Steklov convexification and a trajectory method for global optimization of multivariate quartic polynomials
2020
The Steklov function
$$\mu _f(\cdot ,t)$$
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
$$\mu _f$$
for the case when f is a multivariate quartic polynomial. Then we show 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 an algorithm 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), 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.
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