(5 marks Consider the following general linear program (P). max{c Ar = b, x 2 0}....
please answer question 1) to 3) fully step by step 7 marks (4 marks) (3 marks) (a) Is the operator P(v) y'+2ty linear? (Show workings) (b) Find the null space of the operator in (a) above. Question 2 5 marks Let P be a linear operator. Suppose that y is a particular solution to the equation Ply) = b. Prove that any solution to this equation can be written as y = Yo + yi for yo au clement of...
if so, prove It. i 10. Consider the dual canonical tableau below: X, X, -1 Assume, without loss of generality, that a>0. a. Ifb>0 and c>0, which of the four types of behavior for dual canonical linear programming problems as given by the duality theorem is exhibited above. Prove your assertion. b. Repeat part a under the assumptions that b>0 and c<0. c. Repeat part a under the assumptions that b <0 and c>0 d. Repeat part a under the...
5. Consider a vector space V = {00 + 0 + ar? + 13. 1 40,, 02, 03 € R} and a linear map D:V+V. P(1) - P(1) (a) Write down the matrix representation L of the linear map D under the basis B, = (b) Consider the new basis B2 = 31.5 +5.368.479 10 Find the matrix transformation T such that B, BT. (c) Let L, be the matrix representation of D under the new basis B. Prove L2...
Consider the following linear program: Max 2X + 3Y s.t. 5X +5Y ≤ 400 -1X+ 1Y ≥ 10 1X + 3Y ≥ 90 X, Y ≥ 0 a. Use the graphical solution procedure to find the optimal solution. b. Conduct a sensitivity analysis to determine the range of optimality for the objective function coefficients X & Y. c. What are the binding constraints? d. If the right-hand-side of the binding constraints are marginally increased, what will be the Dual Value?
1. (30 points) Consider the following linear program 220 > > 0 (a) (5 points) we write down the standard form of the above linear program e , peintre (1) (5 ) Wie weet witt artificial variables and write down the stand for o r thod write (c) 20 points) Plote the gallina p noted to fill all of the no variables at each iteration and an optimal solution and the optimale them down. If w e explain.) i mple...
Please do it ASAP. I will upvote immediately. Thanks! Problem 3 (Convex Optimization): Consider a linear programming: min c'e s.t.Ax > b (1) x > 0 Find the dual problem of the linear programming and argue that: (1) If the primal is unbounded, then the dual is infeasible; (2) If the primal is infeasible, then the dual is either infeasible or unbounded. 1 Note that strong duality holds for a linear programming if either the primal or the dual is...
Consider the linear program max z = 5x, + 3x2 + xz st. x + x₂ + x₂ <6 5xı + 3x2 +6xz S15 X, X2, X, 20 and an associated tableau Z X1 X2 X3 S1 S2 RHS 1 0 0 5 0 1 15 0 0 0.4 -0.2 1 -0.2 3 0 1 0.6 1.2 0 0.2 3 (a) What basic solution does this tableau represent? Is this solution optimal? Why or why not? (b) Does this tableau...
Question 3 (Chapter 6) 13+2+3+6 14 marks Fix p EN and consider the following set: : T1 (a) Prove that Cp is convex. (b) Prove that C, is a cone. (c) Compute Ci and C2. (d) Show that x = 0 is an extreme point of CP. Question 3 (Chapter 6) 13+2+3+6 14 marks Fix p EN and consider the following set: : T1 (a) Prove that Cp is convex. (b) Prove that C, is a cone. (c) Compute Ci...
For the linear program Max 3A+2B s.t. A+B>=4 3A+4B<=24 A>=2 A-B<=0 A, B>=0 a. Write the problem in standard form. b. Solve the problem. c. What are the values of the slack and surplus variables at the optimal solution?
Consider the following 2nd order nonhomogeneous linear equation x 00 + 4x 0 + 5x = cos 2t 1. Solve for the fundamental solutions of its associated homogeneous equation. 2. Find a particular solution of the nonhomogeneous equation. 3. Based on your answer to the previous two questions, write down the general solution of the nonhomogeneous equation. Problem II (15 points) Consider the following 2nd order nonhomogeneous linear equation x" + 40' + 5x = cos 2t 1. (6 points)...