B. T. Polyak
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View article: Smooth monotone stochastic variational inequalities and saddle point problems: A survey
Smooth monotone stochastic variational inequalities and saddle point problems: A survey Open
This paper is a survey of methods for solving smooth, (strongly) monotone stochastic variational inequalities. To begin with, we present the deterministic foundation from which the stochastic methods eventually evolved. Then we review meth…
View article: Smooth Monotone Stochastic Variational Inequalities and Saddle Point Problems: A Survey
Smooth Monotone Stochastic Variational Inequalities and Saddle Point Problems: A Survey Open
This paper is a survey of methods for solving smooth (strongly) monotone stochastic variational inequalities. To begin with, we give the deterministic foundation from which the stochastic methods eventually evolved. Then we review methods …
View article: Stopping Rules for Gradient Methods for Non-Convex Problems with Additive Noise in Gradient
Stopping Rules for Gradient Methods for Non-Convex Problems with Additive Noise in Gradient Open
We study the gradient method under the assumption that an additively inexact gradient is available for, generally speaking, non-convex problems. The non-convexity of the objective function, as well as the use of an inexactness specified gr…
View article: Optimizing Static Linear Feedback: Gradient Method
Optimizing Static Linear Feedback: Gradient Method Open
The linear quadratic regulator is the fundamental problem of optimal control. Its state feedback version was set and solved in the early 1960s. However the static output feedback problem has no explicit-form solution. It is suggested to lo…
View article: Gradient Projection and Conditional Gradient Methods for Constrained Nonconvex Minimization
Gradient Projection and Conditional Gradient Methods for Constrained Nonconvex Minimization Open
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 fo…
View article: Adaptive and Robust Control in the USSR
Adaptive and Robust Control in the USSR Open
Control theory in the USSR after WW2 achieved serious successes in such fields as optimal control, absolute stability, delay systems, pulse and relay control. Later there was a huge peak of breakthrough research on adaptation, learning and…
View article: Minimum Fuel-Consumption Stabilization of a Spacecraft at the Lagrangian Points
Minimum Fuel-Consumption Stabilization of a Spacecraft at the Lagrangian Points Open
We consider the motion of a spacecraft described by the differential equations of the three-body problem in the Earth-Moon system. The goal is to stabilize the spacecraft in the neighborhood of the collinear Lagrangian points (which are kn…
View article: New versions of Newton method: step-size choice, convergence domain and under-determined equations
New versions of Newton method: step-size choice, convergence domain and under-determined equations Open
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 d…
View article: Gradient projection and conditional gradient methods for constrained\n nonconvex minimization
Gradient projection and conditional gradient methods for constrained\n nonconvex minimization Open
Minimization of a smooth function on a sphere or, more generally, on a smooth\nmanifold, is the simplest non-convex optimization problem. It has a lot of\napplications. Our goal is to propose a version of the gradient projection\nalgorithm…
View article: Geometry of quadratic maps via convex relaxation
Geometry of quadratic maps via convex relaxation Open
We consider several basic questions pertaining to the geometry of image of a general quadratic map. In general the image of a quadratic map is non-convex, although there are several known classes of quadratic maps when the image is convex.…
View article: Robust Principal Component Analysis: An IRLS Approach
Robust Principal Component Analysis: An IRLS Approach Open
The modern problems of optimization, estimation, signal processing, and image recognition deal with data of huge dimensions. It is important to develop effective methods and algorithms for such problems. An important idea is the constructi…
View article: Lyapunov Functions: An Optimization Theory Perspective
Lyapunov Functions: An Optimization Theory Perspective Open
The problems of unconstrained optimization and establishing asymptotic stability have much in common. Understanding the analogy between these two sheds light on their interconnection and may lead to a number of new results. For instance, i…
View article: On Existence of Equilibria of Multi-Port Linear AC Networks With Constant-Power Loads
On Existence of Equilibria of Multi-Port Linear AC Networks With Constant-Power Loads Open
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View article: Solving underdetermined nonlinear equations by Newton-like method
Solving underdetermined nonlinear equations by Newton-like method Open
Newton method is one of the most powerful methods for finding solution of nonlinear equations. In its classical form it is applied for systems of $n$ equations with $n$ variables. However it can be modified for underdetermined equations (w…