# A Primal-Dual Algorithm with Line Search for General Convex-Concave Saddle Point Problems

20 Oct 2020 Hamedani Erfan Yazdandoost Aybat Necdet Serhat

In this paper, we propose a primal-dual algorithm with a novel momentum term using the partial gradients of the coupling function that can be viewed as a generalization of the method proposed by Chambolle and Pock in 2016 to solve saddle point problems defined by a convex-concave function $\mathcal L(x,y)=f(x)+\Phi(x,y)-h(y)$ with a general coupling term $\Phi(x,y)$ that is not assumed to be bilinear. Assuming $\nabla_x\Phi(\cdot,y)$ is Lipschitz for any fixed $y$, and $\nabla_y\Phi(\cdot,\cdot)$ is Lipschitz, we show that the iterate sequence converges to a saddle point; and for any $(x,y)$, we derive error bounds in terms of $\mathcal L(\bar{x}_k,y)-\mathcal L(x,\bar{y}_k)$ for the ergodic sequence $\{\bar{x}_k,\bar{y}_k\}$... (read more)

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