Let G = Spin[4n+1] be the connected, simply connected complex Lie group of type B2n and let G = Spin(p, q) (p + q = 4n + 1) denote a (connected) real form. If q / ∈ {0, 1}, G has a nontrivial fundamental group and we denote the corresponding nonalgebraic double cover by G̃ = ̃ Spin(p, q). The main purpose of this paper is to describe a symmetry in the set of genuine parameters for the various G̃ ...

Given a hyperplane arrangement in an affine space equipped with a linear functional, we define two finite-dimensional, noncommutative algebras, both of which are motivated by the geometry of hypertoric varieties. We show that these algebras are Koszul dual to each other, and that the roles of the two algebras are reversed by Gale duality. We also study the centers and representation categories ...

Classical Ramsey theory (at least in its simplest form) is concerned with problems of the following kind: given a set X and a colouring of the set [X] of unordered n-tuples from X, find a subset Y ⊆ X such that all n-tuples in [Y ] get the same colour. Subsets with this property are called monochromatic or homogeneous, and a typical positive result in Ramsey theory has the form that when X is l...

This article provides an overview on regularity conditions for Fenchel duality in convex optimization. Our attention is focused, on the one hand, on three generalized interior-point regularity conditions expressed by means of the quasi interior and of the quasi-relative interior and, on the other hand, on two closednesstype conditions that have been recently introduced in the literature. We dis...

In the paper, we describe various applications of the closedness and duality theorems of [7] and [8]. First, the strong separability of a polyhedron and a linear image of a convex set is characterized. Then, it is shown how stability conditions (known from the generalized Fenchel-Rockafellar duality theory) can be reformulated as closedness conditions. Finally, we present a generalized Lagrange...

This article contains the proof of a theorem on orthogonal-Pin duality that was cited without in previous this journal.

This paper is based on the author’s thesis, “On duality theorems for quasiconvex programming”. In this paper, we investigate duality theorems for quasiconvex programming as generalizations of results in convex programming, and consists of three topics. The first topic is about quasiconjugates and polar sets. The second is about three types of set containment characterizations. The third is abou...

Last lecture we studied duality of linear programs (LP), specifically how to construct the dual, the relation between the optimum of an LP and its dual, and some duality applications. In this lecture, we will talk about another application of duality to prove one of the theorems in combinatorics so called Maximum Flow-Minimum Cut Problem. The theorem roughly says that in any graph, the value of...

In this paper we present a robust duality theory for generalized convex programming problems in the face of data uncertainty within the framework of robust optimization. We establish robust strong duality for an uncertain nonlinear programming primal problem and its uncertain Lagrangian dual by showing strong duality between the deterministic counterparts: robust counterpart of the primal model...