## Adaptive Third-Order Methods for Composite Convex Optimization

In this paper we propose third-order methods for composite convex optimization problems in which the smooth part is a three-times continuously differentiable function with Lipschitz continuous third-order derivatives. The methods are adaptive in the sense that they do not require the knowledge of the Lipschitz constant. Trial points are computed by the inexact minimization of … Read more

## On Inexact Solution of Auxiliary Problems in Tensor Methods for Convex Optimization

In this paper we study the auxiliary problems that appear in p-order tensor methods for unconstrained minimization of convex functions with \nu-Holder continuous pth derivatives. This type of auxiliary problems corresponds to the minimization of a (p+\nu)-order regularization of the pth order Taylor approximation of the objective. For the case p=3, we consider the use … Read more

## Tensor Methods for Finding Approximate Stationary Points of Convex Functions

In this paper we consider the problem of finding \epsilon-approximate stationary points of convex functions that are p-times differentiable with \nu-Hölder continuous pth derivatives. We present tensor methods with and without acceleration. Specifically, we show that the non-accelerated schemes take at most O(\epsilon^{-1/(p+\nu-1)}) iterations to reduce the norm of the gradient of the objective below … Read more

## Tensor Methods for Minimizing Convex Functions with Hölder Continuous Higher-Order Derivatives

In this paper we study p-order methods for unconstrained minimization of convex functions that are p-times differentiable with $\nu$-Hölder continuous pth derivatives. We propose tensor schemes with and without acceleration. For the schemes without acceleration, we establish iteration complexity bounds of $\mathcal{O}\left(\epsilon^{-1/(p+\nu-1)}\right)$ for reducing the functional residual below a given $\epsilon\in (0,1)$. Assuming that $\nu$ … Read more

## Computing closest stable non-negative matrices

Problem of finding the closest stable matrix for a dynamical system has many applications. It is well studied both for continuous and discrete-time systems, and the corresponding optimization problems are formulated for various matrix norms. As a rule, non-convexity of these formulations does not allow finding their global solutions. In this paper, we analyze positive … Read more

## Relatively-Smooth Convex Optimization by First-Order Methods, and Applications

The usual approach to developing and analyzing first-order methods for smooth convex optimization assumes that the gradient of the objective function is uniformly smooth with some Lipschitz constant L. However, in many settings the differentiable convex function f(.) is not uniformly smooth — for example in D-optimal design where f(x):=-ln det(HXH^T), or even the univariate … Read more

## Complexity bounds for primal-dual methods minimizing the model of objective function

We provide Frank-Wolfe ($\equiv$ Conditional Gradients) method with a convergence analysis allowing to approach a primal-dual solution of convex optimization problem with composite objective function. Additional properties of complementary part of the objective (strong convexity) significantly accelerate the scheme. We also justify a new variant of this method, which can be seen as a trust-region … Read more