Convex underestimators of polynomials

Convex underestimators of a polynomial on a box. Given a non convex polynomial $${f\in \mathbb{R}[{\rm x}]}$$ and a box $${{\rm B}\subset \mathbb{R}^n}$$, we construct a sequence of convex polynomials $${(f_{dk})\subset \mathbb{R}[{\rm x}]}$$, which converges in a strong sense to the “best” (convex and degree-d) polynomial underestimator $${f^{*}_{d}}$$ of f. Indeed, $${f^{*}_{d}}$$ minimizes the L1-norm $${\Vert f-g\Vert_1}$$ on B, over all convex degree-d polynomial underestimators g of f. On a sample of problems with non convex f, we then compare the lower bounds obtained by minimizing the convex underestimator of f computed as above and computed via the popular αBB method and some of its other refinements. In most of all examples we obtain significantly better results even with the smallest value of k.