ABSTRACT

In this paper, we present two algorithms for solving equilibrium problems on Hadamard manifolds. The two algorithms use the extragradient model and the golden ratio model, respectively, which are two classical models for solving the equilibrium problem in linear space. In each iteration, the step sizes of the two algorithms only depend on the value of the initial parameters and the information of the current iteration. Moreover, compared with the first algorithm, the second algorithm only needs to solve one quadratic programming problem per iteration, which can greatly reduce the computational complexity of the algorithm when the structure of the feasible region is complex. Under mild conditions, we prove that the sequence generated by each algorithm converges to some equilibrium point. Also, we use an experiment to verify the effectiveness of the algorithms.

摘要

在本文中，我们提出了两种算法来解决 Hadamard 流形上的平衡问题。两种算法分别使用了梯度模型和黄金分割率模型，这两个模型是求解线性空间均衡问题的经典模型。在每次迭代中，两种算法的步长仅取决于初始参数的值和当前迭代的信息。而且，与第一种算法相比，第二种算法每次迭代只需要求解一个二次规划问题，当可行域结构复杂时，可以大大降低算法的计算复杂度。在温和的条件下，我们证明了每个算法生成的序列收敛到某个平衡点。还，