Random walk in a simplex and quadratic optimization over convex polytopes

In this paper we develop probabilistic arguments for justifying the quality of an approximate solution for global quadratic minimization problem, obtained as a best point among all points of a uniform grid inside a polyhedral feasible set. Our main tool is a random walk inside the standard simplex, for which it is easy to find explicit probabilistic characteristics. For any integer k = 1 we can generate an approximate solution with relative accuracy 1k provided that the quadratic objective function is non-negative in all nodes of the feasible set. The complexity of the process is polynomial in the number of nodes and in the dimension of the space of variables. We extend some of the results to problems with polynomial objective function. We conclude the paper by showing that some related problems (maximization of cubic or quartic form over the Euclidean ball, and the matrix ellipsoid problem) are NP-hard.