Variance-Based Bregman Extragradient Algorithm with Line Search for Solving Stochastic Variational Inequalities Article Swipe

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Xian-Jun Long , Yue-Hong He , Nan‐jing Huang ·

YOU? · · 2022 · Open Access · · DOI: https://doi.org/10.48550/arxiv.2208.14069

The main purpose of this paper is to propose a variance-based Bregman extragradient algorithm with line search for solving stochastic variational inequalities, which is robust with respect an unknown Lipschitz constant. We prove the almost sure convergence of the algorithm by a more concise and effective method instead of using the supermartingale convergence theorem. Furthermore, we obtain not only the convergence rate $\mathcal{O}(1/k)$ with the gap function when $X$ is bounded, but also the same convergence rate in terms of the natural residual function when $X$ is unbounded. Under the Minty variational inequality condition, we derive the iteration complexity $\mathcal{O}(1/\varepsilon)$ and the oracle complexity $\mathcal{O}(1/\varepsilon^2)$ in both cases. Finally, some numerical results demonstrate the superiority of the proposed algorithm.

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Variational inequality Lipschitz continuity Mathematics Line search Bounded function Rate of convergence Convergence (economics) Bregman divergence Function (biology) Constant (computer programming) Oracle Applied mathematics Residual Variance (accounting) Algorithm Line (geometry) Pure mathematics Mathematical analysis Computer science Evolutionary biology Geometry Computer network Computer security Biology Software engineering Economic growth Accounting Business RADIUS Channel (broadcasting) Economics Programming language

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Type: preprint
Language: en
Landing Page: http://arxiv.org/abs/2208.14069
PDF: https://arxiv.org/pdf/2208.14069
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DOI: https://doi.org/10.48550/arxiv.2208.14069

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Title: Variance-Based Bregman Extragradient Algorithm with Line Search for Solving Stochastic Variational Inequalities

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Publication year: 2022

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Publication date: 2022-08-30

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Authors: Xian-Jun Long, Yue-Hong He, Nan‐jing Huang

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Landing page: https://arxiv.org/abs/2208.14069

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Concepts: Variational inequality, Lipschitz continuity, Mathematics, Line search, Bounded function, Rate of convergence, Convergence (economics), Bregman divergence, Function (biology), Constant (computer programming), Oracle, Applied mathematics, Residual, Variance (accounting), Algorithm, Line (geometry), Pure mathematics, Mathematical analysis, Computer science, Evolutionary biology, Geometry, Computer network, Computer security, Biology, Software engineering, Economic growth, Accounting, Business, RADIUS, Channel (broadcasting), Economics, Programming language

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abstract_inverted_index.natural	81
abstract_inverted_index.propose	8
abstract_inverted_index.purpose	2
abstract_inverted_index.respect	26
abstract_inverted_index.results	111
abstract_inverted_index.solving	18
abstract_inverted_index.unknown	28
abstract_inverted_index.Finally,	108
abstract_inverted_index.bounded,	70
abstract_inverted_index.function	66, 83
abstract_inverted_index.proposed	117
abstract_inverted_index.residual	82
abstract_inverted_index.theorem.	53
abstract_inverted_index.Lipschitz	29
abstract_inverted_index.algorithm	13, 39
abstract_inverted_index.constant.	30
abstract_inverted_index.effective	45
abstract_inverted_index.iteration	97
abstract_inverted_index.numerical	110
abstract_inverted_index.algorithm.	118
abstract_inverted_index.complexity	98, 103
abstract_inverted_index.condition,	93
abstract_inverted_index.inequality	92
abstract_inverted_index.stochastic	19
abstract_inverted_index.unbounded.	87
abstract_inverted_index.convergence	36, 52, 60, 75
abstract_inverted_index.demonstrate	112
abstract_inverted_index.superiority	114
abstract_inverted_index.variational	20, 91
abstract_inverted_index.Furthermore,	54
abstract_inverted_index.extragradient	12
abstract_inverted_index.inequalities,	21
abstract_inverted_index.variance-based	10
abstract_inverted_index.supermartingale	51
abstract_inverted_index.$\mathcal{O}(1/k)$	62
abstract_inverted_index.$\mathcal{O}(1/\varepsilon)$	99
abstract_inverted_index.$\mathcal{O}(1/\varepsilon^2)$	104
cited_by_percentile_year
countries_distinct_count	0
institutions_distinct_count	3
sustainable_development_goals[0].id	https://metadata.un.org/sdg/10
sustainable_development_goals[0].score	0.5199999809265137
sustainable_development_goals[0].display_name	Reduced inequalities
citation_normalized_percentile