局部Lipschitz连续条件下单调变分不等式的自适应无线搜索方法
An Adaptive Linesearch-free Method for Monotone Variational Inequalities under Local Lipschitz Continuity
- Kyushu University（九州大学）
- IMT School for Advanced Studies Lucca（卢卡高等研究学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
针对单调变分不等式，提出一种自适应无线性搜索的FRB变体，无需全局Lipschitz常数，通过新Lyapunov函数保证收敛，数值实验验证其有效性。
AI中文摘要：
前向-反射-后向（FRB）分裂方法用于求解包含一个极大单调算子与一个单调Lipschitz连续算子之和的包含问题。每次迭代执行一次预解步和一次对Lipschitz算子的求值，外加一个系数为1的反射项，该反射项重用上一次的算子值。我们考虑极大单调算子是适当、下半连续、凸函数的次微分的情形。我们确定了常数反射系数的紧致容许范围，即对于足够小的步长，所有大于二分之一的值。随后，我们针对局部Lipschitz连续算子提出了一种FRB的无线搜索自适应变体，其中步长和反射系数在迭代间均发生变化。步长通过简单的局部Lipschitz估计以闭式形式计算，无需全局Lipschitz常数的存在或已知，同时保持了FRB的迭代成本。两种方法的分析均依赖于一个新的Lyapunov函数，该函数结合了到解的距离、迭代点的逐次差分以及一个间隙型项。各项在迭代过程中不必单调递减，但步长条件确保其加权组合递减。此外，除非渐近速率外，我们还推导了生成步长的显式下界，并通过在极小极大问题、均衡模型和正则化回归上的数值实验展示了自适应方法的有效性。
英文摘要：
The forward-reflected-backward (FRB) splitting solves inclusion problems involving the sum of a maximally monotone operator and a monotone Lipschitz continuous operator. Each iteration performs one resolvent step and one evaluation of the Lipschitz operator, plus a reflection term with coefficient one that reuses the previous operator value. We consider the setting where the maximally monotone operator is the subdifferential of a proper, lower semicontinuous, convex function. We identify the tight admissible range of constant reflection coefficients, namely all values larger than one half for a sufficiently small stepsize. We then propose a linesearch-free adaptive variant of FRB for locally Lipschitz continuous operators, in which both the stepsize and the reflection coefficient change across iterations. The stepsize is computed in closed form from a simple local Lipschitz estimate, without requiring the existence or knowledge of a global Lipschitz constant, while maintaining the iteration cost of FRB. The analysis of both methods relies on a new Lyapunov function that combines the distance to a solution, successive differences of iterates, and a gap-type term. The individual terms need not decrease along the iterates, but the stepsize conditions ensure that their weighted combination does. Moreover, alongside nonasymptotic rates, we derive explicit lower bounds on the generated stepsizes, and showcase the effectiveness of the adaptive method through numerical experiments on minimax problems, equilibrium models, and regularized regression.
