For a primal-dual pair of linear programs (where the primal is a minimisation problem and the dual is a maximisation problem), you are told that the optimal value of the dual linear program is 22. Which of the following statements is true?Question 1Answera.The primal linear program is infeasible.b.Any feasible point of the dual linear program has value at least 22.c.The primal linear program is unbounded.d.Any feasible point of the primal linear program has objective value of at least 22.

onsider the following primal problem:max z = −40x1 − 30x2 − 20x3s.t.10x1 + 3x2 + 5x3 ≤ 104x1 + 10x2 + 4x3 ≤ 15x1, x2, x3 ≥ 0 .(i) State its dual problem, and find the optimal solution of the latter using the SimplexMethod

state the weak duality theorem (1 mark)ii) Write the dual of the given primal problem.Min W=10y1 − 15y2Subject toy1 − 4y2 ≥ 3y1 + y2 ≥ 53y1 − 2y2 = 7y1, y2 ≥ 0 (2 marks)iii) Given the linear programming problemMax z = 15y1 + 25y2Subject toy1 + y2 ≤ 450y2 ≤ 3504y1 + 5y2 ≤ 2000y1 ≤ 350y1, y2 ≥ 0Find the range of optimality of y1, y2. (3 marks)Will the optimal basis change when the objective function coefficient of y1, y2change to 17 and 24 respectively? (2 marks)

Write the dual of the following linear programming problem:Maximize 8x1 + 3x2 − 2x3subject to x1 − 6x2 + x3 > 25x1 + 7x2 − 2x3 = −4x1 ≤ 0, x2 ≥ 0, x3 unrestricted.

A primal linear program has the form:minx∈Rnc⊺xsubject toAx≥bmin𝑥∈𝑅𝑛𝑐⊺𝑥subject to𝐴𝑥≥𝑏where A𝐴 is an m×n𝑚×𝑛 matrix. The dual linear program has the form:maxy∈Rmb⊺ysubject to{Dy=cy≥0max𝑦∈𝑅𝑚𝑏⊺𝑦subject to{𝐷𝑦=𝑐𝑦≥0Which of the following is a correct expression for the matrix D𝐷?Question 3Answera.D=I𝐷=𝐼b.D=A⊺𝐷=𝐴⊺c.D=A𝐷=𝐴d.D=A+bc⊺

Consider the quadratic program of the form min ⁡ 𝑥 1 , 𝑥 2 ( 𝑥 1 − 1 ) 2 + ( 𝑥 2 − 1 ) 2 subject to 𝑥 1 ≥ 0 , 𝑥 2 ≥ 0 , 𝑥 1 + 𝑥 2 ≤ 1 min x 1 ​ ,x 2 ​ ​ (x 1 ​ −1) 2 +(x 2 ​ −1) 2 subject tox 1 ​ ≥0,x 2 ​ ≥0,x 1 ​ +x 2 ​ ≤1 The optimal solution of the quadratic program is ( 1 / 2 , 1 / 2 ) (1/2,1/2). If 𝑦 ∗ y ∗ ​ is the optimal dual solution, how many entries of the vector 𝑦 ∗ y ∗ ​ are non-zero? (Hint: draw a picture of the level sets of the objective function and of the feasible region.) 0 1 2 3 4