© 2016 IOP Publishing Ltd.The work describes the vibroacoustic behavior of anti-tetrachiral and auxetic hexagonal gradient sandwich panels using homogenized finite element models to determine the mechanical properties of the auxetic structures, the natural frequencies and radiated sound power level of sandwich panels made by the auxetic cores. The mechanical properties and the vibroacoustic behavior of auxetic hexagonal sandwich panels are investigated as a benchmark. The radiated sound power level of the structure over the frequency range of 0-1000 Hz is minimized by modifying the core geometry of the gradient auxetic sandwich panels. Several excitation cases are considered. First-order and random optimization methods are used for the minimization of radiated sound power level of the structures. The results of this study present significant insights into the design of auxetic structures with respect to their vibroacoustical properties.