Relative Entropy Methods in Constrained Polynomial and Signomial Optimization

Relative entropy programs belong to the class of convex optimization problems. Within techniques based on the arithmetic-geometric mean inequality, they facilitate to compute nonnegativity certificates of polynomials and of signomials. While the initial focus was mostly on unconstrained certificates and unconstrained optimization, recently, Murray, Chandrasekaran and Wierman developed conditional techniques, which provide a natural extension to the case of convex constrained sets. This expository article gives an introduction into these concepts and explains the geometry of the resulting conditional SAGE cone. To this end, we deal with the sublinear circuits of a finite point set in ℝ n $$\mathbb {R}^n$$ , which generalize the simplicial circuits of the affine-linear matroid induced by a finite point set to a constrained setting.