I have a quadratic optimization problem as follow: $$min_{\boldsymbol{x} \in \{0,1 \}^n} \boldsymbol{1}^T\boldsymbol{P}\boldsymbol{x} - \boldsymbol{x}^T \boldsymbol{P} \boldsymbol{x} \quad \quad (1)$$ where $\boldsymbol{x}$ is a n-dimensional vector with binary elements. In order to solve this problem I first relax $\boldsymbol{x}$ into continuous space such that $\boldsymbol{x} \in [0,1]^n$. Then the problem becomes:
$$min_{\boldsymbol{x} \in [0,1 ]^n} \boldsymbol{1}^T\boldsymbol{P}\boldsymbol{x} - \boldsymbol{x}^T \boldsymbol{P} \boldsymbol{x} \quad \quad (2)$$ Unfortunately the matrix $-\boldsymbol{P}$ is not necessarily positive semi-definite, so (2) is not necessarily a convex optimization. My current strategy is to derive the gradient of the objective function ($\mathcal{L}$) of (2) with respect to $\boldsymbol{x}$:
$$\frac{d\mathcal{L}}{d\boldsymbol{x}} = -2\boldsymbol{P}\boldsymbol{x} + \boldsymbol{P}^T\boldsymbol{1},$$ and use project gradient descent to iteratively minimize (2): $$\boldsymbol{x}^{(t+1)} = Proj_{\mathcal{C}}\big(\boldsymbol{x}^{(t)} - \eta\frac{d\mathcal{L}}{d\boldsymbol{x}}\big), $$ where $\eta$ is the learning rate, $\mathcal{C}$ is the set $[0,1]^n$, and $Proj_{\mathcal{C}}(\cdot)$ is an Euclidean projection onto $\mathcal{C}$.
From my experimental results after several iterations the project gradient descent procedure seems to converge (the difference between $\boldsymbol{x}^{(t)}$ and $\boldsymbol{x}^{(t+1)}$ is almost $0$). So I am wondering if there is any technique that can derive some guarantees of the procedure? For example, is it possible to use fixed point theorem to show this procedure is guaranteed to converge to some $\boldsymbol{x}^{\ast}$ (possibly local optimal)? If the answer is yes, is it possible to derive some guarantees about $\mathcal{L}(\boldsymbol{x}^{\ast})$, for example, how close it is to the global optimum.
I appreciate if anyone can point me out some textbooks/papers/tutorials/insights. Thanks!