The variational quantum eigensolver (VQE), which has attracted attention as a promising application of noisy intermediate-scale quantum devices, finds a ground state of a given Hamiltonian by variationally optimizing the parameters of quantum circuits called Ansätze. Since the difficulty of the optimization depends on the complexity of the problem Hamiltonian and the structure of the Ansätze, it has been difficult to systematically analyze the performance of optimizers for the VQE. To resolve this problem, we propose a technique to construct a benchmark problem whose ground state is guaranteed to be achievable with a given Ansatz by using the idea of a parent Hamiltonian of a low-depth parametrized quantum circuit. We compare the convergence of several optimizers by varying the distance of the initial parameters from the solution and find that the converged energies showed a thresholdlike behavior depending on the distance. This work provides a systematic way to analyze optimizers for the VQE and contribute to the design of the Ansatz and its initial parameters.