Use of LinearProgramming

LinearOptimization can accept the objective and constraints in this form

LinearOptimization[t, {2 x + y + 2 < t, -x - 2 y + 3 < t, 
                       -3 x + y < t, 2 x - 3 y < t}, {t, x, y}]

(* {t -> 29/15, x -> -(2/5), y -> 11/15} *)

 N[%]

(* {t -> 1.93333, x -> -0.4, y -> 0.733333} *)

and it can also convert them to the matrix formulation, for example

obj = LinearOptimization[t, {2 x + y + 2 < t, -x - 2 y + 3 < t, 
      -3 x + y < t, 2 x - 3 y < t}, {t, x, y}, "ObjectiveVector"]

(* {1, 0, 0} *)

{a, b} = Normal @ LinearOptimization[t, {2 x + y + 2 < t, -x - 2 y + 3 < t, 
      -3 x + y < t, 2 x - 3 y < t}, {t, x, y}, "LinearInequalityConstraints"]

(* {{{1, -2, -1}, {1, 1, 2}, {1, 3, -1}, {1, -2, 3}}, {-2, -3, 0, 0}} *)

Of course it is possible to feed these into LinearOptimization:

 LinearOptimization[obj, {a, b}]

 (* {29/15, -(2/5), 11/15} *)

or into LinearProgramming:

 LinearProgramming[obj, a, -b, None]

 (* {29/15, -(2/5), 11/15} *)

but in general LinearOptimization is the more modern and flexible function.

Taking your variables in the order {t, x, y}, you can rewrite your constraints as follows, which makes it easier (at least for me) to turn them into matrix form (as shown in comments for each line):

t - 2 x -   y > 2    (* {1, -2, -1} *)
t +   x + 2 y > 3    (* {1,  1,  2} *)
t + 3 x -   y > 0    (* {1,  3, -1} *)
t - 2 x + 3 y > 0    (* {1, -2,  3} *)

Then your target function $t$ is represented by coefficients {1, 0, 0} in the variables {t, x, y}, respectively.

With this, you can write:

LinearProgramming[
  {1, 0, 0},
  {{1, -2, -1},
   {1,  1,  2},
   {1,  3, -1},
   {1, -2,  3}},
  {2, 3, 0, 0}, 
  None
]

(* Out: {29/15, -(2/5), 11/15} *)

The result above is exact, because exact coefficients were provided, but of course it is numerically the same as the one reported by FindMinimum.