Partially finite convex programming, Part I: Quasi relative interiors and duality theory

The authors study convex programs that involve the minimization of a convex function over a convex subset of a topological vector space, subject to a finite number of linear inequalities. They develop the notion of the quasi relative interior of a convex set, an extension of the relative interior in finite dimensions. The authors use this idea in a constraint qualification for a fundamental Fenchel duality result, and then deduce duality results for these problems despite the almost invariable failure of the standard Slater condition. Part II of this work studies applications to more concrete models, whose dual problems are often finite-dimensional and computationally tractable.