Efficiency improvement in an systems approach to polynomial optimization

The problem of finding the global minimum of a so-called Minkowski-norm dominated polynomial can be approached by the matrix method of Stetter and Möller, which reformulates it as a large eigenvalue problem. A drawback of this approach is that the matrix involved is usually very large. However, all that is needed for modern iterative eigenproblem solvers is a routine which computes the action of the matrix on a given vector. This paper focuses on improving the efficiency of computing the action of the matrix on a vector. To avoid building the large matrix one can associate the system of first-order conditions with an system of difference equations. One way to compute the action of the matrix efficiently is by setting up a corresponding shortest path problem and solving it. It turns out that for large n the shortest path problem has a high computational complexity, and therefore some heuristic procedures are developed for arriving cheaply at suboptimal paths with acceptable performance.