This paper studies the projected saddle-point dynamics associated to a convex-concave function, which we term saddle function. The dynamics consists of gradient descent of the saddle function in variables corresponding to convexity and (projected) gradient ascent in variables corresponding to concavity. We examine the role that the local and/or global nature of the convexity-concavity properties of the saddle function plays in guaranteeing convergence and robustness of the dynamics. Under the assumption that the saddle function is twice continuously differentiable, we provide a novel characterization of the omega-limit set of the trajectories of this dynamics in terms of the diagonal blocks of the Hessian. Using this characterization, we establish global asymptotic convergence of the dynamics under local strong convexity-concavity of the saddle function. When strong convexity-concavity holds globally, we establish three results. First, we identify a Lyapunov function (that decreases strictly along the trajectory) for the projected saddle-point dynamics when the saddle function corresponds to the Lagrangian of a general constrained convex optimization problem. Second, for the particular case when the saddle function is the Lagrangian of an equality-constrained optimization problem, we show input-to-state stability (ISS) of the saddle-point dynamics by providing an ISS Lyapunov function. Third, we use the latter result to design an opportunistic state-triggered implementation of the dynamics. Various examples illustrate our results.

中文翻译：

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凸性在鞍点动力学中的作用：李亚普诺夫函数和鲁棒性


本文研究了与凸凹函数（我们称之为鞍函数）相关的投影鞍点动力学。动力学包括对应于凸性的变量中鞍函数的梯度下降和对应于凹性的变量中的（投影）梯度上升。我们研究鞍函数凸凹性质的局部和/或全局性质在保证动力学收敛和鲁棒性方面所起的作用。在鞍函数两次连续可微的假设下，我们以 Hessian 矩阵的对角块的形式提供了该动力学轨迹的欧米伽极限集的新颖表征。利用这种表征，我们在鞍函数的局部强凸凹性下建立了动力学的全局渐近收敛。当强凸凹性全局成立时，我们得出三个结果。首先，当鞍函数对应于一般约束凸优化问题的拉格朗日函数时，我们确定了投影鞍点动力学的李亚普诺夫函数（严格沿轨迹递减）。其次，对于鞍函数是等式约束优化问题的拉格朗日函数的特殊情况，我们通过提供 ISS Lyapunov 函数来显示鞍点动力学的输入状态稳定性 (ISS)。第三，我们使用后一个结果来设计动态的机会主义状态触发实现。各种例子说明了我们的结果。