  Imaging interferometric data in radio astronomy requires the use of
non-linear algorithms that rely on different assumptions on the source
structure and may produce non-unique results. This is especially true for Very
Long Baseline Interferometry (VLBI) observations, where the sampling of Fourier
space is very sparse. A basic tenet in standard VLBI imaging techniques is to
assume that the observed source structure does not evolve during the
observation. However, the recent VLBI results of the supermassive black hole
(SMBH) at our Galactic Center (Sagittarius A$^*$, SgrA*), recently reported by
the Event Horizon Telescope Collaboration (EHTC), require the development of
dynamic imaging algorithms, since it exhibits variability at minute timescales.
In this paper, we introduce a new non-convex optimization problem that extends
the standard Maximum Entropy Method (MEM), for reconstructing intra-observation
dynamical images from interferometric data that evolves in every integration
time. We present a rigorous mathematical formalism to solve the problem via the
primal-dual approach. We build a Newton strategy and we give its numerical
complexity. We also give a strategy to iteratively improve the obtained
solution and finally, we define a novel figure of merit to evaluate the quality
of the recovered solution. Then, we test the algorithm, called ngMEM, in
different synthetic datasets, with increasing difficulty. Finally, we compare
it with another well-established dynamical imaging method. Within this
comparison we identified a significant improvement of the ngMEM
reconstructions. Moreover, the evaluation of the integration time evolution
scheme and the time contribution showed to play a crucial role for obtaining
good dynamic reconstructions.
