Duality in Countably Infinite Linear Programs
Posted: 18 Sep 2014
Date Written: September 16, 2014
Duality results on countably infinite linear programs (CILPs) are scarce. Roughly speaking, one fundamental hurdle is as follows. Subspaces that admit an interior point, which is a sufficient condition for a zero duality gap, yield a dual problem where the constraints cannot be expressed using the ordinary transpose of the primal doubly-infinite constraint matrix. Subspaces that do allow us to write the dual using this ordinary transpose do not admit an interior point. This has recently been called the "Slater conundrum." We find a way around this hurdle.
We propose a pair of primal-dual spaces with three properties: (i) the series in the primal and dual objective functions converge; (ii) the series defined by the rows and columns of the primal constraint matrix converge; and (iii) the order of sums in a particular iterated series of a double sequence defined by the primal constraint matrix can be interchanged so that the dual is defined by the ordinary transpose. Weak duality and complementary slackness are then immediate. Instead of using interior point conditions to establish a zero duality gap, we call upon the planning horizon method. When the series in the primal and dual constraints are continuous, we prove that strong duality holds if a sequence of optimal solutions to finite-dimensional truncations of the primal and dual CILPs has an accumulation point. We show by counterexample that the requirement that such an accumulation point exist cannot be relaxed. We also prove that the Lagrangean of our CILPs provides a linear support to the primal optimal value function. Finally, we establish that saddle points of the Lagrangean are equivalent to complementary feasible solutions of our CILPs. Our results are illustrated using several examples, and are also applied to countable-state Markov decision processes and to a problem in robust optimization.
Keywords: infinite dimensional optimization, linear programming, duality
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