MATH 20070
OPTIMIZATION IN FINANCE
Problem sheet week 9
1 Let f : R2 R be dened by f (x, y) = 2x + y 2 and the constraints
x2 + y 2 10
and
x 0.
(i) Prove that the feasible set is bounded.
(ii) Using the Kuhn-Tucker method, nd the local extrema of f subjec

Optimization in Finance Problem set 6
6.1 Use the simplex method to solve the following LP problem:
Minimize z = x1 7x2 9x3 + x4 subject to x1 + x2 10,
2x1 3x2 + x4 5, x2 x3 15 and x1 , x2 , x3 , x4 0.
6.2 For the following problem, formulate the correspo

Optimization in Finance Problem set 4
Use the simplex method to solve the following LP problems:
4.1 Maximise z = 4x1 + 2x2 subject to x1 + x2 50, 6x1 240, x1 , x2 0.
4.2 Maximise z = 10x1 + 12x2 subject to x1 + x2 150, 3x1 + 6x2 300,
4x1 + 2x2 160, x1 ,

Optimization in Finance Problem set 2
2.1 Find the maximum and minimum of f (x, y) = 2x2 + 4y 2 + 1 subject to
the constraint g(x, y) = 2x2 + y 2 = 6.
2.2 Find the maximum of f (x1 , x2 , x3 ) = 5x1 x2 x3 subject to the constraint
x1 + 2x2 + 3x3 = 24.
2.3

Optimization in Finance Problem set 1
1.1 Find and classify all critical points of the following functions:
x2 x 2
x3
(b) f (x, y) = x2 + y 2 3x + 12
(a) f (x) =
(c) f (x, y) = x2 y 2 + 4x + 8y
(d) f (x, y) = x3 + y 3 3xy
(e) f (x1 , x2 , x3 ) = 2x2 + x1

MATH 20070
OPTIMIZATION IN FINANCE
Quasi-Concave and Quasi-Convex Functions
Prove that the following functions are quasi-concave:
(i)
(ii)
(iii)
f (x, y) = yexy , y > 0;
f (x, y) = xy 4 , x > 0, y > 0;
f (x, y) = xe2xy , x > 0.
(i) The bordered Hessian ma

Convex and Concave Functions
Convex Sets
Convex and
Concave
Functions
Denition
Let
x
and
joining
x
y
be two points in
and
y
Rn .
We dene the
line segment
as the set
cfw_(1 )x + y | 0 1.
This consists of all points on the straight line from
x
to
y
Convex S

Nonlinear Programming
Nonlinear
Programming
Nonlinear Programming
When solving problems with Lagrange Multipliers we considered
the problem of nding the maximum or minimum of a function
subject to equality constraints.
In linear programming we considered

The Dual Problem
The Dual
Problem
The Dual Problem
With every Linear Programming we associate its Dual
Problem.
It is formed from information in the Original Problem and its
solution can be obtained from the nal tableau of the Original
Problem.
This means

The Simplex Method with General Constraints
The Simplex
Method with
General
Constraints
Some observations
Let us begin with some observations from the previous chapter.
1 The entry in row (0) in the
ci
column gives us the current
value of the objective fu

The Simplex Method
The Simplex
Method
Linear Programming
In order to solve a linear programming problem using the
Simplex Method we require that the problem is stated in
Standard Form.
For this
I The right side of a constraint cannot be negative;
II All c

Lagrange Multipliers
Lagrange
Multipliers
Lagrange Multipliers
Let us recall the method of Lagrange multiplies for functions of
two variables and one constraint
Problem
Find the dimensions of the rectangle with maximum area, given
that the perimeter is 10

Extrema of Functions of Several Variables
Extrema of
Functions of
Several
Variables
Extrema of Single Variable Functions
We let R denote the set of real numbers.
For a, b R with a b, we let
[a, b] := cfw_x R : a x b,
closed interval
(a, b) := cfw_x R : a