Abstract: The uncertainty in structural parameters has notable impact on stability analysis of structures. It is of great significance for structural design and safety evaluation to obtain random buckling loads and buckling modes with efficiency and high-precision. The homotopy stochastic finite element method is improved to effectively solve the structural elastic stability problem involving large fluctuation of random parameters, and the statistical properties of the buckling eigenvalue and buckling mode are obtained. The buckling eigenvalue and buckling mode of the structure involving random parameters are expressed using the homotopy series, and the arbitrary order coefficients of the homotopy series are given as explicit recursive relationship formulas. Further, the stochastic residual error with respect to the buckling governing equation is defined, and the optimal form of the homotopy series is determined by minimizing the stochastic residual error. The proposed advanced homotopy stochastic finite element method can automatically realize the optimization process, which overcomes the drawbacks of the existing homotopy stochastic finite element method, named HSFEM, that the computational accuracy is affected by the selected samples and relies on empirical knowledge. In addition, for a structure involving large fluctuation of random parameters, the proposed method has better stability than HSFEM when higher-order terms in homotopy series are employed, while the results from the traditional perturbation method based on Taylor series may diverge. And the proposed method has an excellent computational efficiency compared with the Monte Carlo simulation method. The validity of the proposed method is verified through a strong-nonlinearity function, the stability analysis of a variable cross-section column subjected to axial force and the stability analysis of a 7-story frame structure.