Lower Bounds for Cubic Optimization over the Sphere

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.