Simultaneous minimization of 3 equations to estimate 4 parameters

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ANGELOS GIANNIS
ANGELOS GIANNIS 2020-7-16
评论： ANGELOS GIANNIS 2020-7-21
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)
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Walter Roberson
Walter Roberson 2020-7-21
All three please. I am trying to figure out whether there is an sense under which all three can be minimized simultaneously
ANGELOS GIANNIS
ANGELOS GIANNIS 2020-7-21
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

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采纳的回答

Alan Weiss
Alan Weiss 2020-7-20
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
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Alan Weiss
Alan Weiss 2020-7-21
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

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