The basic Grey Wolf Optimizer (GWO) has some shortcomings, for example, the convergence speed is slow, it is easy to fall into local extremum, and high-dimensional optimization ability is poor and so on. In response to these shortcomings, an improved grey wolf algorithm which combines flower pollination mechanism, teaching mechanism and polynomial variation is proposed in this study. The flower pollination mechanism is integrated with GWO algorithm, Levy distribution is introduced into the global search of grey wolf population. And the double random mechanism is added in the local search, for these improvements, this algorithm’s overall optimization performance is improved. The teaching mechanism is added to α wolf to improve the algorithm’s convergence speed. Polynomial mutation is applied to the individuals with poor optimization effect to improve the algorithm’s accuracy and its ability to jump out of local extremum. Theoretical analysis shows that the time complexity of the improved algorithm is the same as that of the basic algorithm. The test results of five representative comparison algorithms on multiple different characteristics and different dimensions of CEC2017 benchmark functions and two classical engineering problems show that FMGWO algorithm has high optimization accuracy, convergence speed and solution stability. Therefore, it has obvious advantages in global optimization.