Abstract:We propose two approximate versions of the first-order primal-dual algorithm (PDA) to solve a class of convex-concave saddle point problems. The introduced approximate criteria are easy to implement in the sense that they only involve the subgradient of a certain function at the current iterate. The first approximate PDA solves both subproblems inexactly and adopts the absolute error criteria, which are based on non-negative summable sequences. Assuming that one of the PDA subproblems can be solved exactly, the second approximate PDA solves the other subproblem approximately and adopts a relative error criterion. The relative error criterion only involves a single parameter in the range of , which makes the method more applicable. For both versions, we establish the global convergence and convergence rate measured by the iteration complexity, where counts the number of iterations. For the inexact PDA with absolute error criteria, we show the accelerated and linear convergence rate under the assumptions that a part of the underlying functions and both underlying functions are strongly convex, respectively. Then, we prove that these inexact criteria can also be extended to solve a class of more general problems. Finally, we perform some numerical experiments on sparse recovery and image processing problems. The results demonstrate the feasibility and superiority of the proposed methods.

---

中文翻译：

---

鞍点问题的近似一阶原始对偶算法

摘要：我们提出了两个近似的一阶原对偶算法（PDA），以解决一类凸凹鞍点问题。引入的近似标准易于实现，因为它们仅涉及当前迭代中某个函数的次梯度。第一个近似PDA不精确地解决了两个子问题，并采用了基于非负可加序列的绝对误差准则。假设一个PDA子问题可以被精确地解决，第二个近似PDA可以近似地解决另一个子问题，并采用相对误差准则。相对误差准则仅涉及范围内的单个参数，这使该方法更适用。对于这两个版本，我们都建立了全局收敛和用迭代复杂度衡量的收敛速度，其中计算迭代次数。对于具有绝对误差标准的不精确PDA，我们在假设部分基础函数和两个基础函数分别为强凸的情况下显示了加速的收敛速度和线性收敛速度。然后，我们证明了这些不精确的标准也可以扩展为解决一类更一般的问题。最后，我们对稀疏恢复和图像处理问题进行了一些数值实验。结果证明了所提方法的可行性和优越性。

---