Spectral unmixing of outdoor nighttime lighting mixtures is a linear but challenging problem due to spectral variability and contributions from unknown light sources. This paper presents a quadratic programming algorithm for constrained linear unmixing abundance estimation that extends the conventional fully constrained least squares approach through a physically motivated set of three linear constraints: estimated abundances must be non-negative, their sum must not be greater than unity, and the spectral signature reconstructed from estimated abundances must not exceed the measured signal in any spectral band. Unlike alternative optimization or neural network approaches, the proposed formulation requires no hyperparameter tuning or stochastic training, and yields a unique globally optimal solution whenever the endmember spectra are linearly independent, as verified for the case study considered here. The accuracy of the algorithm is evaluated using a physically motivated dataset generation framework grounded in the Generalized Linear Mixing Model, which is capable of modeling wavelength-dependent surface reflections and variable fractions of unknown light sources. Experiments conducted on a database of measured illumination sources demonstrate that the proposed formulation outperforms the classical fully constrained least squares baseline. Specifically, it reduces the abundance RMSE from 0.1539 to 0.1484 and improves the endmember detection F1 score from 0.795 to 0.803, with differences confirmed to be statistically significant. Furthermore, the model remains robust to surface reflections present in up to 80% of the evaluated mixtures. It maintains reliable abundance estimates for unknown spectral contributions up to approximately 15%, with accuracy gradually decreasing beyond this level. Finally, both the optimization method and the dataset generation framework are completely independent of the underlying channel configuration, making them readily adaptable to arbitrary choices of spectral channels and application domains beyond public lighting analysis.