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MIE1621 – Non-Linear Programming

创建日期：2024-05-24 14:53:31

所属分类：数学mathematics

标签： MIE1621

Non-Linear Programming

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MIE1621 – Non-Linear Programming

Part 1: Problem formulation
The problem at hand is the optimization of the risk adjusted return of a financial portfolio of n
assets. The risk adjusted return is governed by the following equations:
Where:
is the return of asset i,
is the covariance of asset I and j,
is the fraction of wealth invested in asset i,
is the risk tolerance of the investor, set to a value between 3.5 and 4.5.
The goal of this optimization will be to maximize the expected return, given some risk
tolerance. The equation is therefore multiplied by -1 in order to apply conventional
minimization techniques.
This equation can be re-written in matrix form for scalability:
=
2
− s.t. = 1 where = ()
In order to reconfigure this equation as an unconstrained optimization, the Lagrangian of this
problem is constructed, ensuring that the optimal solutions meet the constraint:
(, ) =
2
− + ( − 1)
In order to satisfy the FONC, the following partial derivatives of L are found:
= − + = 0
= − 1 = 0
In order to solve these simultaneously, the following equation is constructed, where is an nxn
matrix, and , , are an n-dimensional column vectors:
∇ = (
0
) (
) − (
1
)
Lastly, the hessian can be shown as:
H = (
0
)
The gradient for a simple 2-stock portfolio is shown as below, however, this generalizes for n
stocks.
∇ = (
11 12 1
21 22 1
1 1 0
) (
1
1
) − (
1
2
1
)

Part 2: Naïve Newton, SDM, BFGS Solution Methods
This portion of the report details attempts to solve the above problem using Newton’s Method,
Steepest Descent Method, and BFGS Quasi-Newton Method, with a step length of 1. The
functions to execute this code are shown in Appendix A. Note that these functions cannot be
run independently of other accompanying functions. The full python notebook is submitted
along with this report, or available at this link: https://github.com/jonsmith359/MIE1621-
Project.
These methods all follow the same basic form of iterations:
+1 = −
where is a step length and is a descent direction. For this portion of the project, = 1.
The way that these three methods differ is in their calculation of descent direction.
Newton: = H−1∇
SDM: = ∇
BFGS: = H−1∇
In the case of BFGS, an approximation of the inverse hessian is used, based on the estimation
formula:
−1+1= (I-
)−1+1 (I-
) +
These three methods were performed on the sample dataset given, with starting portfolio
weights of [1,1,1] and risk factor 3.5. Iterations are performed until the current solution vector
is sufficiently close to the previous solution vector. Each element must be within some user
defined tolerance of the previous. The results are summarized below.
Return var_1 var_2 var_3
0 0.1073 0.02778 0.00387 0.00021
1 0.0731 0.00387 0.01112 -0.0002
2 0.0621 0.00021 -0.0002 0.00115
Table 1: Sample Data Set
Method Convergence # Iterations Time
Newton Yes 1 4.05 µs
SDM No 100 5.01 µs
BFGS Yes 17 5.03 µs
Table 2: Part 2 Results
From these results, we can see that Newton’s method performs the best, reaching the optimal
portolio: x = [ 0.45526555 0.1745838 0.37015065] , Return: 7.25% within a single iteration.
This is because of the use of an exact gradient and hessian at initial point x. The steepest
descent method diverges for this, and other starting points. This occurs because the Lagrangian
function is non-convex. Steepest descent is unable to guarantee a solution under these
conditions. It should be noted however, that the iterations come close to the optimum, but
then diverge from it. The BFGS method converges to the solution after 17 iterations. The added
iterations from Newton are because the initial hessian approximation is based on the identity
matrix. Overall, these methods perform as expected.

