Linear programming on reals is "easy"... You can just check all the points. I believe you can follow the shell of the legal polytope and just use a greedy algor

Linear programming on reals is "easy"... You can just check all the points. I believe you can follow the shell of the legal polytope and just use a greedy algorithm to choose the next point that will minimize your goal. If you can get away with a continuous linear program I don't see why you'd use monte carlo. The simplex method will get you an exact answer.