In this paper, we propose a new algorithm for the estimation of multiple time delays (TDs). Since a TD is a fundamental spatial cue for sensor array signal processing techniques, many methods for estimating it have been studied. Most of them, including generalized cross correlation (CC)-based methods, focus on how to estimate a TD between two sensors. These methods can then be easily adapted for multiple TDs by applying them to every pair of a reference sensor and another one. However, these pairwise methods can use only the partial information obtained by the selected sensors, resulting in inconsistent TD estimates and limited estimation accuracy. In contrast, we propose joint optimization of entire TD parameters, where spatial information obtained from all sensors is taken into account. We also introduce a consistent constraint regarding TD parameters to the observation model. We then consider a multidimensional CC (MCC) as the objective function, which is derived on the basis of maximum likelihood estimation. To maximize the MCC, which is a nonconvex function, we derive the auxiliary function for the MCC and design efficient update rules. We additionally estimate the amplitudes of the transfer functions for supporting the TD estimation, where we maximize the Rayleigh quotient under the non-negative constraint. We experimentally analyze essential features of the proposed method and evaluate its effectiveness in TD estimation. Code will be available at https://github.com/onolab-tmu/AuxTDE.