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Andrey Tremba

Publications and source records attributed to Andrey Tremba.

5 recordsLinked to original sources
arXiv2020

Mixed robustness: Analysis of systems with uncertain deterministic and random parameters using the example of linear systems

Robustness of linear systems with constant coefficients is considered. There exist methods and tools for analyzing the stability of systems with random or deterministic uncertainties. At the same time, there are no approaches for the analysis of systems containing both types of parametric uncertainty. The types of robustness are reviewed and new type of "mixed parametric robustness" is introduced. It includes several variations. The proposed formulations of mixed robustness problems can be considered as intermediate type between the classical deterministic and probabilistic approaches to robustness. Several cases are listed in which the tasks are easily solved. In general, tests of the stability of robust systems using the scenario approach are applicable, but these tests can be computationally complex. To calculate the desired stability probability, a simple graphical approach based on a robust D-partition is proposed. This method is suitable for the case of a small number of random parameters. The final estimate of the probability of stability is calculated in a deterministic way and can be found with arbitrary precision. Approximate ways of solving the assigned tasks are described. Examples and generalization of mixed robustness to other types of systems are given.

math.OC↗
arXiv2019

Error bound conditions and convergence of optimization methods on smooth and proximally smooth manifolds

We analyse the convergence of the gradient projection algorithm, which is finalized with the Newton method, to a stationary point for the problem of nonconvex constrained optimization $\min_{x \in S} f(x)$ with a proximally smooth set $S = \{x \in R^n : g(x) = 0 \}, \; g : R^n \rightarrow R^m$ and a smooth function $f$. We propose new Error bound (EB) conditions for the gradient projection method which lead to the convergence domain of the Newton method. We prove that these EB conditions are typical for a wide class of optimization problems. It is possible to reach high convergence rate of the algorithm by switching to the Newton method.

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arXiv2019

New versions of Newton method: step-size choice, convergence domain and under-determined equations

Newton method is one of the most powerful methods for finding solutions of nonlinear equations and for proving their existence. In its "pure" form it has fast convergence near the solution, but small convergence domain. On the other hand damped Newton method has slower convergence rate, but weaker conditions on the initial point. We provide new versions of Newton-like algorithms, resulting in combinations of Newton and damped Newton method with special step-size choice, and estimate its convergence domain. Under some assumptions the convergence is global. Explicit complexity results are also addressed. The adaptive version of the algorithm (with no a priori constants knowledge) is presented. The method is applicable for under-determined equations (with $m<n$, $m$ being the number of equations and $n$ being the number of variables). The results are specified for systems of quadratic equations, for composite mappings and for one-dimensional equations and inequalities.

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arXiv2019

Gradient projection and conditional gradient methods for constrained nonconvex minimization

Minimization of a smooth function on a sphere or, more generally, on a smooth manifold, is the simplest non-convex optimization problem. It has a lot of applications. Our goal is to propose a version of the gradient projection algorithm for its solution and to obtain results that guarantee convergence of the algorithm under some minimal natural assumptions. We use the Lezanski-Polyak-Lojasiewicz condition on a manifold to prove the global linear convergence of the algorithm. Another method well fitted for the problem is the conditional gradient (Frank-Wolfe) algorithm. We examine some conditions which guarantee global convergence of full-step version of the method with linear rate.

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