Inverse Polynomial Optimization

• Authors: Jean B. Lasserre
• Venue: Mathematics of Operations Research
• Year: 2013

We consider the inverse optimization problem associated with the polynomial program $f^*=\min \{f\bx\COLON \bx\in\K\}$ 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 the 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 l1, l2 or l∞-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 fy 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 $\tilde{f}$ takes a simple and explicit canonical form. Some variations are also discussed.