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Gradient-based topology optimization via the adjoint method has been successfully used in nanophotonics to uncover shapes with superior performances compared to what would be possible with traditional design methods. Here, we have extended this technique to optimize nanostructures to engineer their induced multipole moments. As an example, we demonstrate the method's application to realize the first Kerker effect in a silicon nanoparticle. The optimization results show a complex shape with highly suppressed backscattering due to the excitation of in-phase electric and magnetic dipoles with the same amplitude. This promising approach can pave the way for the inverse design of photonic structures based on a set of desired multipole moments, which can exhibit a variety of complex photonic phenomena.