We propose a data aggregation-based algorithm with monotonic convergence to a global optimum for a generalized version of the L1-norm error fitting model with an assumption of the fitting function, by generalizing the previously proposed algorithm by Park and Klabjan (2016). The new algorithm can solve multi-dimensional fitting problems with arbitrary constraints on the fitting coefficients. Any model following the form can be solved optimally using the proposed algorithm. The generalized problem includes popular models such as regression, principal component analysis, and the orthogonal Procrustes problem. The results of the computational experiment show that the proposed algorithms are up to 9,000 times faster than the state-of-the-art benchmarks for the problems and data sets studied.