For strongly convex objectives that are smooth, the classical theory of gradient descent ensures linear convergence relative to the number of gradient evaluations. An analogous nonsmooth theory is challenging. Even when the objective is smooth at every iterate, the corresponding local models are unstable and the number of cutting planes invoked by traditional remedies is … Read more
In this paper, we present new methods for black-box convex minimization. They do not need to know in advance the actual level of smoothness of the objective function. Their only essential input parameter is the required accuracy of the solution. At the same time, for each particular problem class they automatically ensure the best possible … Read more
The main goal of this paper is to develop uniformly optimal first-order methods for large-scale convex programming (CP). By uniform optimality we mean that the first-order methods themselves do not require the input of any problem parameters, but can still achieve the best possible iteration complexity bounds. To this end, we provide a substantial generalization … Read more
The main goal of this paper is to develop uniformly optimal first-order methods for large-scale convex programming (CP). By uniform optimality we mean that the first-order methods themselves do not require the input of any problem parameters, but can still achieve the best possible iteration complexity bounds. To this end, we provide a substantial generalization … Read more
The bundle-level method and their certain variants are known to exhibit an optimal rate of convergence, i.e., ${\cal O}(1/\sqrt{t})$, and also excellent practical performance for solving general non-smooth convex programming (CP) problems. However, this rate of convergence is significantly worse than the optimal one for solving smooth CP problems, i.e., ${\cal O}(1/t^2)$. In this paper, … Read more
In this paper we analyze several new methods for solving optimization problems with the objective function formed as a sum of two convex terms: one is smooth and given by a black-box oracle, and another is general but simple and its structure is known. Despite to the bad properties of the sum, such problems, both … Read more
In this paper we analyze several new methods for solving optimization problems with the objective function formed as a sum of two convex terms: one is smooth and given by a black-box oracle, and another is general but simple and its structure is known. Despite to the bad properties of the sum, such problems, both … Read more