The research of applying evolutionary algorithms (EAs) to digital infinite-impulse response (IIR) filter design has gained much attention in recent years. Previously, most works treated digital IIR filter design as a single objective optimization problem of minimizing the magnitude response error with supplementary conditions. While the lack of considering the linear phase response error and the order may result in the loss of control on the structural flexibility, the distortion of output, and the dependency on pre-knowledge. The aim of this paper is to develop proper IIR filter designing method that (1) can provide relatively more complete optimal solutions with equal consideration of magnitude response, linear phase response and the order of structure; (2) can simultaneously optimize the structure and coefficients of digital IIR filter to obtain relatively better linear phase response and lower order, besides the good magnitude response. To achieve these targets, the digital IIR filter design problem is treated as a multi-objective optimization problem in this paper. A new local search operator enhanced multi-objective evolutionary algorithm (LS-MOEA) is specifically proposed for such kind of multi-objective optimization problems. To evaluate the effectiveness and efficiency of LS-MOEA, we experimentally compare it with classical methods and previously effective EAs for digital IIR filter design on four typical IIR filter design cases. Experimental results show that the proposed method can effectively improve the linear phase response of the designed filter, and can obtain filter of lower order. Besides, it achieves these by relatively much lower computational cost than compared EAs.