Fetching the paper…
Reading the bibliography…
Given $X \subset R^n$, $\varepsilon \in (0,1)$, a parametrized family of probability distributions $(\mu\_{a})\_{a\in A}$ on $\Omega\subset R^p$, we consider the feasible set $X^*\_\varepsilon\subset X$ associated with the {\em distributionally robust} chance-constraint \[X^*\_\varepsilon\,=\,\{x \in X :\:{\rm Prob}\_\mu[f(x,\omega)\,>\,0]> 1-\varepsilon,\,\forall\mu\in M\_a\},\]where $M\_a$ is the set of all possibles mixtures of distributions $\mu\_a$, $a\in A$.For instance and typically, the family$M\_a$ is the set of all mixtures ofGaussian distributions on $R$ with mean and standard deviation $a=(a,\sigma)$ in some compact set $A\subset R^2$.We provide a sequence of inner approximations $X^d\_\varepsilon=\{x\in X: w\_d(x) <\varepsilon\}$, $d\in N$, where $w\_d$ is a polynomial of degree $d$ whosevector of coefficients is an optimal solution of a semidefinite program.The size of the latter increases with the degree $d$.
A. Charnes, W.W. Cooper. Chance constrained programming. Manag. Sci. 6, pp. 73–79, 1959
1959
Earlier work this paper cites.
B. Miller, H. Wagner. Chance-constrained programming with joint constraints. Oper. Res. 13, pp. 930–945, 1965 34
1965
Earlier work this paper cites.
P. Billingsley. Convergence of Probability Measures
1968
Earlier work this paper cites.
R. Ash. Real Analysis and Probability
1972
Earlier work this paper cites.
J.S. Marron, M.P. Wand. Exact mean integrated squared error, The Ann. Statist. 20, pp. 712–736, 1992
1992
Earlier work this paper cites.
M. Putinar, Positive polynomials on compact semi-algebraic sets, Indiana Univ. Math. J. 42, pp. 969–984, 1993
1993
Earlier work this paper cites.
J.L. Doob. Measure Theory
1994
Earlier work this paper cites.
P. Li, M. Wendt, G. Wozny, A Probabilistically Constrained Model Predictive Controller, Automatica 38, pp. 1171–1176, 2002
2002
Earlier work this paper cites.
L. El Ghaoui, M. Oks, F. Oustry. Worst-case value-at-risk and robust portfolio optimization: A conic programming approach, Oper. Res. 51, pp. 543–556, 2003
2003
Earlier work this paper cites.
A. Prékopa, Probabilistic Programming, in Stochastic Programming, A. Ruszczynski and A. Shapiro (Eds.), Handbooks in Operations Research and Management Science Volume 10, pp. 267–351, 2003
2003
Earlier work this paper cites.
G. Calafiore, F. Dabbene, Probabilistic and Randomized Methods for Design under Uncertainty, G. Calafiore and F. Dabbene (Eds.), Springer, 2006
2006
Earlier work this paper cites.
G.C. Calafiore, L. El Ghaoui. On Distributionally Robust Chance-Constrained Linear Programs, J. Optim. Theory Appl. 130, pp. 1-22, 2006
2006
Earlier work this paper cites.
E. Erdogan, G. Iyengar. Ambiguous chance constrained problems and robust optimization. Math. Program. Sér. B 107, pp. 37–61, 2006
2006
Earlier work this paper cites.
J.B. Lasserre, T. Netzer. SOS approximations of nonnegative polynomials via simple high degree perturbations”. Math. Zeitschrift 256, pp. 99–112, 2006
2006
Earlier work this paper cites.
A. Nemirovski, A. Shapiro. Convex approximations of chance constrained programs. SIAM J. Optim. 17, pp. 969–996, 2006
2006
Earlier work this paper cites.
H. Waki, S. Kim, M. Kojima, M. Muramatsu, Sums of squares and semidefinite programming relaxations for polynomial optimization problems with structured sparsity, SIAM J. Optim. 17, pp. 218–242, 2006
2006
Earlier work this paper cites.
M. Di Zioa, U. Guarneraa, R. Roccib. A mixture of mixture models for a classification problem: The unity measure error, Computational Statistics and Data Analysis 51, pp. 2573–2585, 2007
2007
Cited alongside, same era.
R. Henrion. Structural Properties of Linear Probabilistic Constraints, Optimization 56, pp. 425–440, 2007
2007
Cited alongside, same era.
R. Henrion, C. Strugarek. Convexity of chance constraints with independent random variables. Computational Optimization and Applications 41, pp. 263–276, 2008
2008
Cited alongside, same era.
