The network localization problem with convex and non-convex distance constraints may be modeled as a nonlinear optimization problem. In the existing localization techniques, the non-convex distance constraints are either removed or relaxed into convex constraints to use the convex optimization methods like semi-definite programming, least square approximation, etc. for solving the problem. We use the nonlinear Lagrangian technique for non-convex optimization to convert the localization problem with noisy distance measurements to a root finding problem of a single variable continuous function without any modification of these constraints. This problem is then solved using an iterative method. In this iterative method, the computation of the functional values involves a finite mini-max problem. We use smoothing gradient method to compute the functional value. The proposed iterative method converges to a solution of the network localization problem with a desired label of accuracy in real time.

具有凸和非凸距离约束的网络定位问题可以建模为非线性优化问题。在现有的定位技术中，将非凸距离约束去除或放宽为凸约束，以使用半定规划，最小二乘逼近等凸优化方法来解决该问题。我们使用非线性拉格朗日技术进行非凸优化将带有噪声距离测量的定位问题转换为单个变量连续函数的求根问题，而无需对这些约束进行任何修改。然后使用迭代方法解决此问题。在这种迭代方法中，功能值的计算涉及有限的最小-最大问题。我们使用平滑梯度法来计算功能值。所提出的迭代方法收敛于具有期望的实时精度标签的网络定位问题的解决方案。