This paper studies the behavior of the extragradient algorithm when applied to hypomonotone operators, a class of problems that extends beyond the classical monotone setting. While the extragradient method is widely known for its efficacy in solving variational inequalities with monotone and Lipschitz continuous operators, we demonstrate that its convergence is not guaranteed in the hypomonotone setting. We provide a characterization theorem that identifies the conditions under which the extragradient algorithm fails to converge. Our results highlight the necessity of stronger assumptions to guarantee convergence of extragradient and to further develop the existing VI methods for broader problems.
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