Which catalog does an optimization problem with decision variables in indicator functions as following belongs to and how to solve it?
$$
\begin{align*}
&\min_{\mathbf{d}, \mathbf{m} \in \mathbf{R}^{n}}\quad z=\sum_{i=1}^{n} \omega_{i}(\mathbf{E}_{X}[(X-d_i)^{+}\cdot \mathbf{1}_{\{X \leq m_i\}}] – (c-d_{i})^{+}\cdot\mathbf{1}_{\{m_i \geq c\}})\\
& \begin{array}{r@{\quad}r@{}l@{\quad}l}
s.t.&d_i \geq 0, i =1, \cdots,n,\\
& m_i \geq 0, i=1, \cdots,n, \\
\end{array}
\end{align*}
$$
where, $\mathbf{d} = [d_1,\cdots,d_n]' \in \mathbf{R}^n$, $\mathbf{m} = [m_1,\cdots,m_n]' \in \mathbf{R}^n$ and $c$ is a given constent.
Best Answer
The first term in your objective can be modeled with mixed integer optimization. Let us focus on one expression in the objective $$(X-d)^+ \mathbf{1}_{\{X \leq m\}}$$ where $X$ is constant and the variables are $d$ and $m$. This expression can be replaced with the newly introduced variable $t$, and the constraints $$t\geq X-d-(1-b)M, \quad t\geq 0, \quad X-\varepsilon\geq m-bM, \quad b \in \{0,1\}$$ where $M$ is a sufficiently large constant (as large as the largest realization of $X$) and $\varepsilon$ is a small tolerance. The third constraint forces $b$ to $1$ only if $X \leq m$, which in turn forces $t$ to be at least $X-d$.
The second term ($(c-d_{i})^{+}\cdot\mathbf{1}_{\{m_i \geq c\}}$) is not convex, and is considerably harder to model. Try global optimization techniques.