Set-valued duality theory for multiple objective linear programs and application to mathematical finance

Abstract

We develop a duality theory for weakly minimal points of multiple objective linear programs which has several advantages in contrast to other theories. For instance, the dual variables are vectors rather than matrices and the dual feasible set is a polyhedron. We use a set-valued dual objective map the values of which have a very simple structure, in fact they are hyperplanes. As in other set-valued (but not in vector-valued) approaches, there is no duality gap in the case that the right-hand side of the linear constraints is zero. Moreover, we show that the whole theory can be developed by working in a complete lattice. Thus the duality theory has a high degree of analogy to its classical counterpart. Another important feature of our theory is that the infimum of the set-valued dual problem is attained in a finite set of vertices of the dual feasible domain. These advantages open the possibility of various applications such as a dual simplex algorithm. Exemplarily, we discuss an application to a Markowitz-type bicriterial portfolio optimization problem where the risk is measured by the Conditional Value at Risk.