Consider an arbitrary program in standard equation form: maximise cTx subject to Ax = b, x 2 0 Show that if 1) and z are two different optimal solutions to this program, then every convex combination...

Extracted text: Consider an arbitrary program in standard equation form: maximise cTx subject to Ax = b, x 2 0 Show that if 1) and z are two different optimal solutions to this program, then every convex combination of y and z is also an optimal solution of the program. 2) Consider two vectors y and z with y z and consider two values 0 E [0, 1 and 0 e [0, 1] with 0 0. Show that Oy + (1 – 0)z 0'y + (1 – 0')z. In other words, every distinct way of choosing the value A E [0, 1] gives a distinct convex combination Ay + (1 – A)z. Hint: let a = Oy + (1-0)z and b 0'y + (1- 0)z, then consider a - 3) Using the result from parts (a) and (b), show that every linear program has either 0, 1, or infinitely many optimal solutions.