initial point is a local minimum. Optimization completed because the size of the gradient at the initial point is less than the default value of the function tolerance.

I just found the result not the optimum and the parameters obtained are always the initial ones.

global c

xdata = [0
    1
    2
    3
    4
    5
    6
    7
    8
    9
    10
    15
    20
    25
    30
    35
    40
    45
    50
    55
    60
    70
    80
    90
    120
    500
    1000
    10000
    30000];
ydata = [0
    NaN
    NaN
    NaN
    NaN
    1.34288
    NaN
    NaN
    NaN
    NaN
    1.43759
    NaN
    NaN
    1.59058
    1.59786
    1.60515
    1.60515
    1.55415
    1.54687
    1.58329
    NaN
    1.64886
    1.72171
    1.72171
    1.72171
    NaN
    NaN
    NaN
    NaN];
xdata0 = xdata;
ydata0 = ydata;
i = ismissing(ydata);
xdata(i) = [];
ydata(i) = [];
x = xdata0(1:end-4);
y = spline(xdata,ydata,x);

plot(xdata,ydata,'o')
hold on
plot(x,y,'LineWidth',2)
figure
rng(1997)
a = 0;
b = 10;
r = a + (b-a).*rand(6,1);
% r = 10*ones(10,1);
c = r;

options = optimoptions('lsqcurvefit',...
    'Algorithm','levenberg-marquardt',...
    'FunctionTolerance',1e-20,...
    'StepTolerance',1e-20,...
    'OptimalityTolerance',1e-20,...
    'FiniteDifferenceStepSize',1e-20,...
    'FiniteDifferenceType','central',...
    'Display','iter');
lb = [];
ub = [];

coeffs = lsqcurvefit(@fitting,r,x,y,lb,ub,options);
c = coeffs;
figure
plot(xdata,ydata,'o')
hold on
xpredict = linspace(0,max(xdata),1000);
ypredict = fitting(coeffs,xpredict);
ypred=fitting(coeffs,x);

p=0;
min=5;
for b=0:0.01:1
    r = a + (b-a).*rand(6,1);
    % r = 10*ones(10,1);
    c = r;
    
    options = optimoptions('lsqcurvefit',...
        'Algorithm','levenberg-marquardt',...
        'FunctionTolerance',1e-20,...
        'StepTolerance',1e-20,...
        'OptimalityTolerance',1e-20,...
        'FiniteDifferenceStepSize',1e-20,...
        'FiniteDifferenceType','central',...
        'Display','iter');
    lb = [];
    ub = [];
    
    coeffs = lsqcurvefit(@fitting,r,x,y,lb,ub,options);
    c = coeffs;
    figure
    plot(xdata,ydata,'o')
    hold on
    xpredict = linspace(0,max(xdata),1000);
    ypredict = fitting(coeffs,xpredict);
    ypred=fitting(coeffs,x);
    
    if sum((y-ypred).^2,2)<=min
        min=sum((y-ypred).^2,2)
        p=b;
    end
end
b=p;
r = a + (b-a).*rand(6,1);
% r = 10*ones(10,1);
c = r;

options = optimoptions('lsqcurvefit',...
    'Algorithm','levenberg-marquardt',...
    'FunctionTolerance',1e-20,...
    'StepTolerance',1e-20,...
    'OptimalityTolerance',1e-20,...
    'FiniteDifferenceStepSize',1e-20,...
    'FiniteDifferenceType','central',...
    'Display','iter');
lb = [];
ub = [];

coeffs = lsqcurvefit(@fitting,r,x,y,lb,ub,options);
c = coeffs;
figure
plot(xdata,ydata,'o')
hold on
xpredict = linspace(0,max(xdata),1000);
ypredict = fitting(coeffs,xpredict);
plot(xpredict,ypredict,'-x','LineWidth',2)
% xlim([0 20])




function ytotal = fitting(c,t)

tspan = t;
y0 = zeros(2,1);
[~,ys] = ode45(@myodes, tspan, y0)
ytotal = ys(:,1).*(1+c(2)) + ys(:,2) + c(3).*ys(:,2).*(1.5-ys(:,1).*(1+c(2)))./(1+c(3).*ys(:,2))
end

function dydt = myodes(t,y)
global c
dydt = zeros(2,1);
dydt(1) = c(1) .* (2.5 - (1 + c(2)) .* y(1) - y(2) - c(3) .* y(2) .* (1.5 - y(1) .* (1 + c(2))) ./ (1 + c(3) .* y(2))) .* (1.5 - y(1) .* (1 + c(2))) ./ (1 + c(3) .* y(2)) - c(4) .* y(1);
dydt(2) = c(5) .* (2.5 - (1 + c(2)) .* y(1) - y(2) - c(3) .* y(2) .* (1.5 - y(1) .* (1 + c(2))) ./ (1 + c(3) .* y(2))) .* (1.5 - y(2) - c(3) .* y(2) .* (1.5 - y(1) .* (1 + c(2))) ./ (1 + c(3) .* y(2))) - c(6) .* y(2);

end

[Torsten]

I think it's not a good idea to use vectors with lots of NaN values for fitting.

Remove them first before proceeding.

[Walter Roberson]

They already have i = ismissing(ydata); and delete those entries

[Jinglei]

Thank you for pointing that out. I used so many NaN values to decrease the gap make the curve more continuous. But I will still try removing them and see whether this works.@Torsten

[Jinglei]

Thank you for your suggestion and I will try it.@Walter Roberson

Answer 1

[Matt J]

I suspect your FiniteDifferenceStepSize is too small. Be mindful of the guidelines for Optimizing Differential Equations.

[Jinglei]

Thank you for your answer. Do you mean I should increase the FiniteDifferenceStepSize from 1e-20 to a larger one or decrease it to get a larger range? I tried both but still can't get any good result. Values larger than 1e-20 even get to no result, showing 'local minimum possible'. Do you have any idea about that?

[Matt J]

'local minimum possible' means the solver might have succeeded. The exitflag would give a clearer picture of why the optimization stopped, though.