Portfolio optimization with two quasiconvex risk measures

We study a static portfolio optimization problem with two risk measures: a principle risk measure in the objective function and a secondary risk measure whose value is controlled in the constraints. This problem is of interest when it is necessary to consider the risk preferences of two parties, such as a portfolio manager and a regulator, at the same time. A special case of this problem where the risk measures are assumed to be coherent (positively homogeneous) is studied recently in a joint work of the author. The present paper extends the analysis to a more general setting by assuming that the two risk measures are only quasiconvex. First, we study the case where the principal risk measure is convex. We introduce a dual problem, show that there is zero duality gap between the portfolio optimization problem and the dual problem, and finally identify a condition under which the Lagrange multiplier associated to the dual problem at optimality gives an optimal portfolio. Next, we study the general case without the convexity assumption and show that an approximately optimal solution with prescribed optimality gap can be achieved by using the well-known bisection algorithm combined with a duality result that we prove.