The problem of sensor network localization (SNL) can be formulated as a semidefinite programming problem with a rank constraint. We propose a new method for solving such SNL problems. We factorize a semidefinite matrix with the rank constraint into a product of two matrices via the Burer–Monteiro factorization. Then, we add the difference of the two matrices, with a penalty parameter, to the objective function, thereby reformulating SNL as an unconstrained multiconvex optimization problem, to which we apply the block coordinate descent method. In this paper, we also provide theoretical analyses of the proposed method and show that each subproblem that is solved sequentially by the block coordinate descent method can also be solved analytically, with the sequence generated by our proposed algorithm converging to a stationary point of the objective function. We also give a range of the penalty parameter for which the two matrices used in the factorization agree at any accumulation point. Numerical experiments confirm that the proposed method does inherit the rank constraint and that it estimates sensor positions faster than other methods without sacrificing the estimation accuracy, especially when the measured distances contain errors.

传感器网络定位 (SNL) 问题可以表述为具有秩约束的半定规划问题。我们提出了一种解决此类 SNL 问题的新方法。我们通过 Burer-Monteiro 分解将具有秩约束的半定矩阵分解为两个矩阵的乘积。然后，我们将带有惩罚参数的两个矩阵的差值添加到目标函数中，从而将 SNL 重新构建为无约束多凸优化问题，我们对其应用块坐标下降法。在本文中，我们还提供了所提出方法的理论分析，并表明通过块坐标下降法顺序解决的每个子问题也可以解析解决，由我们提出的算法生成的序列收敛到目标函数的一个平稳点。我们还给出了分解中使用的两个矩阵在任何累积点都一致的惩罚参数范围。数值实验证实，所提出的方法确实继承了秩约束，并且它在不牺牲估计精度的情况下比其他方法更快地估计传感器位置，尤其是当测量的距离包含误差时。