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The optimal design of infinite impulse response (IIR) digital filters is usually a nonconvex problem, and thus has many local optimal solutions. A new method that combines Gauss-Newton strategy with the sequential constraint minimization procedure is proposed in this paper to convert the minimax design of IIR digital filters into a sequence of convex sub-problems. Compared with some recent design methods, this method can decrease the frequency response error and the group-delay error near the passband edges. Simulation examples illustrate the effectiveness of the proposed algorithm.