Steklov convexification and a trajectory method for global optimization of multivariate quartic polynomials

The Steklov function μ _f(· ,t) is defined to average a continuous function f at each point of its domain by using a window of size given by t>0 . It has traditionally been used to approximate f smoothly with small values of t . In this paper, we first find a concise and useful expression for μ _f for the case when f is a multivariate quartic polynomial. Then we show that, for large enough t , μ _f(· ,t) is convex; in other words, μ _f(· ,t) convexifies f . We provide an easy-to-compute formula for t with which μ _f convexifies certain classes of polynomials. We present an algorithm which constructs, via an ODE involving μ _f , a trajectory x ( t ) emanating from the minimizer of the convexified f and ending at x (0), an estimate of the global minimizer of f . For a family of quartic polynomials, we provide an estimate for the size of a ball that contains all its global minimizers. Finally, we illustrate the working of our method by means of numerous computational examples.