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We present a new optimization-based approach for synthesizing photonic infinite impulse response filters formed using microresonators. Taking into account the finite internal losses in these resonators, we cast the synthesis of filter coefficients as a semidefinite programming problem designed to restrict root-magnitudes set by the internal loss, while meeting filter-mask specifications. We provide a discussion on the formulation of the algorithm, and show how it can be used to design the optical components in a second-order unit cell. In addition, we provide an example of forming a fourth-order low-pass filters by cascading two unit cells. Our simulations show that fourth-order low-pass filters can achieve band-rejection ratios of 60 dB with integrated microresonators that possess an intrinsic quality-factor Q0 of \text1.1\times\text10^5. [ABSTRACT FROM PUBLISHER]