Before revision:  By a further development of the author's relaxation method, this paper provides a systemic approach to solving this new type of quantile optimization problems }  .  The optimal quantile is expressed via the solution of a free boundary problem for a  second-order nonlinear ordinary differential equation (ODE), which is similar to the Black-Scholes ODE for perpetual American options and has been well studied in literature theoretically and numerically .
After revision:  By a further development of the author's relaxation method, this paper provides a systemic approach to solving this new type of quantile optimization problems } . Mathematically speaking, the model reduces to a quantile optimisation problem with a compatibility constraint. They solved the problem by imposing assumptions on the loss and the probability weighting function. This paper solves the problem completely by a new quantile optimisation approach under general setting. The optimal quantile is expressed via the solution of a free boundary problem for a second-order nonlinear ordinary differential equation (ODE), which is similar to the Black-Scholes ODE for perpetual American options and has been well studied in literature theoretically and numerically .