I can't imagine why you want to do this. BUT, it is an interesting problem, so I'll take a look.
Assume that all functions are continuous and at least once differentiable. It would probably not break things in theory if they are not, but I'll do so anyway. It could cause isses with the root finder.
Next, consider the simple case of two pieces, thus two functions and one break point. You want to find the unknown parameter B, such that
int(f(x),[A,B]) + int(g(x),[B,C])
is maximized. The maximum MUST occur at some point where the two functions cross. Yes, there may be multiple crossings. But the maximum will lie at ONE of those crossings of the two functions. For example, suppose we consider some general location where f(x) > g(x). Assume the optimum value for the break point (B) is at that location. But if f(B)>g(B), then we can trivially increase the value of B to increse the combined integral. And we can continue to increase the break point until exactly the location is found where the two functions are equal. If G is the larger of the two functions, then we can decrease the break by the same logic, until the two functions cross.
The point is, all we need to do is locate the points where f(x) and g(x) cross. The breakpoint must lie at ONE of the crossings.
If we have more then two functions, so now also h(x), then we also need to look for the crossings of the pair of functions g(x) and h(x). In some cases, if h(x) everywhere dominates g(x), so that h(x) > g(x) everywhere, then the problem may reduce to an integral where the length of the interval for g(x) is zero, so the upper and lower limits on the g integral may be the same. Effectively, this means we need to find the crossings of the pair of fnctions f(x) and h(x).
Anyway, all of this just means all we care about is how to locate the points where each of those functions cross, and then the solution will be a combination of those crossings. My point is, this is a purely combinatorial problem, and a fairly simple one at that. First locate all function crossings. Then test each combination, looking at where f and g cross first. Then look for the points where you transition to h, etc. Absolutely no optimization is necessary beyond the initial rootfinding, and that can be done using fzero. In fact, optimization is not appropriate for the breakpoints. Just find the breaks using fzero.
For example, consider these functions on the global interval [0,10].
h = @(x) sin(sqrt(x)/2 + pi/3) - 0.5;
Again, all that matters is where the functions cross. So obtain a list of all such crossings. Then just test them all. Even better, integrate each function between each of those transition points in advance, but that is just a question of extra CPU cycles spent at a cost of coding time here.
Code to find all crossings of a pair of functions in 1-dimension is not that difficult to write. I've attached a simple code to this post that just uses fzero.
Xfg = allcrossings(f,g,support,nsamples)
Xgh = allcrossings(g,h,support,nsamples)
Xfh = allcrossings(f,h,support,nsamples)
Xfh =
0.3195 2.8371 6.4404 9.4033
Now all you need to do is test those locations as potential break points. At this point, it is just a fairly simple search. I won't go any further, since at this point, you have already accepted an answer, even though I would suggest you may be taking the wrong path there. But the solution MUST lie at some set of those breakpoints. We know that in advance. Effectively, there are now just a few integrals we need to compute.
