Recent trends in designing filters involve development of sparse filters with coefficients that not only have real but also zero values. These sparse filters can achieve a high performance through optimizing the selection of the zero coefficients and computing the real (non-zero) coefficients. Designing an infinite impulse response (IIR) sparse filter is more challenging than designing a finite impulse response (FIR) sparse filter. Therefore, studies on the design of IIR sparse filters have been rare. In this study, we consider IIR filters whose coefficients involve zero value, called sparse IIR filter. First, we formulate the design problem as a linear programing problem without imposing any stability condition. Subsequently, we reformulate the design problem by altering the error function and prepare several possible denominator polynomials with stable poles. Finally, by incorporating these methods into successive thinning algorithms, we develop a new design algorithm for the filters. To demonstrate the effectiveness of the proposed method, its performance is compared with that of other existing methods.