  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.
