Enhancing the robustness of complex networks is crucial for the networks to resist external attacks and cascading failures. Existing evolutionary algorithms have limitations in solving network structure optimization problems， especially in terms of convergence and optimization speed. In response to this challenge， a new adaptive complex network robustness optimization algorithm named SU-ANet （SUrrogate-assisted and Adaptive Network optimization algorithm） was proposed. To reduce the huge time overhead of robustness computation， a robustness predictor based on attention mechanism was constructed in SU-ANet as an offline surrogate model to replace the frequent robustness computation in local search operator. In the evolutionary process， the global and local information was considered comprehensively to avoid falling into local optimum and broaden the search space simultaneously. By designing crossover operators， each individual exchanges edges with the global optimum candidate solution and a randomly selected individual to balance the convergence and diversity of the algorithm. Additionally， a parameter self-adaptive mechanism was applied to adjust the operator execution probabilities automatically， thereby alleviating the uncertainty of the algorithm brought by the parameter design. Experimental results on both synthetic networks and real-world networks demonstrate that SU-ANet has better search capabilities and higher evolutionary efficiency.