Consider the following linear program:minx1,x2,x3,x4c1x1+c2x2+c3x3+2x4subject to⎧⎩⎨x1+x2+x3+x4≥103x1+x2+4x3+2x4≥12x1≥0,x2≥0,x3≥0,x4≥0min𝑥1,𝑥2,𝑥3,𝑥4𝑐1𝑥1+𝑐2𝑥2+𝑐3𝑥3+2𝑥4subject to{𝑥1+𝑥2+𝑥3+𝑥4≥103𝑥1+𝑥2+4𝑥3+2𝑥4≥12𝑥1≥0,𝑥2≥0,𝑥3≥0,𝑥4≥0 The parameters c1,c2,c3𝑐1,𝑐2,𝑐3 are real numbers.(a) How many variables does the dual problem have? Answer 1 Question 2Suppose that the optimal solution to the primal is x1=0𝑥1=0, x2=0𝑥2=0, x3=0𝑥3=0, x4=10𝑥4=10.(b) What is the optimal value of the objective function of the dual linear program? Answer 2 Question 2.(c) How many entries of the optimal dual variable are equal to zero?

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.

Consider the quadratic program minx1,x2f(x1,x2)subject toCx≥dmin𝑥1,𝑥2𝑓(𝑥1,𝑥2)subject to𝐶𝑥≥𝑑where C𝐶 is a 5×25×2 matrix and d𝑑 is a length 5 vector. (c) How many variables does the dual quadratic program have? Answer 4 Question 1(d) What is the value of the dual quadratic program at the point y=0𝑦=0?

Consider the following linear program:minx1,x2,x3,x4c1x1+c2x2+c3x3+2x4subject to⎧⎩⎨x1+x2+x3+x4≥103x1+x2+4x3+2x4≥12x1≥0,x2≥0,x3≥0,x4≥0min𝑥1,𝑥2,𝑥3,𝑥4𝑐1𝑥1+𝑐2𝑥2+𝑐3𝑥3+2𝑥4subject to{𝑥1+𝑥2+𝑥3+𝑥4≥103𝑥1+𝑥2+4𝑥3+2𝑥4≥12𝑥1≥0,𝑥2≥0,𝑥3≥0,𝑥4≥0 The parameters c1,c2,c3𝑐1,𝑐2,𝑐3 are real numbers.(a) How many variables does the dual problem have? Answer 1 Question 2Suppose that the optimal solution to the primal is x1=0𝑥1=0, x2=0𝑥2=0, x3=0𝑥3=0, x4=10𝑥4=10.(b) What is the optimal value of the objective function of the dual linear program? Answer 2 Question 2.(c) How many entries of the optimal dual variable are equal to zero?

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

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.