Simultaneous minimization of 3 equations to estimate 4 parameters

I want to find the values of 4 parameters for which three equations reach their minimum value simultaneous.

Aqtually what I want to do is : min F(q) for q where q=[a aplha c k] and F=[F1(q) F2(q) F3(q)].

Then only way that I found online to do it is with gamultiobj. The problem is that every time that I run the code with the same initial conditions without changing anything I get different 'optimal values ' for q.

Is there something that I need to define extra?

Also is there another solver that I could use for the simultaneous minimization of three functions.

I attach my code.

Thank you

Ps=0.49;

% a=q(1);
% alpha=q(2);
% c=q(3);
% k=q(4);

%parameters from the differential susceptibilities on the P-E graph
Ec=1.2e6; 
xc=7.3e-7;
Pr=0.38;
xr=4.4e-8;
Em=6.2e6;
Pm=0.46;
xmp=2.8e-8;
xmm=5e-9;

%call the functions that are included in the functions
% dPPr=dPanPr(Ps,a,alpha,xr,Pr);
% dPEc=dPanEc(Ps,a,alpha,xc,Ec);
% Pan_Em= PanEm(Ps,a,alpha,Em,Pm);
% Pan_Pr= PanPr(Ps,a,alpha,Pr);
% Pan_Ec= PanEc(Ps,a,Ec);



F1=@(q) xr-(1-q(3))*((PanPr(Ps,q(1),q(2),Pr)-Pr)/(-q(4)*(1-q(3))-q(2)*(PanPr(Ps,q(1),q(2),Pr)-Pr)))-q(3)*dPanPr(Ps,q(1),q(2),xr,Pr);
F2 =@(q) q(4) - PanEc(Ps,q(1),Ec)*( (q(1)/(1-q(3))) + (1/(xc-q(3)*dPanEc(Ps,q(1),q(2),xc,Ec))) );
F3= @(q) xmm - ((PanEm(Ps,q(1),q(2),Em,Pm)-Pm)/(q(4)*(1-q(3))-q(2)*(PanEm(Ps,q(1),q(2),Em,Pm)-Pm)));

Ftot=@(q) [F1(q) F2(q) F3(q)];

lb=[4.1e5 3.7e6 0.35  1.8e6];
ub=[5e5 4e6 0.5  2e6];

options = optimoptions('gamultiobj','InitialPopulationRange',[lb;ub]);
       [q,fval,exitflag,output] = gamultiobj(Ftot,4,[],[],[],[],lb,ub,options)

[Walter Roberson]

When you speak of min F(q) does that mean that you are looking for the q vector that has the minimum over any of F1 F2 F3? If so then just min() the parts and use ga or patternsearch. Do not use fmincon or fminsearch however as min() is not differentiateable

[ANGELOS GIANNIS]

At the paper that I am reading they are looking for the q vector that has the minimum over the three F1,F2,F3 simultaneously. Thank you for the advice and your answer.

[Walter Roberson]

Your three Pan* functions are not given?

[ANGELOS GIANNIS]

They are given. The three Pan* equations are of these kind:

function Pan_Ec= PanEc(Ps,a,Ec)

Pan_Ec=Ps*(coth(Ec/a)-a/(Ec));


end

[Walter Roberson]

All three please. I am trying to figure out whether there is an sense under which all three can be minimized simultaneously

[ANGELOS GIANNIS]

function Pan_Pr= PanPr(Ps,a,alpha,Pr)

Pan_Pr=Ps*(coth(alpha*Pr/a)-a/(alpha*Pr));

end


function Pan_Em= PanEm(Ps,a,alpha,Em,Pm)

Pan_Em=Ps*(coth((Em+alpha*Pm)/a)-(a/(Em+alpha*Pm)));

end


function dPPr= dPanPr(Ps,a,alpha,xr,Pr)

dPPr= (Ps/a)*(1+alpha*xr)*(-(csch(alpha*Pr)/a)^2 + (a/(alpha*Pr)));

end

function dPEc=dPanEc(Ps,a,alpha,xc,Ec)

dPEc= (Ps/a)*(1+alpha*xc)*(-(csch(Ec)/a)^2 + (a/(Ec)));

end

Answer 1

[Alan Weiss]

Did you understand that in general there is no such thing as the minimum of a vector-valued function? gamultiobj finds a Pareto optimal set, meaning a surface where you can lower one of the objective function values only by raising another. See What Is Multiobjective Optimization? and x output of gamultiobj.

Alan Weiss

MATLAB mathematical toolbox documentation

[ANGELOS GIANNIS]

Alan thank you for your answer. I got wrong the way that gamultiobj works. Now I see what exactly I get. But do you know another way that I can follow ?

[Alan Weiss]

Sorry, I have no more advice, because I do not know really what you are trying to do. If you use paretosearch instead of gamultiobj you can use the psplotparetof plot function to visualize the tradeoffs.

options = optimoptions('paretosearch','PlotFcn','psplotparetof');
[x,fval] = paretosearch(fun,nvar,A,b,Aeq,beq,lb,ub,nonlcon,options);

Alan Weiss

MATLAB mathematical toolbox documentation

[ANGELOS GIANNIS]

Thank you.