Previously proposed variational techniques for approximate MMAP inference in complex graphical models of high-order factors relax a dual variational objective function to obtain its tractable approximation, and further perform MMAP inference in the resulting simplified graphical model, where the sub-graph with decision variables is assumed to be a disconnected forest. In contrast, we developed novel variational MMAP inference algorithms and proximal convergent solvers, where we can improve the approximation accuracy while better preserving the original MMAP query by designing such a dual variational objective function that an upper bound approximation is applied only to the entropy of decision variables. We evaluate the proposed algorithms on both simulated synthetic datasets and diagnostic Bayesian networks taken from the UAI inference challenge, and our solvers outperform other variational algorithms in a majority of reported cases. Additionally, we demonstrate the important real-life application of the proposed variational approaches to solve complex tasks of policy optimization by MMAP inference, and performance of the implemented approximation algorithms is compared. Here, we demonstrate that the original task of optimizing POMDP controllers can be approached by its reformulation as the equivalent problem of marginal-MAP inference in a novel single-DBN generative model, which guarantees that the control policies computed by probabilistic inference over this model are optimal in the traditional sense. Our motivation for approaching the planning problem through probabilistic inference in graphical models is explained by the fact that by transforming a Markovian planning problem into the task of probabilistic inference (a marginal MAP problem) and applying belief propagation techniques in generative models, we can achieve a computational complexity reduction from PSPACE-complete or NEXP-complete to NPPP-complete in comparison to solving the POMDP and Dec-POMDP models respectively search vs. dynamic programming).