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In this paper, we consider conditional gradient methods for optimizing strongly convex functions. These are methods that use a linear minimization oracle, which, for a given vector $p \in \mathbb{R}^n$, computes the solution of the subproblem

\[ \text{Argmin}_{x\in X}{\langle p,\,x \rangle}. \]There are a variety of conditional gradient methods that have a linear convergence rate in a strongly convex case. However, in all these methods, the dimension of the problem is included in the rate of convergence, which in modern applications can be very large. In this paper, we prove that in the strongly convex case, the convergence rate of the conditional gradient methods in the best case depends on the dimension of the problem $ n $ as $ \widetilde {\Omega} \left(\!\sqrt {n}\right) $. Thus, the conditional gradient methods may turn out to be ineffective for solving strongly convex optimization problems of large dimensions.

Also, the application of conditional gradient methods to minimization problems of a quadratic form is considered. The effectiveness of the Frank – Wolfe method for solving the quadratic optimization problem in the convex case on a simplex (PageRank) has already been proved. This work shows that the use of conditional gradient methods to solve the minimization problem of a quadratic form in a strongly convex case is ineffective due to the presence of dimension in the convergence rate of these methods. Therefore, the Shrinking Conditional Gradient method is considered. Its difference from the conditional gradient methods is that it uses a modified linear minimization oracle. It's an oracle, which, for a given vector $p \in \mathbb{R}^n$, computes the solution of the subproblem \[ \text{Argmin}\{\langle p, \,x \rangle\colon x\in X, \;\|x-x_0^{}\| \leqslant R \}. \] The convergence rate of such an algorithm does not depend on dimension. Using the Shrinking Conditional Gradient method the complexity (the total number of arithmetic operations) of solving the minimization problem of quadratic form on a $ \infty $-ball is obtained. The resulting evaluation of the method is comparable to the complexity of the gradient method.

Finding PageRank vector is of great scientific and practical interest due to its applicability to modern search engines. Despite the fact that this problem is reduced to finding the eigenvector of the stochastic matrix $P$, the need for new algorithms is justified by a large size of the input data. To achieve no more than linear execution time, various randomized methods have been proposed, returning the expected result only with some probability close enough to one. We will consider two of them by reducing the problem of calculating the PageRank vector to the problem of finding equilibrium in an antagonistic matrix game, which is then solved using the Grigoriadis – Khachiyan algorithm. This implementation works effectively under the assumption of sparsity of the input matrix. As far as we know, there are no successful implementations of neither the Grigoriadis – Khachiyan algorithm nor its application to the task of calculating the PageRank vector. The purpose of this paper is to fill this gap. The article describes an algorithm giving pseudocode and some details of the implementation. In addition, it discusses another randomized method of calculating the PageRank vector, namely, Markov chain Monte Carlo (MCMC), in order to compare the results of these algorithms on matrices with different values of the spectral gap. The latter is of particular interest, since the magnitude of the spectral gap strongly affects the convergence rate of MCMC and does not affect the other two approaches at all. The comparison was carried out on two types of generated graphs: chains and $d$-dimensional cubes. The experiments, as predicted by the theory, demonstrated the effectiveness of the Grigoriadis – Khachiyan algorithm in comparison with MCMC for sparse graphs with a small spectral gap value. The written code is publicly available, so everyone can reproduce the results themselves or use this implementation for their own needs. The work has a purely practical orientation, no theoretical results were obtained.