Inverse Polynomial Optimization
针对多项式优化问题，提出一种数值方案计算逆最优解，即找到一个与原目标函数相近的多项式，使得给定可行解成为其全局极小点，并给出最优值偏差的界。
We consider the inverse optimization problem associated with the polynomial program [Formula: see text] and a given current feasible solution [Formula: see text]. We provide a systematic numerical scheme to compute an inverse optimal solution. That is, we compute a polynomial [Formula: see text] (which may be of the same degree as f, if desired) with the following properties: (a) y is a global minimizer of [Formula: see text] on K with a Putinar's certificate with an a priori degree bound d fixed, and (b) [Formula: see text] minimizes [Formula: see text] (which can be the l 1 , l 2 or l ∞ -norm of the coefficients) over all polynomials with such properties. Computing [Formula: see text] 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 l 1 -norm, then [Formula: see text] takes a simple and explicit canonical form. Some variations are also discussed.