Inverse Polynomial Optimization

Abstract

We consider the inverse optimization problem associated with the polynomial program [Formula: see text] and a given current feasible solution [Formula: see text]. We provide a systematic numerical scheme to compute an inverse optimal solution. That is, we compute a polynomial [Formula: see text] (which may be of the same degree as f, if desired) with the following properties: (a) y is a global minimizer of [Formula: see text] on K with a Putinar's certificate with an a priori degree bound d fixed, and (b) [Formula: see text] minimizes [Formula: see text] (which can be the l1, l2 or l∞-norm of the coefficients) over all polynomials with such properties. Computing [Formula: see text] reduces to solving a semidefinite program whose optimal value also provides a bound on how far f(y) is from the unknown optimal value f*. The size of the semidefinite program can be adapted to the available computational capabilities. Moreover, if one uses the l1-norm, then [Formula: see text] takes a simple and explicit canonical form. Some variations are also discussed.