This paper presents an efficient algorithm for finding all solutions of nonlinear equations using linear programming. This algorithm is based on a simple test (called the LP test) for nonexistence of a solution to a system of nonlinear equations in a given region. In the conventional LP test, a system of nonlinear equations is formulated as a linear programming problem by surrounding component nonlinear functions by rectangles. In the proposed algorithm, we first use rectangles, and when the nonlinearity of functions becomes weak, we switch to parallelograms. It is shown that we can use the dual simplex method throughout the algorithm by applying the variable transformation to the oblique coordinate system, by which the LP test becomes more efficient. Moreover, since polygons with proper sizes are used, the LP test becomes more powerful. By numerical examples, it is shown that the proposed algorithm is more efficient than the conventional algorithms using rectangles only or parallelograms only. We also consider the special case where component nonlinear functions are locally convex and monotone, and propose an efficient LP test algorithm using rectangles and triangles.

本文提出了一种有效的算法，用于使用线性规划来查找非线性方程的所有解。该算法基于简单测试（称为LP测试），用于确定给定区域中非线性方程组的解不存在。在传统的LP测试中，通过用矩形包围非线性组件将非线性方程组公式化为线性规划问题。在提出的算法中，我们首先使用矩形，当函数的非线性变得很弱时，我们切换到平行四边形。结果表明，通过将变量变换应用于斜坐标系，可以在整个算法中使用对偶单纯形法，从而使LP测试更加有效。而且，由于使用了适当大小的多边形，LP测试变得更加强大。通过数值示例表明，所提出的算法比仅使用矩形或仅平行四边形的常规算法更有效。我们还考虑了组件非线性函数为局部凸和单调的特殊情况，并提出了使用矩形和三角形的有效LP测试算法。