An $\varepsilon$ improved moth-flame optimization algorithm ($\varepsilon$IMFO) is proposed to solve constrained optimization problems in this paper. Considering the change of the constraint violation degree of the population, this algorithm uses improved $\varepsilon$ constraint method to deal with the constraints, and develops a threshold $\varepsilon$ formula based on the constraint violation degree of the flame population. The update method of the flame population is improved. According to the relationship between the constraint violation degree of individuals in the population and $\varepsilon$, they are divided into two categories: one is the individuals whose constraint violation degree is less than or equal to $\varepsilon$, sorted according to the objective function value; the other is the individuals whose constraint violation degree is bigger than $\varepsilon$, sorted by constraint violation degree. Then the individuals in the first category are firstly selected. If the number does not meet the population size, individuals are selected from the second category to build a new flame population. Finally, an improved moth mutation strategy is proposed. Based on the original moth mutation strategy, two random flame individuals are introduced to affect moth mutation, and the guiding effect of the excellent flame individuals on moth mutation is increased. The comparison of algorithm performance tests with other 13 algorithms among 25 benchmark test functions and 2 actual engineering optimization problems shows that the $\varepsilon$IMFO algorithm is superior in terms of the accuracy and stability.