Part 3 Newton, SDM, BFGS Solution Methods with Backtracking
Backtracking is implemented in order to find an optimal step length in the equation
+1 = −
The backtrack lines search works by assuming a step length of 1, and decreasing it by half until
the following condition is satisfied:
(+1) ≤ () + ∗ ∗ ∗ ∇()
This is intended to ensure that the step length is not too large so as to miss the critical point,
but still large enough to make meaningful progress towards the point. An additional constraint
is implemented in the code to ensure that the step length does not approach zero on each
iteration
The methods in part 2 were repeated with this line search to achieve the following results:
Method Convergence # Iterations Time
Newton Yes 1 3.81 µs
SDM No 100 7.15 µs
BFGS Yes 26 4.05 µs
Table 3: Part 3 Results
As was the case in part 2, Newton converged in one step due to the exact gradient and hessians
used in the computation. SDM still diverges under these conditions due to the non-convexity of
the function. As was the case with part 2, The BFGS method still converges to the correct
answer, but this happens in more iterations when backtracking is used. This occurs because a
smaller step length is used, leading to a more conservative step between iterations.
MIE1621 Course Project 2017 Jonathan Smith
5
Part 4: Scaling up
The following dataset was created using daily data for the stocks: AAPL, GOOG, SAF.PA, AIR.DE,
LMT, BA, UTC1.DE, BBD-B.TO, GE, RR.L in the year leading up to Dec 5 2017. The return is the
average daily return throughout the year, while the covariance’s are calculated based on the
daily changes of each stock with respect to one another.
Return var_1 var_2 var_3 var_4 var_5 var_6 var_7 var_8 var_9 var_10
0 0.1465 0.0118 0.0082 0.0098 0.0033 0.0073 0.0174 -0.0014 0.0071 -0.0132 0.0110
1 0.9031 0.0082 0.0071 0.0082 0.0040 0.0055 0.0130 -0.0006 0.0059 -0.0101 0.0087
2 0.0769 0.0098 0.0082 0.0110 0.0038 0.0074 0.0177 -0.0014 0.0064 -0.0128 0.0111
3 0.0940 0.0033 0.0040 0.0038 0.0341 0.0018 0.0045 0.0003 0.0043 -0.0054 0.0031
4 0.2831 0.0073 0.0055 0.0074 0.0018 0.0058 0.0139 -0.0017 0.0051 -0.0097 0.0074
5 0.2056 0.0174 0.0130 0.0177 0.0045 0.0139 0.0346 -0.0046 0.0137 -0.0243 0.0174
6 0.1037 -0.0014 -0.0006 -0.0014 0.0003 -0.0017 -0.0046 0.0016 -0.0015 0.0028 -0.0009
7 0.0024 0.0071 0.0059 0.0064 0.0043 0.0051 0.0137 -0.0015 0.0153 -0.0122 0.0057
8 0.0270 -0.0132 -0.0101 -0.0128 -0.0054 -0.0097 -0.0243 0.0028 -0.0122 0.0192 -0.0120
9 0.8350 0.0110 0.0087 0.0111 0.0031 0.0074 0.0174 -0.0009 0.0057 -0.0120 0.0135
Table 4: Scaled-Up Data
When these results are fed into the same models described above, with starting weights of 1’s
the following results are obtained:
Method Convergence # Iterations Time
Newton Yes 1 4.05 µs
SDM No 100 11.00 µs
BFGS Yes 70 4.05 µs
Table 5: Part 4 Results without backtracking
Method Convergence # Iterations Time
Newton Yes 558 8.70 µs
SDM No 100 5.52 µs
BFGS Yes 497 5.25 µs
Table 6: Part 4 Results with backtracking
As we see from these results, the newton and quasi-newton methods still converge to the
optimal portfolio:
x = [-185.74551335, 494.79125851, -372.1543059, -6.20940063, 200.18461618,
-40.70871509, -173.04802813, -29.32457964, 6.62083792, 106.59275841]
In this case, some weights are negative because short selling is not restricted in this
optimization model.
Iterations typically took longer with this data, since there are more degrees of freedom to
optimize, and extra variables that have to meet the stopping criteria. Interestingly, as the
dataset increases in size, the BFGS model converges faster than the regular newton method
when backtracking is implemented.

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