Fsolve- equation solved, inaccuracy [possible

This is my function:

function F = root3d(x, alpha_1, alpha_2, beta_a1, beta_a2, beta_c1, beta_c2, gama_1, gama_2, theta_1, theta_2)
F(1) = alpha_1*x(1) - x(7)^-2;
F(2) = 1000 - x(2) - (x(7)/(alpha_1^2));
F(3) = (x(10)*(((x(9)/x(8))*(beta_a1/(1-beta_a1)))^(1-beta_a1))/(gama_1*x(3)));
F(4) = (x(10)*(((((1-beta_a1)/beta_a1)*(x(8)/x(9)))^beta_a1)/(gama_1*x(5))));
F(5) = (x(11)*(((x(9)/x(8))*(beta_c1/(1-beta_c1)))^(1-beta_c1))/(theta_1*x(4)));
F(6) = (x(11)*(((x(8)/x(9))*((1-beta_c1)/beta_c1))^(beta_c1))/(theta_1*x(6)));
F(7) = x(7)*x(10) -x(8)*x(3)-x(9)*x(5);
F(8) = x(5)+x(6)-650;
F(9) = x(3)+x(4)-650;
F(10) = x(1)-x(10);
F(11) = x(2)-x(11);
F(12) = alpha_2*x(12) - x(18)^-2;
F(13) = 1000 - x(13) - (x(18)/(alpha_2^2));
F(14) = (x(21)*(((x(20)/x(19))*(beta_a2/(1-beta_a2)))^(1-beta_a2))/(gama_2*x(14)));
F(15) = (x(21)*(((((1-beta_a2)/beta_a2)*(x(19)/x(20)))^beta_a2)/(gama_2*x(16))));
F(16) = (x(22)*(((x(20)/x(19))*(beta_c2/(1-beta_c2)))^(1-beta_c2))/(theta_2*x(15)));
F(17) = (x(22)*(((x(19)/x(20))*((1-beta_c2)/beta_c2))^(beta_c2))/(theta_2*x(17)));
F(18) = x(18)*x(21) -x(19)*x(14)-x(20)*x(16);
F(19) = x(16)+x(17)-650;
F(20) = x(14)+x(15)-650;
F(21) = x(12)-x(21);
F(22) = x(13)-x(22) 
end

And my code is this:

alpha_1 = 0.1;
beta_a1 = 0.2;
beta_c1 = 0.5;
theta_1 = 0.4;
gama_1 = 0.2;
alpha_2 = 0.1;
beta_a2 = 0.2;
beta_c2 = 0.25;
theta_2 = 0.4;
gama_2 = 0.2
x0 = [70,70,70,70,70,70,70,70,70,70,70,70,70,70,70,70,70,70,70,70,70,70];
options=optimset('disp','iter','LargeScale','off','TolFun',.001,'MaxIter',100000,'MaxFunEvals',100000);
[SS, fval, exitflag, output]=fsolve(@(x) root3d(x,alpha_1,alpha_2, beta_a1, beta_a2, beta_c1, beta_c2, theta_1, theta_2, gama_1, gama_2),x0,options);
SS

However I get this message in Matlab- Equation solved, inaccuracy possible.

The vector of function values is near zero, as measured by the value
of the function tolerance. However, the last step was ineffective.

<stopping criteria details>

Can you give me some ideas that I can try? Thanks in advance !

-Arpita

[Matt J]

It doesn't appear that the thetas or gamas can have any effect on the solution. They simply scale the right hand sides of F([3:6,14;17]) which will have no effect on a set of homogeneous equations.

[Arpita Nehra]

But then, does it mean that it should not be included or is it creating any issue? These equations are coming from a set of optimization problems.

Answer 1

[Raunak Gupta]

Hi,

This problem can be arrived if fsolve found a point where sum of squares of function values is less than the square root of the FunctionTolerance. This means the optimization should stop at this step however, the sum of squares changed very little in the last step even though gradient of the sum was larger than OptimalityTolerance (that is the first order condition to stop the optimization process). Thus, this suggests that the point found out by fsolve is not near an actual solution. This can happen if the optimization function is not smooth or fsolve have arrived at a Local minimum.

It’s suggested to proceed with the step given in Local Minimum Possible to check whether the solver has succeeded in finding a solution. Also, the exitflag gives good indication of where the solver has stopped.

[Arpita Nehra]

So, I am getting an exitflag of 3 or 4. Does that mean that I don't have a solution?

Answer 2

[Alex Sha]

a set of approximate solutions：

x1 0.0999971283199049

x2 -0.0143540751799935

x3 -1439252.41946883

x4 1439902.41946883

x5 -527282.99010789

x6 527932.99010789

x7 10.0001435407518

x8 -2.74840022211214E-7

x9 -1.14629426928716E-6

x10 0.0999971283124376

x11 -0.0143540751790788

x12 0.0999998530092107

x13 -2.71657239163275E-7

x14 -2444.18053646169

x15 3094.18053646169

x16 122870.151252869

x17 -122220.151252869

x18 10.0000000027164

x19 0.239014034293751

x20 0.00476269821181569

x21 0.0999998515396455

x22 -2.71660836684954E-7

[Arpita Nehra]

So, the answer needs to be positive- may be in tens or hundreds- I tried changing parameters and initial values, however few of my variables are always close to zero which is not correct- can anyone suggest me something that I can try?

Answer 3

[Alex Sha]

If want all are positive, refer the results follow:

x1: 0.0990335590812945

x2: 1.27592369827812E-11

x3: 650.000000000022

x4: 2.72057930783573E-10

x5: 650.000000001122

x6: 5.53147931778252E-11

x7: 10.0000000000192

x8: 0.000507814844408164

x9: 0.00101562968901066

x10: 0.0990238946721448

x11: 5.86346295731603E-21

x12: 0.099033559078221

x13: 3.9554649355099E-12

x14: 650.000000000205

x15: 8.92562451814302E-11

x16: 650.000000001166

x17: 1.17219410734425E-11

x18: 10.0000000000193

x19: 0.000507814844347538

x20: 0.00101562968902337

x21: 0.0990238946690532

x22: 6.03438668649515E-26