Vol. 62, No. 1, 1976

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ISSN: 0030-8730
Stochastic convex programming: basic duality

Ralph Tyrrell Rockafellar and Roger Jean-Baptiste Robert Wets

Vol. 62 (1976), No. 1, 173–195

A duality theory is developed for stochastic programs with convex objective and convex constraints. The problem consists in selecting x1 Rn1 and x2 ∈ℒ(S,Σ;Rn2) so as to satisfy the constraints and minimize total expected cost, where σ is a probability measure and the constraints as well as the objective are functions of the random elements of the problem. Under the additional restriction that x1 and x2(s) belong to compact subsets of Rn1 and Rn2 respectively, it is shown that the problem is equivalent to the more common dynamic formulation for stochastic programs with recourse, a basic duality theorem — of the type min = sup — is proved and qualitative results on the existence of dual solutions are derived.

Mathematical Subject Classification 2000
Primary: 90C15
Received: 5 March 1975
Published: 1 January 1976
Ralph Tyrrell Rockafellar
Roger Jean-Baptiste Robert Wets