On the sufficiency of finite support duals in semi-infinite linear programming

TL;DR

This paper proves that for semi-infinite linear programs with countably many constraints, the finite support dual is equivalent to the algebraic Lagrangian dual in the sequence space, clarifying duality gap issues.

Contribution

It establishes the equivalence of finite support and algebraic Lagrangian duals in the sequence space, resolving an open question from prior research.

Findings

Finite support dual equals algebraic Lagrangian dual in sequence space.

Duality gap cannot be closed by larger dual variable spaces.

Different subspaces may exhibit positive duality gaps with finite support dual.

Abstract

We consider semi-infinite linear programs with countably many constraints indexed by the natural numbers. When the constraint space is the vector space of all real valued sequences, we show the finite support (Haar) dual is equivalent to the algebraic Lagrangian dual of the linear program. This settles a question left open by Anderson and Nash [Linear programming in infinite dimensional spaces : theory and applications, Wiley 1987]. This result implies that if there is a duality gap between the primal linear program and its finite support dual, then this duality gap cannot be closed by considering the larger space of dual variables that define the algebraic Lagrangian dual. However, if the constraint space corresponds to certain subspaces of all real-valued sequences, there may be a strictly positive duality gap with the finite support dual, but a zero duality gap with the algebraic Lagrangian dual.