Maximization of a 3-variable function

So I define a 3-variable function $h$ and I want to find the maximum of $h$ in a region $\mathcal{R}$:

h[x_, y_, z_] := y (a - z) - z (a - t) + (y + x) (Max[0, b - (y - z)] - x + 1)
Maximize[{h[x, y, z], 0 <= z <= t, z <= y <= a, 0 <= x <= Max[0, b - (y - z)]}, {x, y, z}]


Here I'm treating $a,b,t$ as constants. Shouldn't Mathematica return the maximum of $h$ as a function of $a,b,t$?

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At first I thought Mathematica was being defeated by the presence of Max. So I took them out and tried h[x_, y_, z_] := y (a - z) - z (a - t) + (y + x) (b - (y - z) - x + 1); Maximize[{h[x, y, z], 0 <= z <= t && z <= y <= a && 0 <= x <= b - (y - z)}, {x, y, z}], just to see if it would work. Now it's been chugging away for an hour with no result. –  Rahul Narain Jun 9 at 20:12
if you specify numerical values for a,b,t you readily get a solution. Not sure how hard you want to work at getting an analytic expression which will inevitably be very complicated. –  george2079 Jun 9 at 21:21

h[x_, y_, z_] := y (a - z) - z (a - t) + (y + x) (Max[0, b - (y - z)] - x + 1)

The solutions that Mathematica finds depend on the variables $x,y,z$ because the program returns the maximum depending on the different faces/edges of the boundary. I want the answer in terms of $a,b,t$ only. In other words I need Mathematica to reiterate the procedure and maximize the solutions it has found on the corresponding boundaries. –  the_fox Apr 10 at 20:30
Inputting values seems to give. x->0.5 (b - a + 1)) - Max[(a - t)/2, 0] y->a z->a||t for your boundary conditions. It gives indeterminate for certain values, but the code is obviously correct, how irritating, something must stall. –  Feyre Apr 11 at 22:20