44 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
The natural ordering cone of a polyhedral convex set-valued objective mapping
For a given polyhedral convex set-valued mapping we define a polyhedral
convex cone which we call the natural ordering cone. We show that the solution
behavior of a polyhedral convex set optimization problem can be characterized
by this cone. Under appropriate assumptions the natural ordering cone is proven
to be the smallest ordering cone which makes a polyhedral convex set
optimization problem solvable.Comment: definition of regularity simplifie
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
Existence of solutions for polyhedral convex set optimization problems
Polyhedral convex set optimization problems are the simplest optimization
problems with set-valued objective function. Their role in set optimization is
comparable to the role of linear programs in scalar optimization. Vector linear
programs and multiple objective linear programs provide proper subclasses. In
this article we choose a solution concept for arbitrary polyhedral convex set
optimization problems out of several alternatives, show existence of solutions
and characterize the existence of solutions in different ways. Two known
results are obtained as particular cases, both with proofs being easier than
the original ones: The existence of solutions of bounded polyhedral convex set
optimization problems and a characterization of the existence of solutions of
vector linear programs.Comment: 10 page