ALT Ivan Oseledets (Skolkovo Institute of Science and Technology) Optimization with low-rank matrix and tensor constraints with applications Low-rank decompositions of matrices and tensors play important role in data analysis, machine learning and solution of high-dimensional problems. Many of such problems can be formulated as a minimization problem with constraints. There are many competitive ways of dealing with such kind of methods, starting from straightforward approaches that utilize classical optimization over the parameters of the factorization, and going into advanced approaches such as Riemmanian optimization and augmented Lagrangian methods. Slow adoption of such kind of methods is hindered by limited availability of efficient implementations in modern machine learning frameworks such as Pytorch and Tensorflow. I will discuss existing and several recent results for improving algorithms for optimization with low-rank constraints and also show some applications of tensor decomp.