The Slater Conundrum: Duality and Pricing in Infinite-Dimensional Optimization

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 Riesz spaces (ordered vector spaces with a lattice structure) that either are σ-order complete or satisfy the projection property. 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) implies the existence 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). Riesz spaces are the most parsimonious vector-space structure sufficient to characterize the Slater conundrum. Finally, we provide sufficient conditions that “resolve” the Slater conundrum; that is, guarantee that in every solvable dual there exists an optimal dual solution that is not singular.