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.