Optimization of multiple variables without any toolbox

I have 5 variables, say l1,l2,g1,g2,g3. I want to optimize the gi (i=1,2,3) inside the two loops of l1 and l2 i.e. for each combination of l1 and l2, I want an optimal set of g_i for which the computed value should show a maximum and so on for the next values of l1 and l2. I don't have any tool box for this and I want to optimize without those.

fid = fopen( sprintf(.......))
for l1=0:0.01:1
    for l2:0:0.01:1
        % optimization of gi
            for g1=0:0.01:0.5
                x2=[];
                for g2=0:0.01:0.5
                       for g3=0:0.01:0.5
                               maxen=.....%some calculations
                        
                            x2=[x2;g2 g3 maxen];
                       end
                            [p, q]=max(x2(:,3));% max with indices 
                end
          
                     x3=x2(q,1);%g2
                     x4=x2(q,2);%g3
                     x5=x2(q,3);%maxen
            end
fprintf(fid,'%f %f %f %f %f\t %f %f \r\n',l1,l2,g1,x3,x4,x5,maxen);

    end
end
 fclose(fid);

I tried for a simple case for fixed l2 and fixed g1 but the result of maxen and x5 doesnt match while writing x2=[] outside g2 to get proper indices of g2 and g3 for that maximum value of maxen (writing in the .dat file). However, if write x2=[] inside g2, I lost the precise value of g2 as it shows end values only but with optimal g3 and equal maxen and x5 there. How can I optimize gi and find this optimal set there for maximum of maxen for each value of two outer loops (l1,l2)?.

if optimization is not possible with this way, do I need to use any algorithms for this?

Please suggest other alternates for the same.

Answer 1

[Jeff Miller]

If I understand the question, your basic approach seems OK but your code isn't structured quite right. I think this is closer to what you are after (but haven't checked it).

for l1=0:0.01:1
    for l2=0:0.01:1
        % optimization of gi
        x2=[];
        for g1=0:0.01:0.5
            for g2=0:0.01:0.5
                for g3=0:0.01:0.5
                    maxen = g1^2-g2^3+g3^4 + l1 - l2;  % example calculation
                    x2=[x2; g1 g2 g3 maxen];
                end
            end % end g2 loop
        end % end g1 loop
        % Now check the results for this l1/l2 combination
        [p, q]=max(x2(:,4));% max with indices
        g1 = x2(q,1); % g1
        g2 = x2(q,2);%g2
        g3 = x2(q,3);%g3
        maxen =x2(q,4);%maxen
        fprintf('%f %f %f %f %f\t %f %f \r\n',l1,l2,g1,g2, g3,maxen);
    end
end

[Anil Kumar]

Dear Jeff, Thanks a lot.

I am curious, if I write

                    x2=[x2; g1 g2 g3 maxen1 maxen2  maxen3];

and then if I can retreive the optimal indices (gi) for a combination of maximum and the minnimum of the calculated values there. For example, if I want an optimal set of gi for

(max(x2(:,4)) && max(x2(:,5)) && min(x2(:,6)))

or

(max(x2(:,4)) && min(x2(:,5)) && min(x2(:,6)))

or for simillar combinatios.

Though, I can save each max or min in independet q as follows [~,q]=max(x2(:,4)), [~,q1]=max(x2(:,5)) and [~,q2]=min(x2(:,6)), then I will have 3 sets of gi but not one set for a combination of max and min which I am looking for.

is it possible to get an optimal set of indices gi for such combinations of maximum and minimum using

[~,q]=(....)

?

Thanks.

[Jeff Miller]

Not sure whether I understand what you are asking, but maybe this will help:

I don't think you can necessarily find optimal gi indices for a combination of critera [e.g., max(maxen1) & max(maxen2) & min(maxen3)], because the indices that give you--say--max(maxen1) may be different than the indices that give you min(maxen3). With multiple criteria like this, I think you need to define a criterion function that combines all three scores into one, something like maxoverall = maxen1 + maxen2 - maxen3 (since you want large maxen1 and maxen2 but small maxen3). Then, you can compute maxoverall at each combination of indices and find the combination that gives you the best maxoverall.

It may be tough to define an overall combination function that gives the appropriate weights to the different components, but I don't see how you can proceed without such a function to define what is best overall.

[Anil Kumar]

Dear Jeff, I agree with you. Thanks.