Optimality. Consider the following optimization problemminimize 12 xT P x + qT x + rsubject to −1 ≤ xi ≤ 1, i = 1, 2, 3whereP = 26 24 −424 34 12−4 12 24 , q =−44−2926 , r = 1.(a) Prove that x? = (1, 0.5, −1) is optimal.(b) Are any of the constraints inactive at x?? If so, which one(s)?

Consider the following optimization problem.Minimizex∈IR2 3x1 − (x2)3Subject to (x1)2 + (x2)2 ≤ 2,−x1 + x2 ≤ −1.(a) Show that the MFCQ holds at every feasible point.[5 marks](b) Write down the KKT conditions.

Using complementary slackness to determine the optimality or otherwise of thecandidate solution (x1, x2, x3, x4) = (0, 14, 0, 5) for the following LP:max 4x1 + x2 + 5x3 + 3x4s.t.x1 − x2 − x3 + 3x4 ≤ 1,5x1 + x2 + 3x3 + 8x4 ≤ 55−x1 + 2x2 + 3x3 − 5x4 ≤ 3x1, x2, x3, x4 ≥ 0.1

Consider the following optimization problem:Minimizex∈IR3 2x21 + x22 + 2x23Subject to x1 + 2x2 + x3 ≤ −3.(a) (10 points) For each c > 0, defineqc(x) := 2x21 + x22 + 2x23 + c2 (x1 + 2x2 + x3 + 3)2+Argue that qc is convex and find the global minimizer of qc.(b) (10 points) For each µ > 0, defineℓµ(x) := 2x21 + x22 + 2x23 − µ ln (−3 − x1 − 2x2 − x3) .Argue that ℓµ is convex and find the global minimizer of ℓµ. You may use without proof thefact that the function t 7 → − ln(−3 − t) is convex (as an extended real-valued function).

(b) The optimal solution to the original problem, in the variables x1,x2𝑥1,𝑥2 and x3𝑥3, is given by:

Suppose we have the following LP model.Maximize Z = 10X + 5YSubject to:3X + 3Y ≤ 6 (Constraint 1)2X + 4Y ≥ 6 (Constraint 2)X, Y ≥ 0It came to be that the optimal solution is X = 1 and Y = 1. Which of the following options is a correct statement?Group of answer choicesConstraint 1 is a binding constraint, and constraint 2 is a non-binding constraint.Constraint 1 is a non-binding constraint, and constraint 2 is a binding constraint.Both constraints 1 and 2 are binding constraints.Both constraints 1 and 2 are non-binding constraints.