,et al.  Estimating the moments of a random vector with applications , 2003 .

[5]  G. Calafiore,et al.  Loop gain under random feedback , 2001, Proceedings of the 40th IEEE Conference on Decision and Control (Cat. No.01CH37228).

[6]  Michael I. Jordan,et al.  Minimax Probability Machine , 2001, NIPS.

[7]  Xiang Li,et al.  Application of probabilistically constrained linear programs to risk-adjusted controller design , 2001, Proceedings of the 2001 American Control Conference. (Cat. No.01CH37148).

[8]  Boris Polyak,et al.  Random spherical uncertainty in estimation and robustness , 2000, Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No.00CH37187).

[9]  Stan Uryasev,et al.  Probabilistic Constrained Optimization , 2000 .

[10]  Constantino Lagoa,et al.  On the convexity of probabilistically constrained linear programs , 1999, Proceedings of the 38th IEEE Conference on Decision and Control (Cat. No.99CH36304).

[11]  A. Rukhin.  Matrix Variate Distributions , 1999, The Multivariate Normal Distribution.

[12]  Arkadi Nemirovski,et al.  Robust solutions of uncertain linear programs , 1999, Oper. Res. Lett..

[13]  R. Tempo,et al.  Radially truncated uniform distributions for probabilistic robustness of control systems , 1997, Proceedings of the 1997 American Control Conference (Cat. No.97CH36041).

[14]  B. Ross Barmish,et al.  The uniform distribution: A rigorous justification for its use in robustness analysis , 1996, Math. Control. Signals Syst..

[15]  R. Wets,et al.  Stochastic programming , 1989 .

[16]  John G. Proakis,et al.  Probability, random variables and stochastic processes , 1985, IEEE Trans. Acoust. Speech Signal Process..

[17]  C. Panne,et al.  Minimum-Cost Cattle Feed Under Probabilistic Protein Constraints , 1963 .

[18]  A. Charnes,et al.  Deterministic Equivalents for Optimizing and Satisficing under Chance Constraints , 1963 .

[19]  W. Hoeffding.  Probability inequalities for sum of bounded random variables , 1963 .

[20]  I. Olkin,et al.  Multivariate Chebyshev Inequalities , 1960 .