The following problem is common in financial economics $$ \min_{m \in L^2} \mathbb{E}[ \phi(y(\theta)-m)] \quad \text{s.t. } \mathbb{E}[ mx ]= q $$ That is, given a random variable $y(\theta)$ ($\theta$ is some parameter vector), a random vector $x$ and a known vector $q$, find $m \in L^2$ which is closest to $y(\theta)$ under the convex loss function $\phi$.
Variations of this problem usually consider changing the loss function $\phi$. When $\phi(x) =x^2$ it is well known that the solution above is $m = y(\theta) + \lambda'x$, where $\lambda$ can be determined from the constraint. Such problems are conveniently solved using the dual representation using the results of Borwein & Lewis (1991). For my research I need to solve a similar problem with asymmetric loss function $\phi(x) = x(\tau - I(x<0))$ (well known from quantile regression).
However, following the standard approach of solving the dual problem seems to lead to a problem that is not well defined. Hence my question: is it possible to solve this optimization problem using duality? If not, are there any other results in the literature that allow one to solve a problem like this?
Below I describe my reasoning for the dual problem. Using the results of Borwein & Lewis (1991) leads to the dual problem \begin{align} \max_{\lambda \in \mathbb{R}^n} \lambda^\top q - \mathbb{E}[\rho_\tau^*(\lambda^\top x)], \label{eq:dual} \\ \rho_\tau^*(z) = \sup_{w \in L_2} zw - \rho_\tau(y(\theta)-w) \label{eq:conj} \end{align} For suitable $z$ in the domain of the convex conjugate $\rho_\tau^*$ I obtain $\rho_\tau^*(z) = zy(\theta)$ and using this in the objection function renders \begin{equation*} \max_{\lambda \in \mathbb{R}^n} \lambda^\top q - \lambda ^\top\mathbb{E}[ x y ] \end{equation*} This of course, doesn't have a well defined solution. Hence it seems that the dual problem is not equal to the primal problem (potentially since the loss function is not strictly convex).
