NAME: 4 MT263 problems Hand in your solutions, with this coversheet, for feedback to the MT263 homework drop box in the foyer of the Bedford Building (beside the door of room 1-33) by 11am on Wednesday 30th October. I will keep a record of how many questions you made a decent attempt on. I can provide you with additional feedback if you want it. Many students don't collect their feedback so if you don't want additional feedback, please tick the relevant box below. Otherwise, if you make a decent attempt of N questions, you can request additional feedback on | ~] questions (note the use of the floor function here). Please indicate which questions you'd like feedback on in the relevant box below (if you don't indicate what feedback you want, I'll assume you don't want any). • Put a tick in this box if you don't want additional written feedback on your solutions. · I attempted questions. Please provide me with feedback on questions YOUR FEEDBACK TO THE LECTURER (e.g., what was hard, interesting, fun, etc. this week): FEEDBACK ON YOUR SOLUTIONS FROM THE MARKER: 1. Which of the following sets are convex? Justify your answers. (a) {(x, y) E R2 : 1 < x < 2, y = 1}. (b) {(x, y) E R2 : x > 0, y > 0, y ? 1/x}. (c) { (x, y) E R2 : 0 ? x ? 1, 0 ? y ? 2} {(x, y) E R2 : 0 ? x ? 2,0? y ?1}. 2. Which of the following statements are true? Justify your answers (i.e., give a proof of, or a counter example to, each statement). (a) The intersection of two convex sets in R" is convex.
(b) The union of two convex sets is convex. (c) The feasible region of the linear program max cTx subject to Ax = b, x > 0 is convex. 3. Let C be a convex set in Rn, and x E C. Prove the following are equivalent. (a) x is an extreme point of C. (b) There do not exist X1, X2 € C such that x1 # X2 and x=??1 + (1 - ?)x2, with 0 < ? < 1. (c) C\{x} is convex. 4. Solve the following LPP graphically and with the branch and bound method. (You may use a computer to solve the LPP relaxation problems if you wish.) Draw the tree diagram of the branch and bound process. (a) maximize subject to (b) maximize z=4x + y subject to 3x + 2y ? 5 2x + 6y ?7 3x + 7y ?6 x, y ? 0 integer z=x+y 2x + 3y ? 12 2x+y ?6 x, y ? 0 integer 5. Which of the following statements are true? Justify your answer. (a) If all the entries of A, b and c are integral (i.e., are integers) then there exist an optimal integral solution x (i.e., an optimal solution with only integer entries) to the linear program: maximize