Thermoacoustic instabilities pose a major threat to modern gas turbines. The use of acoustic dampers, like Helmholtz resonators, has proven useful for the mitigation of such instabilities. However, assessing the effect of acoustic dampers on thermoacoustic modes in annular combustion chambers remains an intricate task. This results from the implicit nature of the thermoacoustic Helmholtz equation associated with the high number of possible parameter values for the positioning of the dampers and their impedance design. In the present work, the principal challenges of the effective placement and the design of the impedance of acoustic dampers in annular chambers are discussed. This includes the choice of an appropriate objective function for the optimization, the combinatorial challenges when dealing with different possible damper arrangements, and the numerical complexities when using the thermoacoustic Helmholtz equation to approach this issue. As a key aspect, the paper proposes a new adjoint-based approach to tackle these problems. The new algorithm establishes algebraic models that predict the effect of acoustic dampers on the growth rates of the thermoacoustic modes. The theory is exemplified on the basis of a generic annular combustor model with 12 burners.