Inverse polynomial optimization

Abstract

We consider the inverse optimization problem associated with the polynomial program f^*=\min \{f(x): x\in K\}$ and a given current feasible solution $y\in K$. We provide a systematic numerical scheme to compute an inverse optimal solution. That is, we compute a polynomial $\tilde{f}$ (which may be of same degree as $f$ if desired) with the following properties: (a) $y$ is a global minimizer of $\tilde{f}$ on $K$ with a Putinar's certificate with an a priori degree bound $d$ fixed, and (b), $\tilde{f}$ minimizes $\Vert f-\tilde{f}\Vert$ (which can be the $\ell_1$, $\ell_2$ or $\ell_\infty$-norm of the coefficients) over all polynomials with such properties. Computing $\tilde{f}_d$ reduces to solving a semidefinite program whose optimal value also provides a bound on how far is $f(\y)$ from the unknown optimal value $f^*$. The size of the semidefinite program can be adapted to the computational capabilities available. Moreover, if one uses the $\ell_1$-norm, then $\tilde{f}$ takes a simple and explicit canonical form. Some variations are also discussed.