Frank Heyde

Duality for Set-Valued Measures of Risk

Extending the approach of Jouini, Meddeb, and Touzi [Finance Stoch., 8 (2004), pp. 531-552] we define set-valued (convex) measures of risk and their acceptance sets, and we give dual representation theorems. A scalarization concept is introduced that has a meaning in terms of internal prices of portfolios of reference instruments. Using primal and dual descriptions, we introduce new examples for set-valued measures of risk, e.g., set-valued upper expectations, value at risk, average value at risk, and entropic risk measure.

Geometric Duality in Multiple Objective Linear Programming

We develop in this article a geometric approach to duality in multiple objective linear programming. This approach is based on a very old idea, the duality of polytopes, which can be traced back to the old Greeks. We show that there is an inclusion-reversing one-to-one map between the minimal faces of the image of the primal objective and the maximal faces of the image of the dual objective map.

Solution concepts in vector optimization: a fresh look at an old story

Over the past decades various solution concepts for vector optimization problems have been established and used: among them are efficient, weakly efficient and properly efficient solutions. In contrast to the classical approach, we define a solution to be a set of efficient solutions on which the infimum of the objective function with respect to an appropriate complete lattice (the space of self-infimal sets) is attained. The set of weakly efficient solutions is not considered to be a solution, but weak efficiency is essential in the construction of the complete lattice. In this way, two classic concepts are involved in a common approach. Several different notions of semicontinuity are compared. Using the space of self-infimal sets, we can show that various originally different concepts coincide. A Weierstrass existence result is proved for our solution concept. A slight relaxation of the solution concept yields a relationship to properly efficient solutions.

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

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.

The Attainment of the Solution of the Dual Program in Vertices for Vectorial Linear Programs

This article is a continuation of LA, Tammer C (2007: A new ap- proach to duality in vector optimization. Optimization 56(1-2):221-239) (14). We developed in (14) a duality theory for convex vector optimization problems, which is different from other approaches in the literature. The main idea is to embed the image space R q of the objective function into an appropriate complete lattice, which is a subset of the power set of R q . This leads to a duality theory which is very anal- ogous to that of scalar convex problems. We applied these results to linear problems and showed duality assertions. However, in (14) we could not answer the question, whether the supremum of the dual linear program is attained in vertices of the dual feasible set. We show in this paper that this is, in general, not true but, it is true under additional assumptions.

A Characterization of Approximate Solutions of Multiobjective Stochastic Optimal Control Problems

We introduce several concepts of approximate solutions of multiobjective optimization problems, prove existence results and an k -minimum principle for multiobjective stochastic optimal control problems.

Exploitation of necessary and sufficient conditions for suboptimal solutions of multiobjective stochastic control problems

In this paper necessary and sufficient conditions for a multicriteria stochastic control problem are stated. Moreover the convergence of a finite difference approximation method for the solution of the resulting system of second order PDEs is shown.

Proximal Point Algorithm for an Approximated Stochastic Optimal Control Problem

In this paper a numerical method for solving a stochastic optimal control problem under control restrictions is introduced. For this purpose a special kind of Markov chain approximation is used in order to discretize the problem. For the solution of the discrete Bellman equation a primal dual proximal point algorithm is derived.