Seydamet Ablaev (CFU) - Adaptive Methods for Convex Optimization Problems with Relative Accuracy

This talk is devoted to some first-order methods for convex optimization problems with relative accuracy with respect to the objective functional. We develop the previously known works of Yu. E. Nesterov on this topic. According to this approach, it is reasonable to reduce convex minimization problems with relative accuracy to problems with positively homogeneous functions. Such functions are non-smooth generally. Therefore, we consider 2 approaches known for non-smooth optimization problems. The first approach is devoted to modifications of gradient-typr methods with inexact oracle, bit for problems with relative accuracy. The second approach is related to subgradient methods with B. T. Polyak stepsize. Result on the linear convergence rate for some methods of this type with adaptive step adjustment is obtained for some class of non-smooth problems. Some generalization to a special class of non-convex non-smooth problems is also considered.