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Variational quantum eigensolver (VQE), aiming at determining the ground state energy of a quantum system described by a Hamiltonian on noisy intermediate scale quantum (NISQ) devices, is among the most significant applications of variational quantum algorithms (VQAs). However, the accuracy and trainability of the current VQE algorithm are significantly influenced due to the barren plateau (BP), the non-negligible gate error and limited coherence time in NISQ devices. To tackle these issues, a gradient-sensitive alternate framework with variable ansatz is proposed in this paper to enhance the performance of the VQE. We first propose a theoretical framework for VA-VQE via alternately solving a multi-objective optimization problem and the original VQE, where the multi-objective optimization problem is defined with respect to cost function values and gradient magnitudes. Then, we propose a novel implementation method based on the double $ε$-greedy strategy with the candidate tree and modified multi-objective genetic algorithm. As a result, the local optima are avoided both in ansatz and parameter perspectives, and the stability of output ansatz is enhanced. The experimental results indicate that our framework shows considerably improvement of the error of the found solution by up to 87.9% compared with the hardware-efficient ansatz. Furthermore, compared with the full-randomized VA-VQE implementation, our framework is able to obtain the improvement of the error and the stability by up to 36.0% and 58.7%, respectively, with similar quantum costs.