Optimization Optimization under constraints in the form of equations (Lagrange factors). Quadratic programming with inequalities. Kuhn-Tucker conditions. Danzig-Wolf method. https://ek.biem.sumdu.edu.ua/

1. Topic 4. Optimization
Kashcha Mariya

2. Contents
• Introduction
• Linear programming
• Constrained optimization
• Quadratic programming
• Kuhn-Tucker condition
• Danzing-Wolfe methods

3. Introduction
• We are concerned with optimization – in
particular we are concerned with determined the
structure of “optimal portfolios” where optimal
is defined as having a minimum variance for a
given level of return
• We could use calculus to find optimum points;
use Lagrange multipliers to find optimum with
constraints.

4. Definitions
• The objective function sets out the task which
the optimization process is to achieve.
• For example, a typical objective function for a
portfolio of risky assets would be
• Z – is total risk, W – weightings of the assets in
the portfolio
• Z – is quadratic function.

5. Constraints
1. Some funds have to be invested in each asset
2. All the funds must be fully invested
3. The minimum risk must be achieved subject
to achieving a minimum level of return

6. A mathematical programming problem is a
problem in which a function of many variables
(the objective function) is to be optimized
subject to a number of constraints.
A linear programming problem is one in which
objective function and constrains is linear.
A quadratic programming problem is one in
which objective function is a quadratic
function of the variables, i.e. where some of
the variables have squared values. However,
the constrains are linear.

7. Linear programming
• When we combine assets in a portfolio the
returns of each asset combine in a linear form
and the risk of the portfolio as represented by
the portfolio β is also a linear combination, in
this case a weighted average of the βs of the
individual assets.

8. Task
Problem to constructing a portfolio with the
objective of achieving maximum expected
return, subject to the constraint that the β of
the portfolio is to be no higher 1,1.
1. Assume that have three assets to chose form: A,
B and C.
2. Their expected returns are 0,11; 0,15 and 0,08.
3. The βs: 1; 1,2; 0,9.
4. Assets are given by Wa, Wb and Wc.

9. • Objective function:
0,11Wa+0,15Wb+0,08Wc → maximum
• The constraints:
Wa+Wb+Wc ≤ 1,1
0≤Wa≤1
0≤Wb≤1
0≤Wc≤1
Wa+Wb+Wc=1

10. Graphical solution
Step 1. Eliminate Wc and thus produce a two-
variable problem.
• Wc=1-Wa-Wb
• 0≤Wa,Wb ≤1
• 0 ≤Wa+Wb ≤1
• 0,1Wa+0,3Wb ≤0,2
Z=0,03Wa+0,07Wb+0,08

11. • Step 2.
Wa
Wb
1
1
2
Wa+3Wb ≤2
Wa ≤1
Wb ≤1
Wa+Wb ≤1
O
Z=0,03Wa+0,07Wb+0,08
K
L
M
L: Wa+Wb=1;
Wa+3Wb=2.
L(0,5;0,5)

12. The simplex method
Wa Wb S1 S2 RHS
1 1 1 1 0 1
2 0,1 0,3 0 1 0,2
Objective
function
-0,03 -0,07 0 0 0
Z-0,03Wa-0,07Wb+0s1+0s2=0
• Wc=1-Wa-Wb; 0≤Wa,Wb ≤1
• Wa+Wb+s1=1
• 0,1Wa+0,3Wb +s2=0,2

13. Wa Wb S1 S2 RHS RHS/Wa
1 1 1 1 0 1 1 - min
2 0,1 0,3 0 1 0,2 2
Objective
function
-0,03 -0,07 0 0 0

14. Wa Wb S1 S2 RHS RHS/Wa
1 1 1 1 0 1 1 - min
2 0,1 0,3 0 1 0,2 2
Objective
function
-0,03 -0,07 0 0 0

15. Wa Wb S1 S2 RHS
1 1 1 1 0 1
2 0 0,2 -0,1 1 0,1
Objective
function
0 -0,04 -0,03 0 -0,03

16. Wa Wb S1 S2 RHS RHS/Wb
1 1 1 1 0 1 1
2 0 0,2 -0,1 1 0,1 0.5 - min
Objective
function
0 -0,04 -0,03 0 -0,03

17. Wa Wb S1 S2 RHS
1 1 1 1 0 1
2 0 1 -0,5 5 0,5
Objective
function
0 -0,04 -0,03 0 -0,03

18. • Wa+1,5s1-5s2=0,5
• Wb-0,5s1+5s2=0,5
Z=0,01s1+0,2s2+0,05+0,08 → max
• s1=s1=0
• Wa=0,5
• Wb=0,5
Wa Wb S1 S2 RHS
1 1 0 1,5 -5 0,5
2 0 1 -0,5 5 0,5
Objective
function
0 0 0,01 0,2 0,05

19. The portfolio optimization
• Problem is to determine what proportion of the
portfolio should be allocated to each investment so
that the amount of expected return and the level of
risk optimally meet the investor’s objectives.
• For example, the objective function may be to
minimize risk, but subject to a minimum level to
return. There may also be constraints on the
minimum and maximum proportions which may be
invested in each asset.

20. Portfolio problem: a three-asset
portfolio
• The expected return:
E(r) = Wara+Wbrb+Wcrc
• The portfolio variance:
Z= Waσ2
a+Wbσ2
b+Wcσ2
c+2WaWbcov(a,b)+
+2WcWacov(a,c)+2WbWccov(b,c) → minimize
Subject to:
Wa+Wb+Wc≤1
WaE(ra)+WbE(rb)+WcE(rc)≥R, where R – is the minimum
acceptable level of return

21. The application of quadratic programming a
three asset portfolio selection problem – finding
the optimal(minimum variance) portfolio
We have three assets A, B, C with expected returns of
0,11; 0,15; 0,08. The covariance matrix is:
Required return of 11%

22. Optimization problem is to:
Z= 0,00015Wa+0,00025Wb+0,0001Wc+0,0001WaWb- -
0,00014WcWa-0,00006WbWc → minimize
Subject to:
Wa+Wb+Wc=1
0,11Wa+0,15Wb+0,08Wc=0,11

23. Optimization under equality
constraints: using Lagrange multipliers
• We construct the Lagrangian:

26. Kuhn – Taker conditions
• Step 1. To manipulate the inequalities so that the
problem is presented in standard form:

27. • Step 2. Building Lagrangian + “Kuhn – Taker
conditions”

29. Dantzig - Wolfe
Our task is to find feasible point as against an optimal
point – this is first step in the simplex algorithm.
Creating solution in which real variables equal 0, and
in which artificial equal right-hand sides.
So, is to take initial part simplex algorithm and add in
the complementary conditions.

30. 1) Wc=0,25;Wb=0,405;Wa=0,436.
Wa+Wb+Wc=0,966 – not all money is used
2) If we want on investing all of our cash, changing
Wa+Wb+Wc=1

31. • Solving consumer choice problem using Kuhn-Tucker
conditions (part 1).
• Solving consumer choice problem using Kuhn-Tucker
conditions (part 2).
• Solving minimization costs problem using Kuhn-
Tucker conditions (part 1).
• Solving minimization costs problem using Kuhn-
Tucker conditions (part 2).