This paper proposes a second-order cone programming (SOCP) relaxation for the generalized trust-region problem by exploiting the property that any symmetric matrix and identity matrix can be simultaneously diagonalizable. We show that our proposed SOCP relaxation can provide a lower bound as tight as that of the standard semidefinite programming (SDP) relaxation. Moreover, we provide a sufficient condition under which the proposed SOCP relaxation is exact. Since the standard SDP relaxation suffers from a much heavier computing burden, the proposed SOCP relaxation has a much higher efficiency in solving process. Then we design a branch-and-bound algorithm based on this SOCP relaxation to obtain the global optimal solution for a general problem. Three types of numerical experiments are carried out to demonstrate the effectiveness and efficiency of our proposed SOCP relaxation.

中文翻译：

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广义信赖域子问题的基于SOCP松弛的分支定界方法

通过利用对称矩阵和恒等矩阵可以同时对角化的性质，提出了广义信赖域问题的二阶锥规划（SOCP）松弛。我们表明，我们提出的SOCP松弛可以提供与标准半定编程（SDP）松弛一样严格的下限。此外，我们提供了一个足够的条件，在该条件下，拟议的SOCP松弛是准确的。由于标准的SDP松弛会承受更大的计算负担，因此建议的SOCP松弛在求解过程中具有更高的效率。然后，我们基于这种SOCP松弛设计了一种分支定界算法，以获得针对一般问题的全局最优解。