Inverse polynomial optimization

TL;DR

This paper develops a numerical method to find an inverse polynomial optimization problem solution, ensuring a given feasible point is a global minimizer while minimizing the difference from the original polynomial.

Contribution

It introduces a systematic semidefinite programming approach to compute inverse optimal polynomials with guarantees and explicit forms under certain norms.

Findings

The method computes inverse solutions via semidefinite programming.

The approach provides bounds on the optimality gap.

Explicit solutions are available when using the -norm.

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.