A precursor to the canonical QAOA is the Quantum Adiabatic Algorithm (QAA). Since we want to end up in the ground state of the Cost Hamiltonian ($H_C$) but don't know how to construct it, we exploit adiabaticity by starting with the ground state $|+\rangle^{\otimes n}$ of the (mixing) Hamiltonian $H_M = \sum_i \sigma_i^x$.
Now, if we slowly change a Hamiltonian, $H = (1 - s) H_M + s H_C$ according to some parametrization $s \in [0, 1]$, the state would continue to remain in the instantaneous eigenstate and you would end up in the ground state of the Cost Hamiltonian when $s = 1$.
This is a precursor to QAOA, since, the time evolution of the initial state can be written as a Trotterization of $H_M$ and $H_C$ and instead of having the unitary time evolution depend on $s$, new degrees of freedom are added and $H_M$ is evolved by $\beta_i$ and $H_C$ by $\gamma_i$ for each Trotterization step, $i \in [0, p]$ (which is what QAOA does). This connection with QAA is the closest thing I can think of to a guarantee that the QAOA should find the global optima.
QAA fails when the instantaneous eigenvalues of H are too close, and the adiabaticity condition on the sweeping speed from $H_m \rightarrow H_C$ is so slow that the algorithm is ultimately inefficient. QAOA, however, circumvents that problem by adding additional dimensions to the evolution and "sweeping" around the critical points/avoided crossings.
In practise, when experimenting with QAOA, there's no guarantee that the algorithm has converged to a global optima for a given value of $p$. You could very well only have found a local minima. But, as $p \rightarrow \infty$, QAOA approaches the QAA and the final result should approach the global minimum.
