Inverse polynomial optimization

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.

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