Inverse Polynomial Optimization

We consider the inverse optimization problem associated with the polynomial program and a given current feasible solution . We provide a systematic numerical scheme to compute an inverse optimal solution. That is, we compute a polynomial (which may be of the same degree as f, if desired) with the following properties: (a) y is a global minimizer of on K with a Putinar's certificate with an a priori degree bound d fixed, and (b) minimizes (which can be the l₁, l₂ or l_∞-norm of the coefficients) over all polynomials with such properties. Computing 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 takes a simple and explicit canonical form. Some variations are also discussed.