Consider the optimisation problem minx∈R3x1−2x2+ex3subject toAx=0 where A is a 2×3 matrix with rank two. Moreover, the sum of the entries in each row of A is zero. (a) This problem can be rewritten as an unconstrained convex optimisation problem in how many variables? Answer 1 Question 1 0 (b) The optimal solution to the original problem, in the variables x1,x2 and x3 , is given by:

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?

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 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).

Consider the following optimization problem.Minimizex∈IR2 x1 + x2Subject to 3x21 + x22 ≤ 3,x2 + 2x21 ≤ 0.(a) (5 points) Show that the MFCQ holds at every feasible point.(b) (30 points) Write down the KKT conditions and find all the stationary points

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.