5 research outputs found
Inexact Model: A Framework for Optimization and Variational Inequalities
In this paper we propose a general algorithmic framework for first-order
methods in optimization in a broad sense, including minimization problems,
saddle-point problems and variational inequalities. This framework allows to
obtain many known methods as a special case, the list including accelerated
gradient method, composite optimization methods, level-set methods, proximal
methods. The idea of the framework is based on constructing an inexact model of
the main problem component, i.e. objective function in optimization or operator
in variational inequalities. Besides reproducing known results, our framework
allows to construct new methods, which we illustrate by constructing a
universal method for variational inequalities with composite structure. This
method works for smooth and non-smooth problems with optimal complexity without
a priori knowledge of the problem smoothness. We also generalize our framework
for strongly convex objectives and strongly monotone variational inequalities.Comment: 41 page
Inexact model: A framework for optimization and variational inequalities
In this paper we propose a general algorithmic framework for first-order methods in optimization in a broad sense, including minimization problems, saddle-point problems and variational inequalities. This framework allows to obtain many known methods as a special case, the list including accelerated gradient method, composite optimization methods, level-set methods, proximal methods. The idea of the framework is based on constructing an inexact model of the main problem component, i.e. objective function in optimization or operator in variational inequalities. Besides reproducing known results, our framework allows to construct new methods, which we illustrate by constructing a universal method for variational inequalities with composite structure. This method works for smooth and non-smooth problems with optimal complexity without a priori knowledge of the problem smoothness. We also generalize our framework for strongly convex objectives and strongly monotone variational inequalities
Inexact relative smoothness and strong convexity for optimization and variational inequalities by inexact model
In this paper we propose a general algorithmic framework for first-order methods in optimization in a broad sense, including minimization problems, saddle-point problems and variational inequalities. This framework allows to obtain many known methods as a special case, the list including accelerated gradient method, composite optimization methods, level-set methods, Bregman proximal methods. The idea of the framework is based on constructing an inexact model of the main problem component, i.e. objective function in optimization or operator in variational inequalities. Besides reproducing known results, our framework allows to construct new methods, which we illustrate by constructing a universal conditional gradient method and universal method for variational inequalities with composite structure. These method works for smooth and non-smooth problems with optimal complexity without a priori knowledge of the problem smoothness. As a particular case of our general framework, we introduce relative smoothness for operators and propose an algorithm for VIs with such operator. We also generalize our framework for relatively strongly convex objectives and strongly monotone variational inequalities
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Inexact relative smoothness and strong convexity for optimization and variational inequalities by inexact model
In this paper we propose a general algorithmic framework for first-order methods in optimization in a broad sense, including minimization problems, saddle-point problems and variational inequalities. This framework allows to obtain many known methods as a special case, the list including accelerated gradient method, composite optimization methods, level-set methods, Bregman proximal methods. The idea of the framework is based on constructing an inexact model of the main problem component, i.e. objective function in optimization or operator in variational inequalities. Besides reproducing known results, our framework allows to construct new methods, which we illustrate by constructing a universal conditional gradient method and universal method for variational inequalities with composite structure. These method works for smooth and non-smooth problems with optimal complexity without a priori knowledge of the problem smoothness. As a particular case of our general framework, we introduce relative smoothness for operators and propose an algorithm for VIs with such operator. We also generalize our framework for relatively strongly convex objectives and strongly monotone variational inequalities
Inexact Relative Smoothness and Strong Convexity for Optimization and Variational Inequalities by Inexact Model
In this paper, we propose a general algorithmic framework for first-order
methods in optimization in a broad sense, including minimization problems,
saddle-point problems, and variational inequalities. This framework allows
obtaining many known methods as a special case, the list including accelerated
gradient method, composite optimization methods, level-set methods, Bregman
proximal methods. The idea of the framework is based on constructing an inexact
model of the main problem component, i.e. objective function in optimization or
operator in variational inequalities. Besides reproducing known results, our
framework allows constructing new methods, which we illustrate by constructing
a universal conditional gradient method and a universal method for variational
inequalities with a composite structure. This method works for smooth and
non-smooth problems with optimal complexity without a priori knowledge of the
problem's smoothness. As a particular case of our general framework, we
introduce relative smoothness for operators and propose an algorithm for
variational inequalities (VIs) with such operators. We also generalize our
framework for relatively strongly convex objectives and strongly monotone
variational inequalities.
This paper is an extended and updated version of [arXiv:1902.00990]. In
particular, we add an extension of relative strong convexity for optimization
and variational inequalities.Comment: arXiv admin note: text overlap with arXiv:1902.00990. To appear in
Optimization Methods and Software,
https://doi.org/10.1080/10556788.2021.192471