We study efficient optimization of the Hamiltonians of multi-species spherical spin glasses. Our results characterize the maximum value attained by algorithms that are suitably Lipschitz with respect to the disorder through a variational principle that we study in detail. We rely on the branching overlap gap property introduced in our previous work and develop a new method to establish it that does not require the interpolation method. Consequently our results apply even for models with non-convex covariance, where the Parisi formula for the true ground state remains open. As a special case, we obtain the algorithmic threshold for all single-species spherical spin glasses, which was previously known only for even models. We also obtain closed-form formulas for pure models which coincide with the $E_{\infty}$ value previously determined by the Kac-Rice formula.

翻译：我们研究了多物种球面自旋玻璃哈密顿量的高效优化问题。通过一个我们详细研究的变分原理，我们的结果刻画了那些对无序性满足适当Lipschitz条件的算法所能达到的最大值。我们依赖于先前工作中引入的分支重叠间隙性质，并发展了一种无需插值方法即可建立该性质的新方法。因此，我们的结果甚至适用于非凸协方差模型，而这类模型真实基态的Parisi公式仍未解决。作为一个特例，我们获得了所有单物种球面自旋玻璃的算法阈值——此前该阈值仅对偶模型成立。我们还获得了纯模型的闭式公式，该公式与先前由Kac-Rice公式确定的$E_{\infty}$值一致。