• Bot, Radu Ioan (Chemnitz University of Technology Faculty of Mathematics) ;
  • Lorenz, Nicole (Chemnitz University of Technology Faculty of Mathematics) ;
  • Wanka, Gert (Chemnitz University of Technology Faculty of Mathematics)
  • Published : 2010.01.01


In this paper we deal with linear chance-constrained optimization problems, a class of problems which naturally arise in practical applications in finance, engineering, transportation and scheduling, where decisions are made in presence of uncertainty. After giving the deterministic equivalent formulation of a linear chance-constrained optimization problem we construct a conjugate dual problem to it. Then we provide for this primal-dual pair weak sufficient conditions which ensure strong duality. In this way we generalize some results recently given in the literature. We also apply the general duality scheme to a portfolio optimization problem, a fact that allows us to derive necessary and sufficient optimality conditions for it.


  1. F. M. Allen, R. N. Braswell, and P. V. Rao, Distribution-free approximations for chance constraints, Oper. Res. 22 (1974), no. 3, 610-621
  2. R. I. Bot¸, I. B. Hodrea, and G. Wanka, Some new Farkas-type results for inequality systems with DC functions, J. Global Optim. 39 (2007), no. 4, 595-608
  3. R. I. Bot¸, N. Lorenz, and G. Wanka, Dual Representations for Convex Risk Measures via Conjugate Duality, to appear in Journal of Optimization Theory and Applications, 2009, DOI: 10.1007/s10957-009-9595-3
  4. P. Bonami and M. A. Lejeune, An exact solution approach for portfolio optimization problems under stochastic and integer constraints,, 2007
  5. G. Calafiore and L. El Ghaouni, On distributionally robust chance-constrained linear programs, J. Optim. Theory Appl. 130 (2006), no. 1, 1-22
  6. A. Charnes and W. W. Cooper, Chance-constrained programming, Management Sci. 6 (1959/1960), 73-79
  7. A. Charnes and W. W. Cooper, Deterministic equivalents for optimizing and satisficing under chance constraints, Oper. Res. 11 (1963), 18-39
  8. A. Charnes, W. W. Cooper, and G. H. Symonds, Cost horizons and certainty equivalents: An approach to stochastic programming of heating oil, Management Sci. 4 (1958), no. 3, 235-263
  9. W. K. K. Haneveld and M. H. van der Vlerk, Integrated chance constraints: reduced forms and an algorithm, Comput. Manag. Sci. 3 (2006), no. 4, 245-269
  10. R. Henrion, Structural properties of linear probabilistic constraints, Optimization 56 (2007), no. 4, 425-440
  11. P. Kall and S. W. Wallace, Stochastic programming, John Wiley & Sons, 1994
  12. S. Kataoka, A stochastic programming model, Econometrica 31 (1963), 181-196
  13. C. M. Lagoa, X. Li, and M. Sznaier, Probabilistically constrained linear programs and risk-adjusted controller design, SIAM J. Optim. 15 (2005), no. 3, 938-951
  14. A. Prekopa, Stochastic Programming, Kluwer Academic Publishers, 1995
  15. R. T. Rockafellar, Convex Analysis, Princeton University Press, 1970
  16. A. Ruszczynski and A. Shapiro (Eds.), Stochastic Programming, Handbooks in Operations Research and Management Science, Vol. 10, Elsevier, Amsterdam, 2003
  17. C. H. Scott and T. R. Jefferson, On duality for square root convex programs, Math. Methods Oper. Res. 65 (2007), no. 1, 75-84
  18. C. van de Panne and W. Popp, Minimum-cost cattle feed under probabilistic protein constraints, Management Sci. 9 (1963), no. 3, 405-430

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