We present a novel efficient theoretical and numerical framework for solving global non-convex polynomial optimization problems. We analytically demonstrate that such problems can be efficiently reformulated using a non-linear objective over a convex set; further, these reformulated problems possess no spurious local minima (i.e., every local minimum is a global minimum). We introduce an algorithm for solving these resulting problems using the augmented Lagrangian and the method of Burer and Monteiro. We show through numerical experiments that polynomial scaling in dimension and degree is achievable for computing the optimal value and location of previously intractable global polynomial optimization problems in high dimension.

翻译：我们提出了一种新颖且高效的理论与数值框架，用于求解全局非凸多项式优化问题。我们通过分析证明，此类问题可被有效地重新表述为凸集上的非线性目标函数优化；进一步地，这些重新表述的问题不存在虚假局部最小值（即每个局部最小值均为全局最小值）。我们引入一种算法，利用增广拉格朗日函数以及Burer和Monteiro的方法来求解这些转化后的问题。通过数值实验表明，在维度与多项式度数的多项式缩放条件下，可计算高维中先前难以处理的全局多项式优化问题的最优值及其位置。