This looks like a job for fmincon. Variables x (3-D) and t = x(4).
Objective function t = x(4).
Lower bound lb = [0,0,0,-Inf].
Linear constraint Aeq = [1 1 1 0], beq = 1.
Nonlinear constraint function as you have, with c(x) = a three-element vector F(x) - x(4). ceq = .
I would take the initial point something like x0 = [1 1 1 30]/3.
Is that clear enough?
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