Inverse polynomial optimization

Abstract

We consider the inverse optimization problem associated with the polynomial program f <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">*</sup> = min{f(x) : x ∈ K} and a given current feasible solution y ∈ K. We provide a systematic numerical scheme to compute an inverse optimal solution. That is, we compute a polynomial f̃ (which may be of same degree as f if desired) with the following properties: (a) y is a global minimizer of f̃ on K with a Putinar's certificate with an a priori degree bound d fixed, and (b), f̃ minimizes ||f - f̃|| <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> over all polynomials with such properties. The size of the semidefinite program can be adapted to the computational capabilities available. Moreover, f takes a simple canonical form, and computing f̃ 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 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">*</sup> . Some variations are also discussed.