On Lagrangian Duality in Vector Optimization. Applications to the linear case.

The paper deals with vector constrained extremum problems. A separation scheme is recalled; starting from it, a vector Lagrangian duality theory is developed. The linear duality due to Isermann can be embedded in this separation approach. Some classical applications are extended to the multiobjective framework in the linear case, exploiting the duality theory of Isermann.Vector Optimization, Separation, Image Space Analysis, Lagrangian Duality, Set-Valued Function.

Multiobjective Lagrangian duality for portfolio optimization with risk measures

In this paper we present an application for a multiobjective optimization problem. The objective functions of the primal problem are the risk and the expected pain associated to a portfolio vector. Then, we present a Lagrangian dual problem for it. In order to formulate this problem, we introduce the theory about risk measures for a vector of random variables. The definition of this kind of measures is a very evolving topic; moreover, we want to measure the risk in the multidimensional case without exploiting any scalarization technique of the random vector. We refer to the approach of the image space analysis in order to recall weak and strong Lagrangian duality results obtained through separation arguments. Finally, we interpret the shadow prices of the dual problem providing new definitions for risk aversion and non-satiability in the linear case.Multivariate risk measures, Vector Optimization, Lagrangian Duality, Shadow prices, Image Space Analysis.