We consider a financial market with one riskless and one risky asset. The super-replication theorem states that there is no duality gap in the problem of super-replicating a contingent claim under transaction costs and the associated dual problem. We give two versions of this theorem. The first theorem relates a numéraire-based admissibility condition in the primal problem to the notion of a lo...

A gauge function f(.) is a nonnegative convex function that is positively homogeneous and satisfies f(O)=O. Norms and pseudonorms are specific instances of a gauge function. This paper presents a gauge duality theory for a gauge program, which is the problem of minimizing the value of a gauge function f(.) over a convex set. The gauge dual program is also a gauge program, unlike the standard La...

We study the optimal transport between two probability measures on the real line, where the transport plans are laws of one-step martingales. A quasi-sure formulation of the dual problem is introduced and shown to yield a complete duality theory for general marginals and measurable reward (cost) functions: absence of a duality gap and existence of dual optimizers. Both properties are shown to f...

Wei’s celebrated Duality Theorem is generalized in several ways, expressed as duality theorems for linear codes over division rings and, more generally, duality theorems for matroids. These results are further generalized, resulting in two Wei-type duality theorems for new combinatorial structures that are introduced and named demi-matroids. These generalize matroids and are the appropriate com...

We study the duality between corings and ring extensions. We construct a new category with a self-dual functor acting on it, which extends that duality. This construction can be seen as the non-commutative case of another duality extension: the duality between finite dimensional algebras and coalgebra. Both these duality extensions have some similarities with the Pontryagin-van Kampen duality t...

In solving certain optimization problems, the corresponding Lagrangian dual problem is sometimes solved simply because in these problems the dual problem is easier to solve than the original primal problem. Another reason for their solution is the implication of the weak duality theorem which suggests that for all optimization problems the optimal dual function value is smaller than or equal to...

Robust optimization problems, which have uncertain data, are considered. We prove surrogate duality theorems for robust quasiconvex optimization problems and surrogate min-max duality theorems for robust convex optimization problems. We give necessary and sufficient constraint qualifications for surrogate duality and surrogate min-max duality, and show some examples at which such duality result...