You might look at the literature on the upper envelope (or equivalently, the lower envelope) of a collection of surfaces in $\mathbb{R}^d$. Such upper envelopes arise in a variety of computational geometry contexts, and so have been heavily studied. A good source is:
Daniel Halperin, "Arrangments," Handbook of Discrete and Computational Geometry, (ed. O’Rourke & Goodman) CRC Press, pp. 529-562, 2004(ed. O’Rourke & Goodman) CRC Press, pp. 529-562, 2004.
For example, the upper envelope of $n$ surfaces in $\mathbb{R}^d$, under certain assumptions on the surfaces (e.g., that they are algebraic of constant maximum degree) has complexity about $O(n^{d-1})$. Perhaps closer to your problem, the upper envelope of $n$ $(d{-}1)$-simplices in $\mathbb{R}^d$ also has complexity near $O(n^{d-1})$. For the latter, there are efficient algorithms to construct the envelope. See:
Herbert Edelsbrunner, Leonidas J. Guibas and Micha Sharir, "The upper envelope of piecewise linear functions: Algorithms and applications," Volume 4, Number 1, 311-336, 1989Volume 4, Number 1, 311-336, 1989. Discrete & Computational Geometry.
It may be that you could adapt these techniques to your computation, avoiding construction of the full envelope, and focusing on the maxima.
