You can use linprog (help page has good examples):
x = linprog(f,A,b,Aeq,beq,lb,ub) defines a set of lower and upper bounds on the design variables, x, so that the solution is always in the range lb ≤ x ≤ ub. Set Aeq = [] and beq = [] if no equalities exist.
f, x, b, beq, lb, and ub are vectors, and A and Aeq are matrices.
So
f = [a; b; c]
lb = []
ub = []
Aeq = []
beq = []
We need to setup the constraints in a matrix format $A{\mathbf x} \le b -\epsilon$. Separate $0 < x < c$ into $-x < 0,$ and $x < c,$ etc. Subtract eps to $-x < 0 -\epsilon,$ etc. So
$$
\begin{pmatrix}
1 & 0 & 0 \\
0 & -1 & 0 \\
0 & 1 & 1 \\
0 & -1 & -1
\end{pmatrix}
\begin{pmatrix}
x \\ y \\ z
\end{pmatrix}
\le
\begin{pmatrix}
c_1 -\epsilon \\ 0 -\epsilon \\ c_2 -\epsilon \\ 0 -\epsilon
\end{pmatrix}
$$
Now, we can invoke: linprog.