Nonlinear optimization with fmincon - How to write an objective function with variables that affects optimised variables

I would like to maximize the sum of six variables in the vector "x" which depends on the value of 6 more variables and 12 nonlinear equalities. I know is hard, but complexity of the equations does not allow to have only six equations for the system. The algortihms are not capable of performing iterations and complying all equalities. I have already calculated the first gradients of the objective and constraints to have efficient iteration. However, I get an unfeasible point when tolerance constraint is reached.

I hope anyone would come up with a suggestion. I am using algortihms such as sqp and active-set getting quite similar results. It is true that there is one solution of the 12 variable equation system that reaches the maximum value; however, I do not know whether it is far better to isolate the problem of the first six variable, with only bound constraints, and then perform fsolve to calculate the other six variables.

F=-sum(x(1:6))
s.t.
eq(1)=b(1)*x(1)^2+a(1)*x(1)+P+x(7)+dp(1)+14.69-Pder(1);
eq(2)=b(2)*x(2)^2+a(2)*x(2)+P+x(7)+dp(2)+14.69-Pder(2);
eq(3)=b(3)*x(3)^2+a(3)*x(3)+P+x(7)+dp(3)+14.69-Pder(3);
eq(4)=b(4)*x(4)^2+a(4)*x(4)+P+x(7)+x(8)+dp(4)+14.69-Pder(4);
eq(5)=b(5)*x(5)^2+a(5)*x(5)+P+x(7)+x(8)+dp(5)+14.69-Pder(5);
eq(6)=b(6)*x(6)^2+a(6)*x(6)+P+x(7)+x(8)+x(9)+dp(6)+14.69-Pder(6);
eq(7)=dv(1,1)*x(10)^2+dv(2,1)*x(10)+dv(3,1)-x(7);
eq(8)=dv(1,2)*x(11)^2+dv(2,2)*x(11)+dv(3,2)-x(8);
eq(9)=dv(1,3)*x(12)^2+dv(2,3)*x(12)+dv(3,3)-x(9);
eq(10)=sum(x(1:6))*cv(1)-x(10)*(P3+0.5*x(7)+14.69);
eq(11)=sum(x(4:6))*cv(2)-x(11)*(P3+x(7)+0.5*x(8)+14.69);
eq(12)=x(6)*cv(3)-x(12)*(P3+x(7)+x(8)+0.5*x(9)+14.69);

Thanks in advance.

[Walter Roberson]

To confirm, you need to adjust 12 variables, and there are 12 nonlinear equalities, and the final cost is based upon the sum of six variables?

[Torsten]

And you start from a feasible point of your system of 12 equations in 12 unknowns ?

[Ramiro Canchucaja]

Walter Roberson, yes. The final cost function is based upon the sum of six out of the 12.

Torsten, I could not manage to start from a feasible point. Do you believe it is the main reason of unfeasibility of the final optimised answer?

[Torsten]

If all the other variables in the system of equations except x1,...,x12 are given constants, the system usually has a unique solution. So I don't understand what you want to minimize here if there usually is only one feasible point.

Try to solve your system using "fsolve" first and see what the solver returns.

[Bruno Luong]

Agree with Tosten, such system usually give a set of discrete solution(s) that can be empty as well.

Nothing smart to optimize a function on top of it beside enumerate all the solutions (of the 12 constraints) then check the score.

[Ramiro Canchucaja]

Hello guys, thanks for your answers. I have spotted a serious mistake in my formulation. 3 of them are actually inequalities, so a unique response would not satisfy the bound constraints for variables and also the 12 equalities. I will solve it and let you know the results.