On the meaning of a minimization procedure applied to a degenerate astrophysical problem

We apply the MINOS optimization system to the population synthesis of galaxy nuclei using as basis a grid of star cluster spectral features as a function of age and metallicity. In inverse numerical problems, a minimization procedure is usually applied to provide a single optimal result which ignores, as a rule, a multitude of other equally good solutions whenever intrinsic errors are to be considered. We avoid this drawback by transforming the population synthesis into a series of optimization problems, each one corresponding to a fixed contribution of an individual basis component. In this way we sweep the vector space of solutions, generated by the basis components, rejecting the minimization results for which the differences between observed and synthetic equivalent widths are larger than the observational uncertainties. Consequently we map the space of mathematically similar solutions, determining the degeneracy degree of the problem. The astrophysical implications of the synthesis results are discussed. In particular, a noteworthy result is that for some classes of red galaxy nuclei, our constraint free population synthesis, consisting of 35 components which are widely distributed in the plane age versus metallicity, reproduces the chemical evolution scenario predicted by theoretical evolutionary models. The results for bluer nuclei tend to scatter more in the plane age versus metallicity, suggesting that additional information from complementary spectral ranges is necessary to produce a better focused solution in the plane. However, strong bursts of star formation are easily detected