Mathematical Programming

, Volume 89, Issue 1, pp 55–77

Concavity and efficient points of discrete distributions in probabilistic programming


  • Darinka Dentcheva
    • Department of Mathematical Sciences, Stevens Institute of Technology, Hoboken, NJ 07030, USA, and RUTCOR (Rutgers University Center for Operations Research), Piscataway, NJ 08854, USA, e-mail:
  • András Prékopa
    • RUTCOR, e-mail:
  • Andrzej Ruszczynski
    • RUTCOR and Department of Management Science and Information Systems, Rutgers University, Piscataway, NJ 08854, USA, e-mail:

DOI: 10.1007/PL00011393

Cite this article as:
Dentcheva, D., Prékopa, A. & Ruszczynski, A. Math. Program. (2000) 89: 55. doi:10.1007/PL00011393


We consider stochastic programming problems with probabilistic constraints involving integer-valued random variables. The concept of a p-efficient point of a probability distribution is used to derive various equivalent problem formulations. Next we introduce the concept of r-concave discrete probability distributions and analyse its relevance for problems under consideration. These notions are used to derive lower and upper bounds for the optimal value of probabilistically constrained stochastic programming problems with discrete random variables. The results are illustrated with numerical examples.

Key words: probabilistic programming – discrete distributions – generalized concavity – column generation Mathematics Subject Classification (1991): 90C15, 90C11, 65K05, 49M27

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© Springer-Verlag Berlin Heidelberg 2000