We propose a new variant of Chubanov's method for solving the feasibility problem over the symmetric cone by extending Roos's method (2018) of solving the feasibility problem over the nonnegative orthant. The proposed method considers a feasibility problem associated with a norm induced by the maximum eigenvalue of an element and uses a rescaling focusing on the upper bound for the sum of eigenvalues of any feasible solution to the problem. Its computational bound is (i) equivalent to that of Roos's original method (2018) and superior to that of Louren\c{c}o et al.'s method (2019) when the symmetric cone is the nonnegative orthant, (ii) superior to that of Louren\c{c}o et al.'s method (2019) when the symmetric cone is a Cartesian product of second-order cones, (iii) equivalent to that of Louren\c{c}o et al.'s method (2019) when the symmetric cone is the simple positive semidefinite cone, and (iv) superior to that of Pena and Soheili's method (2017) for any simple symmetric cones under the feasibility assumption of the problem imposed in Pena and Soheili's method (2017). We also conduct numerical experiments that compare the performance of our method with existing methods by generating instances in three types: strongly (but ill-conditioned) feasible instances, weakly feasible instances, and infeasible instances. For any of these instances, the proposed method is rather more efficient than the existing methods in terms of accuracy and execution time.

翻译：我们通过扩展Roos方法（2018）求解非负象限可行性问题的思路，提出一种求解对称锥可行性问题的Chubanov方法新变体。所提方法关注与元素最大特征值诱导范数相关的可行性问题，并采用基于可行解特征值之和上界的重缩放策略。其计算复杂度表现为：（i）当对称锥为非负象限时，与Roos原始方法（2018）等价且优于Lourenço等人方法（2019）；（ii）当对称锥为二阶锥笛卡尔积时，优于Lourenço等人方法（2019）；（iii）当对称锥为简单半正定锥时，与Lourenço等人方法（2019）等价；（iv）在Pena和Soheili方法（2017）所施加的可行性假设条件下，对于任何简单对称锥均优于Pena和Soheili方法（2017）。我们通过生成强可行（但病态）实例、弱可行实例与不可行实例三类数值实验，比较了所提方法与现有方法的性能。对于所有测试实例，本方法在求解精度和运行时间方面均显著优于现有方法。