Vol. 62, No. 2, 1976

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Stochastic convex programming: singular multipliers and extended duality singular multipliers and duality

Ralph Tyrrell Rockafellar and Roger Jean-Baptiste Robert Wets

Vol. 62 (1976), No. 2, 507–522

A two-stage stochastic programming problem with recourse is studied here in terms of an extended Lagrangian function which allows certain multipliers to be elements of a dual space (), rather than an 1 space. Such multipliers can be decomposed into an 1-component and a “singular” component. The generalization makes it possible to characterize solutions to the problem in terms of a saddle-point, if the problem is strictly feasible. The Kuhn-Tucker conditions for the basic duality framework are modified to admit singular multipliers. It is shown that the optimal multiplier vectors in the extended dual problem are, in at least one broad case, ideal limits of maximizing sequences of multiplier vectors in the basic dual problem.

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