We consider the problem of minimizing a polynomial function of degree three over the boundary of the sphere. If the objective is quadratic instead of cubic, this is the well-studied trust region subproblem, which is known to be tractable. In the cubic case, the problem turns out to be NP-hard. In this paper, we derive and evaluate different approaches for computing lower bounds for the cubic problem. Alternatively to semidefinite programming relaxations proposed in the literature, our approaches do not lift the problem to higher dimensions. The strongest bounds are obtained by Lagrangian decomposition, resulting in a number of parameterized quadratic problems for which the above-mentioned results can be exploited, in particular the existence of a tractable dual problem. In an experimental evaluation, we consider the cubic one-spherical optimization problem, with homogeneous objective function, and compare the bounds generated with the different approaches proposed, for small examples from the literature and for randomly generated instances of varied dimensions.

我们考虑使球面边界上的三次多项式函数最小化的问题。如果目标是二次方而不是三次方，则这是经过充分研究的信任区域子问题，众所周知这是易处理的。在立方情况下，问题证明是NP难题。在本文中，我们推导并评估了计算立方问题下界的不同方法。除了文献中提出的半定规划松弛之外，我们的方法也无法将问题提至更高的维度。通过拉格朗日分解获得最强的边界，从而导致许多参数化的二次问题，对于这些问题，可以利用上述结果，尤其是存在易处理的对偶问题。在实验评估中，