p = MixedIntegerLinearProgram()
v = p.new_variable(real=True, nonnegative=False)
x, y, z = v['x'], v['y'], v['z']
p.set_objective(z)
p.add_constraint(z <= 2*x +   y)
p.add_constraint(z <=  -x + 2*y)
p.add_constraint(x + y == 1)
p.add_constraint(x >= 0)
p.add_constraint(y >= 0)
p.solve()
p.get_values(z,x,y)

p = MixedIntegerLinearProgram()
v = p.new_variable(real=True, nonnegative=False)
s, t, z = v['s'], v['t'], v['z']
p.set_objective(z)
p.add_constraint(z <= -2*s + t)
p.add_constraint(z <=  -s - 2*t)
p.add_constraint(s + t == 1)
p.add_constraint(s >= 0)
p.add_constraint(t >= 0)
p.solve()
p.get_values(z,s,t)

p = MixedIntegerLinearProgram()
v = p.new_variable(real=True, nonnegative=False)
a, b, c, z = v['a'], v['b'], v['c'], v['z']
p.set_objective(z)
p.add_constraint(z <= a   + b + 3*c)
p.add_constraint(z <= 2*a     + c)
p.add_constraint(z <= a + 2*b)
p.add_constraint(a + b + c == 1)
p.add_constraint(a >= 0)
p.add_constraint(b >= 0)
p.add_constraint(c >= 0)
p.solve()
p.get_values(z,a,b,c)