Portfolio optimization: not necessarily concave utility and constraints on wealth and allocation

We consider a portfolio optimization problem for a utility maximizing investor who is simultaneously restricted by convex constraints on portfolio allocation and upper and lower bounds on terminal wealth. After introducing a capped version of the Legendre–Fenchel-transformation, we use it to suitably extend the well-known auxiliary market framework for convex allocation constraints to derive equivalent optimality conditions for our setting with additional bounds on terminal wealth. The considered utility does not have to be strictly concave or smooth, as long as it can be concavified.