Marc Lassonde
It is known that for convex sets, the KKM condition is equivalent to the finite intersection property. We use this equivalence to obtain a characterisation of monotone operators in terms of convex KKM maps and in terms of the existence of solutions to Minty variational inequalities. The latter result provides a converse to the seminal theorem of Minty.
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Non-Monotone Variational Inequalities
本文基于逆映射理论推导非单调变分不等式解存在的充分条件，证明外梯度法在Minty解存在时的收敛性与效率，并将结果推广到博弈论，得到弱耦合下纯拟纳什均衡及纯纳什均衡的存在条件。
On Non-Monotone Variational Inequalities
本文针对非单调变分不等式解的存在性问题，基于逆映射理论与度理论给出若干充分条件，提出Minty解存在的充分条件，并证明Minty解存在时Korpelevich和Popov方法收敛到解。
Beyond Monotone Variational Inequalities: Solution Methods and Iteration Complexities
本文从投影型算法收敛性角度研究非单调变分不等式问题，提出了包括Minty条件在内的多种充分条件，并通过示例与收敛分析证明这些条件能刻画保证算法收敛的不同VI问题类别。
We show that a point is solution of the Minty variational inequality of subdifferential type for a given function if and only if the function is increasing along rays starting from that point. This provides a characterization of the monotone polar of subdifferentials of lower semicontinuous functions, which happens to be a common subset of their graphs depending only on the function.
This paper introduces a family of stochastic extragradient-type algorithms for a class of nonconvex-nonconcave problems characterized by the weak Minty variational inequality (MVI). Unlike existing results on extragradient methods in the monotone setting, employing diminishing stepsizes is no longer possible in the weak MVI setting. This has led to approaches such as increasing batch sizes per iteration which can however be prohibitively expensive. In contrast, our proposed methods involves two stepsizes and only requires one additional oracle evaluation per iteration. We show that it is possi…
The notion of a firmly nonexpansive mapping is central in fixed point theory because of attractive convergence properties for iterates and the correspondence with maximal monotone operators due to Minty. In this paper, we systematically analyze the relationship between properties of firmly nonexpansive mappings and associated maximal monotone operators. Dual and self-dual properties are also identified. The results are illustrated through several examples.
We introduce the notion of quasimonotone polar of a multivalued operator, in a similar way as the well-known monotone polar due to Martinez-Legaz and Svaiter. We first recover several properties similar to the monotone polar, including a characterization in terms of normal cones. Next, we use it to analyze certain aspects of maximal (in the sense of graph inclusion) quasimonotonicity, and its relation to the notion of maximal quasimonotonicity introduced by Aussel and Eberhard. Furthermore, we study the connections between quasimonotonicity and Minty Variational Inequality Problems.
A Block Coordinate and Variance-Reduced Method for Generalized Variational Inequalities of Minty Type
针对单调算子变分不等式缺乏块坐标更新优势的问题，提出一种随机块坐标方法，在块Lipschitz参数高度不均匀时显著改进复杂度，并扩展至有限和算子实现方差缩减。
Variance-reduction for Variational Inequality Problems with Bregman Distance Function
本文针对有限和结构变分不等式，提出结合Bregman距离函数的单循环随机方差缩减算法，在单调及弱Minty非单调情形下取得最优收敛保证并显著优于现有方法。
New Classes of Non-monotone Variational Inequality Problems Solvable via Proximal Gradient on Smooth Gap Functions
本文研究近端梯度下降算法在非单调变分不等式问题参数化间隙函数重构上的局部线性收敛，通过引入误差界条件与同伦延拓方案，实现了无需Minty型解假设下的局部线性收敛与全局收敛，有效求解Nash均衡、交通控制及GAN等非单调VI问题。
Beyond First-Order Methods for $\ell_p$-Structured Non-Monotone Variational Inequalities
本文提出高阶算法解决ℓ_𝑝结构非单调变分不等式问题，针对弱 Minty 变分不等式参数ρ>0范围内的ℓ_𝑝-范数 stationary 点收敛，扩展了单调算子的高阶光滑性。
Single-Call Stochastic Extragradient Methods for Structured Non-monotone Variational Inequalities: Improved Analysis under Weaker Conditions
本研究针对单次调用随机外梯度方法，在更弱条件下（无需方差有界）为两类结构化非单调变分不等式提供收敛保证，引入期望残差条件，并涵盖多种步长与采样策略。
In this paper, we study convex analysis and its theoretical applications. We first apply important tools of convex analysis to Optimization and to Analysis. We then show various deep applications of convex analysis and especially infimal convolution in Monotone Operator Theory. Among other things, we recapture the Minty surjectivity theorem in Hilbert space, and present a new proof of the sum theorem in reflexive spaces. More technically, we also discuss autoconjugate representers for maximally monotone operators. Finally, we consider various other applications in mathematical analysis.
