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The Slater Conundrum: Duality and Pricing in Infinite Dimensional Optimization

Kipp Martin (kipp.martin***at***chicagobooth.edu)
Christopher Thomas Ryan (chris.ryan***at***chicagobooth.edu)
Matthew Stern (matt.stern***at***chicagobooth.edu)

Abstract: Duality theory is pervasive in finite dimensional optimization. There is growing interest in solving infinite-dimensional optimization problems and hence a corresponding interest in duality theory in infinite dimensions. Unfortunately, many of the intuitions and interpretations common to finite dimensions do not extend to infinite dimensions. In finite dimensions, a dual solution is represented by a vector of dual prices that index the primal constraints and have a natural economic interpretation. In infinite dimensions, we show that this simple dual structure, and its associated economic interpretation, may fail to hold for a broad class of problems with constraint vector spaces that are \sigma-order complete Riesz spaces (ordered vector spaces with a lattice structure). In these spaces we show that the existence of interior points required by common constraint qualifications for zero duality gap (such as Slater's condition) imply the ex- istence of singular dual solutions that are difficult to find and interpret. We call this phenomenon the Slater conundrum: interior points ensure zero duality gap (a desirable property), but interior points also imply the existence of singular dual solutions (an undesirable property). A Riesz space is the most parsimonious vector-space structure sufficient to characterize the Slater conundrum. Topological vector spaces common to the vast majority of infinite dimensional optimization literature are not necessary.

Keywords: Infinite dimensional optimization, duality theory, dual prices, constraint qualifications, Riesz spaces

Category 1: Infinite Dimensional Optimization

Citation:

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Entry Submitted: 04/04/2014
Entry Accepted: 04/05/2014
Entry Last Modified: 04/20/2015

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