Past climates provide a means for evaluating the response of the climate system to large perturbations. Our ultimate goal is to constrain climate models rigorously by paleoclimate data. For illustration, we used a conceptual climate model (a classical energy balance model) and applied the so-called “adjoint method” to minimize the misfit between our model and sea-surface temperature data for the Last Glacial Maximum (LGM, between 19,000-23,000 years before present). The “adjoint model” (derivative code) was generated by an “adjoint compiler”. We optimized parameters controlling the thermal diffusion and the sensitivity of the outgoing longwave radiation to changes in the zonal-mean surface temperature and the atmospheric CO2 concentration. As a result, we estimated that an equilibrium climate sensitivity between 2.2 degC and 2.5 degC was consistent with the reconstructed glacial cooling, and we were able to infer structural deficits of the simple model where the fit to current observations and paleo data was not successful.