D. Henrion, J.B. Lasserre, J. Lofberg. Gloptipoly 3: moments, optimization and semidefinite programming, Optim. Methods and Softwares 24, pp. 761–779, 2009
2009
Cited alongside, same era.
D. Henrion, J. B. Lasserre, C. Savorgnan, Approximate volume and integration for basic semi-algebraic sets, SIAM Review 51, pp. 722–743, 2009
A. M. Jasour, N.S. Aybat, C. M. Lagoa, Semidefinite programming for chance constrained optimization over semi-algebraic sets, SIAM J. Optim. 25, No 3, pp. 1411–1440, 2015
2015
Later among the works it cites.
S.I. Wang, A.T. Chaganti, P. Liang. Estimating Mixture Models via Mixtures of Polynomials Proceedings of the 28th International Conference on Neural Information Processing Systems (NIPS’15), Montreal, December 2015, pp. 487–495
2015
Later among the works it cites.
R. Jiang, Y. Guan. Data-driven chance constrained stochastic program. Math. Program. 158, 291-327, 2016
2016
Later among the works it cites.
J.B. Lasserre, Lebesgue decomposition in action via semidefinite relaxations, Adv. Comput. Math. 42, pp. 1129–1148, 2016
2016
Later among the works it cites.
Wenzhuo Yang, Huan Xu. Distributionally Robust Chance Constraints for Non-Linear Uncertainties, Math. Program. 155, pp. 231–265, 2016
alphaXiv searches the wider corpus for related work and actual follow-ups.
alphaXiv is searching for related work…
2009
Cited alongside, same era.
D. Xu. The Applications of Mixtures of Normal Distributions in Empirical Finance: A Selected Survey, Working Papers 0904, University of Waterloo, Department of Economics, revised September 2009
2009
Cited alongside, same era.
W. Chen, M. Sim, J. Sun, C.P. Teo. From CVaR to uncertainty set: Implications in joint chance-constrained optimization. Operations research, 58(2), 470-485, 2010
2010
Cited alongside, same era.
E. Delage, Y. Ye. Distributionally robust optimization under moment uncertainty with applications to data-driven problems. Operations Research 58, pp. 595–612, 2010
2010
Cited alongside, same era.
J.B. Lasserre, A “Joint+Marginal” approach to parametric polynomial optimization, SIAM J. Optim. 20, pp. 1995–2022, 2010
2010
Cited alongside, same era.
J. B. Lasserre, Moments, positive polynomials and their applications. Imperial College Press, London, 2010
2010
Cited alongside, same era.
J.B. Lasserre, The K-moment problem for continuous linear functionals, Trans. Amer. Math. Soc. 365, pp. 2489–2504, 2013
2013
Cited alongside, same era.
2013
Cited alongside, same era.
2016
Later among the works it cites.
Zhang, Y., Shen, S., J. Mathieu. Distributionally robust chance-constrained optimal power flow with uncertain renewables and uncertain reserves provided by loads, IEEE Trans. Power Systems 32, pp. 1378–1388, 2016
2016
Later among the works it cites.
G.A. Hanasusanto, V. Roitch, D. Kuhn, W. Wiesemann. Ambiguous joint chance constraints under mean and dispersion information. Oper. Res. 65, pp. 751-767, 2017
2017
Later among the works it cites.
2017
Later among the works it cites.
J.B. Lasserre, Representation of chance-constraints with strong asymptotic properties, IEEE Control Systems Letters 1, pp. 50–55, 2017
2017
Later among the works it cites.
J.B. Lasserre, Computing gaussian and exponential measures of semi-algebraic sets, Adv. Appl. Math. 91, pp. 137–163, 2017
2017
Later among the works it cites.
Mosek Aps, Mosek Matlab Toolbox, 2017
2017
Later among the works it cites.
S. Wang, J. Li, C. Peng. Distributionally robust chance-constrained program surgery planning with downstream resource, Proceedings 2017 International Conference on Service Systems and Service Management, Dalian, China, June 2017
2017
Later among the works it cites.
W. Xie, S. Ahmed. Distributionally robust chance constrained optimal power flow with renewables: A conic reformulation, IEEE Trans. Power Systems, 2017
2017
Later among the works it cites.
W. van Ackooij, J. Malick. Eventual convexity of probability constraints with elliptical distributions, Math. Program., pp. 1–27, 2018
2018
Closest in time.
X. Tong, H. Sun, X. Luo, Q. Zheng, Distributionally robust chance constrained optimization for economic dispatch in renewable energy integrated systems, J. Global Optim. 70, pp. 131–158, 2018
2018
Closest in time.
W. Xie, S. Ahmed. On deterministic reformulations of distributionally robust joint chance constrained optimization problems. SIAM J. Optim. 28, pp. 1151-1182, 2018
2018
Closest in time.