  Tensor train (TT) format is a common approach for computationally efficient
work with multidimensional arrays, vectors, matrices, and discretized functions
in a wide range of applications, including computational mathematics and
machine learning. In this work, we propose a new algorithm for TT-tensor
optimization, which leads to very accurate approximations for the minimum and
maximum tensor element. The method consists in sequential tensor
multiplications of the TT-cores with an intelligent selection of candidates for
the optimum. We propose the probabilistic interpretation of the method, and
make estimates on its complexity and convergence. We perform extensive
numerical experiments with random tensors and various multivariable benchmark
functions with the number of input dimensions up to $100$. Our approach
generates a solution close to the exact optimum for all model problems, while
the running time is no more than $50$ seconds on a regular laptop.
