Minimalization problem LinearConstraint and conjugate gradient optimizer

Problem, input data and equations are described in details in attachment. This matrix is called Ms in the below mentioned equation.

The equation is the function F(ω). Omega (ω) are the seven wages which I’m looking for by minimize values of the second equation. The condition is that ω1 + ω2 + ω3 + ω4 + ω5 + ω6 + ω7 = 1.

When using Scipy.stats, the LinearConstraint and Conjugate gradient optimizer were used.

The obtained results were: 0.20141944, 0.1590185 , 0.13852083, 0.08702209, 0.13283426, 0.14539815, 0.14247747. Sum of these wages equals 1.

I very appreciate if someone help me out to write code or use Optimization tool to obtain these results.The input matrix Ms is in attached file.

Best Regards,

Tomi

[Torsten]

What are you trying to minimize ? What are your constraints ? I don't get it from your decription.

[Tomi]

I'm sorry for confusion.

Please see the python code used with Scipy - this code produce the results which I want to write in MatLab script and receive the same results. The second equation is used to minimalize omega wages.

I attached python code.

Best,

Tomi

Answer 1

[Torsten]

According to the Python code, F is maximized, not minimized. Change in the below code if appropriate.

M = [0.170543	0.327434	0.174194	0	    0.421053	0.307167	0.297659
0.155039	0.504425	0.664516	0.530612	0.102493	0.05802	    0.053512
0.255814	0.318584	0.212903	0	        0.445983	0.337884	0.311037
0.224806	0.548673	0.664516	0.591837	0.141274	0.068259	0.053512
0.383721	0.389381	0.303226	0	        0.573407	0.433447	0.41806
0.360465	0.716814	0.883871	0.755102	0.227147	0.078498	0.073579
0.449612	0.566372	0.36129	    0	        0.775623	0.573379	0.498328
0.484496	0.920354	0.948387	1	        0.265928	0.109215	0.107023
0.375969	0.539823	0.303226	0	        0.648199	0.481229	0.438127
0.399225	0.769912	0.716129	0.857143	0.224377	0.102389	0.100334
0.356589	0.39823	    0.264516	0	        0.717452	0.498294	0.444816
0.391473	0.761062	0.703226	0.795918	0.218837	0.098976	0.09699
0.290698	0.327434	0.251613	0	        0.770083	0.518771	0.464883
0.395349	0.761062	0.767742	0.795918	0.207756	0.085324	0.09699
0.352713	0.380531	0.277419	0	        0.797784	0.501706	0.501672
0.426357	0.778761	0.870968	0.877551	0.265928	0.112628	0.100334
0.403101	0.336283	0.309677	0	        0.761773	0.467577	0.491639
0.468992	0.743363	0.877419	0.897959	0.224377	0.119454	0.090301
0.387597	0.345133	0.341935	0	        0.775623	0.518771	0.551839
0.496124	0.787611	0.877419	0.857143	0.263158	0.122867	0.113712
0.333333	0.380531	0.341935	0	        0.759003	0.566553	0.585284
0.624031	0.80531	    0.780645	0.795918	0.293629	0.12628	    0.130435
0.534884	0.40708	    0.419355	0	        0.894737	0.641638	0.628763
0.786822	0.938053	1	        0.632653	0.379501	0.197952	0.120401
0.453488	0.380531	0.419355	0	        0.842105	0.607509	0.628763
0.554264	0.876106	0.741935	0.877551	0.254848	0.334471	0.130435
0.639535	0.646018	0.593548	0	        1	        0.8157	    0.73913
0.689922	1	        0.735484	0.693878	0.351801	0.337884	0.137124
1	        0.867257	0.354839	0	        0.617729	1	        1
0.546512	0.876106	0.703226	0.877551	0.254848	0.334471	0.130435];
w0 = [1/7;1/7;1/7;1/7;1/7;1/7;1/7];
Aeq = ones(1,7);
beq = 1.0;
lb = zeros(7,1);
ub = ones(7,1);
options = optimset('TolFun',1e-10,'TolX',1e-10);
format long
[w,fval] = fmincon(@(w)fun(w,M),w0,[],[],Aeq,beq,lb,ub,[],options)

Local minimum possible. Constraints satisfied.

fmincon stopped because the size of the current step is less than
the value of the step size tolerance and constraints are 
satisfied to within the value of the constraint tolerance.

w = 7×1

0.200081504852845 0.157959395465906 0.137599668689948 0.086444832698773 0.131950619523270 0.144443169868022 0.141520808901237

fval = 
  -3.039398091210332

function value = fun(w,M)
  cM = zeros(7,1);
  Mw = M*w;
  Mwm = mean(Mw); 
  Mim = mean(M,1);
  for i = 1:7
    Mi = M(:,i);
    cM(i) = sum((Mi-Mim(i)).*(Mw-Mwm))/sqrt(sum((Mi-Mim(i)).^2)*sum((Mw-Mwm).^2));
  end
  value = -sum(cM);
end

[Torsten]

@Tomi comment moved here:

Dear Torsten,

I'm trying to understand code - need some help.

