299 research outputs found
On the Dual of the Solvency Cone
A solvency cone is a polyhedral convex cone which is used in Mathematical
Finance to model proportional transaction costs. It consists of those
portfolios which can be traded into nonnegative positions. In this note, we
provide a characterization of its dual cone in terms of extreme directions and
discuss some consequences, among them: (i) an algorithm to construct extreme
directions of the dual cone when a corresponding "contribution scheme" is
given; (ii) estimates for the number of extreme directions; (iii) an explicit
representation of the dual cone for special cases. The validation of the
algorithm is based on the following easy-to-state but difficult-to-solve result
on bipartite graphs: Running over all spanning trees of a bipartite graph, the
number of left degree sequences equals the number of right degree sequences.Comment: 15 page
Primal and Dual Approximation Algorithms for Convex Vector Optimization Problems
Two approximation algorithms for solving convex vector optimization problems
(CVOPs) are provided. Both algorithms solve the CVOP and its geometric dual
problem simultaneously. The first algorithm is an extension of Benson's outer
approximation algorithm, and the second one is a dual variant of it. Both
algorithms provide an inner as well as an outer approximation of the (upper and
lower) images. Only one scalar convex program has to be solved in each
iteration. We allow objective and constraint functions that are not necessarily
differentiable, allow solid pointed polyhedral ordering cones, and relate the
approximations to an appropriate \epsilon-solution concept. Numerical examples
are provided
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