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The Steklov function $\mu_f(\cdot,t)$ is defined to average a continuousfunction $f$ 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 smallvalues of $t$. In this paper, we first find a concise and useful expression for$\mu_f$ for the case when $f$ is a multivariate quartic polynomial. Then weshow that, for large enough $t$, $\mu_f(\cdot,t)$ is convex; in other words,$\mu_f(\cdot,t)$ convexifies $f$. We provide an easy-to-compute formula for $t$with which $\mu_f$ convexifies certain classes of polynomials. We present analgorithm which constructs, via an ODE involving $\mu_f$, a trajectory $x(t)$emanating from the minimizer of the convexified $f$ and ending at $x(0)$, anestimate 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 globalminimizers. Finally, we illustrate the working of our method by means ofnumerous computational examples.

Steklov Convexification and a Trajectory Method for Global Optimization of Multivariate Quartic Polynomials