Concave programming and numerical methods for searching for real solutions of nonlinear systems. (Q1432461)

Let \(g_i(x), i=1, \dots, n\) be smooth convex functions on \(E^n\) and \(R^0\) a closed compact subset of \(E^n\). The paper is concerned with the problem of finding all solutions to the system \(x \in R^0\), \(g_i(x)=0, i=1, \dots, n\). Assuming that the problem satisfies a regularity condition, the authors suggest algorithms based on auxiliary optimization problems of the form \(\min \{ \| x\|^2 +N \sum_{i=1}^n g_i(x): x \in R^k \}\), where \(N>0\) is sufficiently large and \(R^k \subset E^n\) approximates the subset \(\{x \in R^0: g_i(x)=0, i=1, \dots, n \}\).