1. Problem on unconstrained optimization
For each of the following optimization problems either verify that the given x is a
stationary point or find a direction d that locally improves at x.
(a) max 10x2 + 12 ln(x2), x = (1, 2)
(b) max x1x2 - 10x1 + 4x2, x = (-4, 10)
2. Local optimality
For each of the following functions f , classify the specified x as a definitely a local
maximum, possibly local maximum, definitely local minimum, possibly local minimum
or none of the above.
(a) f (x) = -x21- 6x1x2 - 9x2, x = (-3, 1)
(b) f (x) = 12x2 - x2 + 3x1x2 - 3x2, x = (12, 8)
(c) f (x) = 6x1 + ln(x1) + x2, x = (1, 2)
(d) f (x) = 4x21+ 3/x2 - 8x1 + 3x2, x = (1, 1)
3.Problem on recognizing convex functions/sets
Determine whether each of the following is a convex program
(a) min{x1 + x2 : x1x2 ≤ 9,|x1| ≤ 5, |x2| ≤5}
(b) max{62x1 + 123x2 : ln(x1) + ln(x2) = 8, 7x1 + 2x2 ≤ 9 00,x1 ³≥ 0, x2 ≥ 0}

set for quadratic programming
Solve the following QP starting from the feasible solution x(0) = (2, 1)T .

T
-3 1 -6
max xT x + x,
1 -1 20

1 -1 1
0 1 5
s.t.
-1 0 x ≤ 0
0 -1 0

6. Simple problems on efficient frontiers

1






(a) Efficient portfolios with 2 assets
Consider a market with N = 2 assets with the parameters


µ1
µ = ,
µ2


s2 rs1s2
∑ = ,
rs1s2 s2

with µ1 = µ2. In this market a portfolio of the form (x, 1 -x) where x denotes the fraction invested in asset 1. Substituting this into the expected return condition

we get

µ1x + µ2(1 - x) = a Þ x = a - µ2
µ1 - µ2

Use this expression to plot out the efficient frontier for the following three cases: r = -1, r = 0, r = +1. In each case argue that the shape that you get is reasonable.
(b) Generating the portfolio frontier
Suppose a1 = a2. Let fai , i = 1, 2 denote the frontier portfolio corresponding to the ai, i = 1, 2. Show that the entire e cient frontier is given by the set {f = gf a1 + (1 - g )fa2 : g Î R}. Compute the g that corresponds to the expected return a.
7. Problems on constructing the efficient frontier
In this problem we will construct the efficient frontier for the market data given on the worksheet Problem 5 on the EXCEL workbook hw3.xls
(a) Plot the efficient frontier corresponding to the classical Markowitz portfolio prob
lem, i.e. portfolios f are unrestricted.
(b) Plot the efficient frontier when the following margin constraint holds:

n $n
1.5 ∑ xi ∑ x+i
i=1 i=1
This constraints ensures that the long positions must at least be 1.5 times the short positions.