How to represent sums of piecewise quadratic functions for efficient optimisation?
I'd suggest storing this in a tree data structure.  The tree represents a piecewise quadratic function (namely, the sum), and each leaf of the tree represents the domain of one of the pieces.  Each internal node corresponds to a range of intervals (the union of the intervals associated with the leaves under it).  Store in each node the maximum of the function on that interval.  Also, you will store in each node some of the summands (each of which is a quadratic function on some limited domain).  The quadratic function associated with a leaf is the sum of all of the summands found on the path from the root to that leaf.

If you maintain the tree using a self-balancing binary tree data structure, the height of the tree will be $O(\log n)$, where $n$ is the number of pieces in the sum.

Now, suppose you want to update the tree by adding a quadratic function $f$ with a small domain.  First, split the two leaves $f$'s endpoints fall into, if necessary.  Then, you can express the domain of $f$ as a union of $O(\log n)$ intervals, each associated with one of the nodes of the tree.  So, update those $O(\log n)$ nodes by adding in $f$ as another summand.  Also, for each leaf whose interval is contained in the domain of $f$, compute the maximum value of the quadratic function that leaf represents.  Now propagate these maxima up the tree, until you have eventually updated the maximum at the root; this lets you compute the maximum of the sum.  If the domain of $f$ contains $m$ leaves, then the running time for this update will be $O(m + \log n)$.  So, it will be quite slow if the domain of $f$ is large, but it will be fast if the domain of $f$ is very small.

I could not figure out how to make this run in $O(\log n)$ time regardless of the size of the domain of the summand added.

I suspect there may be a simpler data structure that also achieves $O(m + \log n)$ time updates -- I suspect the machinery of self-balancing binary tree isn't needed.