I have a quadratic programming problem with a non-convex feasible region. The problem is as follow
\begin{align}&\min_{x\in \mathbb{R}^M_+} x^T\pmb{A}x+\pmb{b}^Tx \\ &\; \text{s.t.} \quad 1^Tx\le c\\ &\;\; \qquad d_kx_k \le \frac{m_k}{1+\exp\left(-g_k(x_1-h_k)\right)}, \qquad \forall k=2,\cdots,M \end{align}
Where $\forall k, d_k,g_k,h_k,m_k,c\in \mathbb{R}_+$ are given positive scalers and $\pmb{A}\in S^M_{++} ,\pmb{b}\in \mathbb{R}^M_+$ are given positive definite matrix and positive vector of parameters, respectively.
Obviously the objective and the first constraint are convex. As for the second batch of constraints, if $\forall k: x_1\ge h_k$ then the constraints are convex and we can use Lagrangian and KKT conditions to solve it.
$$\mathcal{L}\left( x,\lambda,\pmb{\nu} \right) = x^TAx+b^Tx+\lambda\left(1^Tx - c\right) + \sum_{k=2}^{M}\nu_k\left( d_kx_k - \frac{m_k}{1+\exp\left(-g_k(x_1-h_k)\right)} \right)$$
where I omitted the KKT conditions here for brevity. However when the conditions $\forall k: x_1\ge h_k$ are not satisfied, i.e. $\exists k: 0\le x_1\le h_k $ then the problem is no longer convex.
I know that for constraints of the form $f(x)=\text{const}$ where $f(x)$ is a non-linear function, the problem is generally non-convex and one way to deal with them is to relax the constraint to $f(x)\le \text{const}$ and hopefully if $f(x)$ is convex, then an approximate solution can be obtained this way. However in my problem the constraint is not convex unfortunately!
-Can a change of variable be introduced?
-Is it possible to introduce new slack variable to make the problem convex?
-Is it possible to break the feasible region into convex regions and solve each separately?
-Is there any special class of optimization problems (like LP, QCQP, SDP, etc.) to solve such a problem?
I don't know if the above questions are the right kind of question to solve this problem and also appreciate any other ideas and hints on how to deal with such constraints!
Thanks in advance!