A Projection-Free Algorithm for Variational Inequalities in Hilbert Spaces with Strong Convergence
提出一种无投影算法，通过半空间投影和Slater修正处理变分不等式，实现强收敛到初始点在解集上的投影，并给出残差估计。
Popov Mirror-Prox Method for Variational Inequalities
本文提出无参数的Popov mirror-prox算法，在多项式增长条件下求解随机和确定性变分不等式，实现最优收敛速率，并扩展至非单调情形，通过实验验证。
Convergence of the Preconditioned Proximal Point Method and Douglas-Rachford Splitting in the Absence of Monotonicity
本文在无单调性条件下研究预处理近端点法和Douglas-Rachford分裂算法的收敛性，通过引入半单调算子类建立DRS与预处理PPA的联系，并给出全局收敛的充分条件。
Because of Minty's classical correspondence between firmly nonexpansive mappings and maximally monotone operators, the notion of a firmly nonexpansive mapping has proven to be of basic importance in fixed point theory, monotone operator theory, and convex optimization. In this note, we show that if finitely many firmly nonexpansive mappings defined on a real Hilbert space are given and each of these mappings is asymptotically regular, which is equivalent to saying that they have or "almost have" fixed points, then the same is true for their composition. This significantly generalizes the resul…
Perseus: A Simple and Optimal High-Order Method for Variational Inequalities
本文提出Perseus，一种无需线搜索的简单p阶方法，以最优速率O(ε^{-2/(p+1)})求解光滑单调变分不等式，并通过匹配下界证明其最优性，同时扩展至非单调Minty条件情形。
Adaptive Delayed-Update Cyclic Algorithm for Variational Inequalities
本文提出自适应延迟更新循环算法（ADUCA），用于求解广义Minty变分不等式，无需全局或块级Lipschitz常数，具有近最优全局Oracle复杂度。
Primal Methods for Variational Inequality Problems with Functional Constraints
针对带函数约束的变分不等式问题现有原始-对偶算法需预知最优拉格朗日乘子的问题，提出无需该信息的约束梯度法，证明其非渐近收敛性，复杂度匹配投影法且预言机成本更低，通过数值算例验证了有效性。
On the Analysis of Misspecified Variational Inequalities with Nonlinear Constraints
本文研究了具有非线性约束的误指定变分不等式，提出单循环近似增广拉格朗日方法，同时更新主变量、对偶乘子和误指定参数，结合前向-反射-后向步长与增广拉格朗日惩罚，证明了$\mathcal{O}(1/K)$的收敛速率。
Convergence of the Chambolle-Pock Algorithm in the Absence of Monotonicity
本文将Chambolle-Pock算法的收敛性分析从凸问题扩展到非单调问题，通过斜弱Minty条件和半单调算子类，给出了依赖奇异值的新步长与松弛参数范围，恢复并扩展了现有收敛结果。
Extragradient Method for $(L_0, L_1)$-Lipschitz Root-finding Problems
针对现代机器学习中更宽松的$\alpha$-对称$(L_0, L_1)$-Lipschitz条件，本文为外梯度方法提出新颖步长策略，求解求根问题，实现单调算子的次线性收敛和强单调算子的线性收敛，并验证了其有效性。
Decentralized Local Stochastic Extra-Gradient for Variational Inequalities
针对异构数据与时变网络下的分布式随机变分不等式问题，提出去中心化局部随机额外梯度法，理论分析了多种单调性设置下的收敛速率，并在GAN去中心化训练中验证了有效性。
Solving Stochastic Variational Inequalities without the Bounded Variance Assumption
本文提出无需方差有界假设的算法，解决无界约束集的随机变分不等式问题，达到最优复杂度O(ε^{-4})。
Beyond Maximum Likelihood: Variational Inequality Estimation for Generalized Linear Models
本文提出基于变分不等式的估计框架，用于解决广义线性模型中非典型或非光滑链接函数带来的计算挑战，通过操作符平衡条件替代最大似然估计，提升数值稳定性与有限样本性能。
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