On robust numerical methods of semidefinite relaxation for polynomial optimization problems

建立了圆锥线性优化问题的面约简算法(FRA)理论，证明了圆锥展开法是FRA的对偶算法。对于多项式优化问题的SDP松弛，有时我们可以通过求解SDP松弛问题来计算多项式优化问题的最优值，尽管SDP问题是不可行的。我们解释了这种现象的原因，并利用这一性质提出了一种新的SDP松弛方案。

We established the theory of facial reduction algorithm (FRA) for conic linear optimization problems, and prove that the conic expansion approach is dual to FRA. For SDP relaxation for polynomial optimization problems (POPs), sometimes we can compute the optimal value of POP by solving the SDP relaxation problem although the SDP problem is infeasible. We elucidated the reason of this phenomena, and proposed a new SDP relaxation scheme using this property.