In this paper, we study a first-order inexact primal-dual algorithm (I-PDA) for solving a class of convex-concave saddle point problems. The I-PDA, which involves a relative error criterion and generalizes the classical PDA, has the advantage of solving one subproblem inexactly when it does not have a closed-form solution. We show that the whole sequence generated by I-PDA converges to a saddle point solution with \(\mathcal {O}(1/N)\) ergodic convergence rate, where N is the iteration number. In addition, under a mild calmness condition, we establish the global Q-linear convergence rate of the distance between the iterates generated by I-PDA and the solution set, and the R-linear convergence speed of the nonergodic iterates. Furthermore, we demonstrate that many problems arising from practical applications satisfy this calmness condition. Finally, some numerical experiments are performed to show the superiority and linear convergence behaviors of I-PDA.

在本文中，我们研究了解决一类凸凹鞍点问题的一阶不精确原始对偶算法（I-PDA）。I-PDA涉及一个相对误差准则，并且对经典PDA进行了概括，它的优点是当它没有闭式解时，可以不精确地解决一个子问题。我们表明，由I-PDA生成的整个序列以\（\ mathcal {O}（1 / N）\）遍历收敛速度收敛到鞍点解，其中N是迭代次数。此外，在适度的镇定条件下，我们建立了I-PDA生成的迭代与解决方案集之间的距离的全局Q线性收敛速度，以及非遍历迭代的R线性收敛速度。此外，我们证明了由实际应用引起的许多问题都满足了这种平静条件。最后，进行了一些数值实验，以证明I-PDA的优越性和线性收敛行为。