  We aim to find a solution $\bm{x}\in\mathbb{C}^n$ to a system of quadratic
equations of the form $b_i=\lvert\bm{a}_i^*\bm{x}\rvert^2$, $i=1,2,\ldots,m$,
e.g., the well-known NP-hard phase retrieval problem. As opposed to recently
proposed state-of-the-art nonconvex methods, we revert to the semidefinite
relaxation (SDR) PhaseLift convex formulation and propose a successive and
incremental nonconvex optimization algorithm, termed as \texttt{IncrePR}, to
indirectly minimize the resulting convex problem on the cone of positive
semidefinite matrices. Our proposed method overcomes the excessive
computational cost of typical SDP solvers as well as the need of a good
initialization for typical nonconvex methods. For Gaussian measurements, which
is usually needed for provable convergence of nonconvex methods,
\texttt{IncrePR} with restart strategy outperforms state-of-the-art nonconvex
solvers with a sharper phase transition of perfect recovery and typical convex
solvers in terms of computational cost and storage. For more challenging
structured (non-Gaussian) measurements often occurred in real applications,
such as transmission matrix and oversampling Fourier transform,
\texttt{IncrePR} with several restarts can be used to find a good initial
guess. With further refinement by local nonconvex solvers, one can achieve a
better solution than that obtained by applying nonconvex solvers directly when
the number of measurements is relatively small. Extensive numerical tests are
performed to demonstrate the effectiveness of the proposed method.