I very appreciate if you could tell me how the cM(i) equation was created?

Best,

Tomi

[Torsten]

It's the correlation coefficient used in the Python code:

https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.pearsonr.html

Answer 2

[Tomi]

Thank you for your time and help.

I very aprpeciate that.

Best,

Tomi

Answer 3

[Tomi]

Dear Torsten,

I'm trying to understand code - need some help.

I very appreciate if you could tell me how the cM(i) equation was created?

Best,

Tomi

[Torsten]

Don't use answers when you want to make a comment.

See my answer above.

[Tomi]

Thank you for help and time.

Best Regards,

Tomi

[Tomi]

Is there any chance to rewrite code without "function" use?

Tomi

[Torsten]

M = [0.170543	0.327434	0.174194	0	    0.421053	0.307167	0.297659
    0.155039	0.504425	0.664516	0.530612	0.102493	0.05802	    0.053512
    0.255814	0.318584	0.212903	0	        0.445983	0.337884	0.311037
    0.224806	0.548673	0.664516	0.591837	0.141274	0.068259	0.053512
    0.383721	0.389381	0.303226	0	        0.573407	0.433447	0.41806
    0.360465	0.716814	0.883871	0.755102	0.227147	0.078498	0.073579
    0.449612	0.566372	0.36129	    0	        0.775623	0.573379	0.498328
    0.484496	0.920354	0.948387	1	        0.265928	0.109215	0.107023
    0.375969	0.539823	0.303226	0	        0.648199	0.481229	0.438127
    0.399225	0.769912	0.716129	0.857143	0.224377	0.102389	0.100334
    0.356589	0.39823	    0.264516	0	        0.717452	0.498294	0.444816
    0.391473	0.761062	0.703226	0.795918	0.218837	0.098976	0.09699
    0.290698	0.327434	0.251613	0	        0.770083	0.518771	0.464883
    0.395349	0.761062	0.767742	0.795918	0.207756	0.085324	0.09699
    0.352713	0.380531	0.277419	0	        0.797784	0.501706	0.501672
    0.426357	0.778761	0.870968	0.877551	0.265928	0.112628	0.100334
    0.403101	0.336283	0.309677	0	        0.761773	0.467577	0.491639
    0.468992	0.743363	0.877419	0.897959	0.224377	0.119454	0.090301
    0.387597	0.345133	0.341935	0	        0.775623	0.518771	0.551839
    0.496124	0.787611	0.877419	0.857143	0.263158	0.122867	0.113712
    0.333333	0.380531	0.341935	0	        0.759003	0.566553	0.585284
    0.624031	0.80531	    0.780645	0.795918	0.293629	0.12628	    0.130435
    0.534884	0.40708	    0.419355	0	        0.894737	0.641638	0.628763
    0.786822	0.938053	1	        0.632653	0.379501	0.197952	0.120401
    0.453488	0.380531	0.419355	0	        0.842105	0.607509	0.628763
    0.554264	0.876106	0.741935	0.877551	0.254848	0.334471	0.130435
    0.639535	0.646018	0.593548	0	        1	        0.8157	    0.73913
    0.689922	1	        0.735484	0.693878	0.351801	0.337884	0.137124
    1	        0.867257	0.354839	0	        0.617729	1	        1
    0.546512	0.876106	0.703226	0.877551	0.254848	0.334471	0.130435];
w0 = [1/7;1/7;1/7;1/7;1/7;1/7;1/7];
Aeq = ones(1,7);
beq = 1.0;
lb = zeros(7,1);
ub = ones(7,1);
options = optimset('TolFun',1e-10,'TolX',1e-10);
Mim = mean(M,1);
fun = @(w) -sum(arrayfun(@(i)sum((M(:,i)-Mim(i)).*(M*w-mean(M*w)))/sqrt(sum((M(:,i)-Mim(i)).^2)*sum((M*w-mean(M*w)).^2)),1:7));

fun = function_handle with value:
    @(w)-sum(arrayfun(@(i)sum((M(:,i)-Mim(i)).*(M*w-mean(M*w)))/sqrt(sum((M(:,i)-Mim(i)).^2)*sum((M*w-mean(M*w)).^2)),1:7))

format long
[w,fval] = fmincon(fun,w0,[],[],Aeq,beq,lb,ub,[],options)

Local minimum possible. Constraints satisfied.

fmincon stopped because the size of the current step is less than
the value of the step size tolerance and constraints are 
satisfied to within the value of the constraint tolerance.

w = 7×1

0.200081504852845 0.157959395465906 0.137599668689948 0.086444832698773 0.131950619523270 0.144443169868022 0.141520808901237

fval = 
  -3.039398091210332

[Tomi]

Thank you very much for your help.

Best,

Tomi