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的高效理论与数值框架，用于求解全局非凸多项式优化问题。解析证明表明，此类问题可通过凸集上的非线性目标函数进行高效重构；并且重构后的问题不存在虚假局部极小值（即每个局部极小值均为全局极小值）。我们引入了一种算法，该算法利用增广拉格朗日函数及Burer-Monteiro方法求解上述重构问题。数值实验显示，该算法在计算高维先前难以处理的全局多项式优化问题的最优值与最优位置时，其性能随维度和度数的增加呈多项式级标度。