Consider the optimisation problemminx∈R3x1−2x2+ex3subject toAx=0min𝑥∈𝑅3𝑥1−2𝑥2+𝑒𝑥3subject to𝐴𝑥=0where A𝐴 is a 2×32×3 matrix with rank two. Moreover, the sum of the entries in each row of A𝐴 is zero.(a) This problem can be rewritten as an unconstrained convex optimisation problem in how many variables? Answer 1 Question 1(b) The optimal solution to the original problem, in the variables x1,x2𝑥1,𝑥2 and x3𝑥3, is given by:

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 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.

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: