Is there any existing optimization solver can solve this Mixed Integer Nonlinear Programming (MINLP) problem?

Question

The MINLP problme is: $$\max_{p,x} f(x,p)=ap-bx^Te \tag{1}$$ $$ \text{s.t. } x^Tt\le c \tag{2}$$ $$p=\frac12 \left[ 1+ \text{erf}\left( \frac{t_i-\mu_i}{\sigma_i\sqrt{2}} \right)\right] \tag{3}$$

Here, $p\in \mathbb R$,$x=\left[ x_1,x_2,...,x_n \right]^T$ in which $x_i=0 \text{ or } 1$ is a binary integer. $e=[1,1,...,1]^T$ is a vector with all elements equals $1$. And $a,b,c\in \mathbb R$ are coefficients. $t=[t_1,t_2,...,t_n]^T\in\mathbb R^n$. For any $t_i$, we have the equality constraint in $(3)$, in which $\mu_i$ and $\sigma_i$ are known coefficients and $\text{erf()}$ are error function.

I have a strong sense that this problem can not be solved with existing optimization solver as I checked MATLAB and Gurobi. It seems that MINLP with quadratic objective function or constraints are the limit of these solvers. But I am not sure about this conclusion. Please let me know if you have any ideas on this.

I would like to share the background of this problem if you are interested.

Answer 1

Since $-1 < \text{erf} < 1$, $p \in (0,1)$. $\mu_i$ and $\sigma_i$ being known, the value of $p$ will determine the $t_i$, and then the remaining optimization over $x$ is a binary integer linear programming problem. For an approximate solution, I would start by trying some grid of values of $p$, and then refine the result by looking more closely near the best $p$ values.