# Optimization of 9 unknown with 9 nonlinear constraints by fmincom

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YI RUI SHIH on 3 Oct 2019
Answered: Alex Sha on 1 Mar 2020
Hi
I want to find the optimal solution of 9 unknown ( x(1) - x(9) ) with 9 nonlinear constraints, and subject to the minimization of x(9) so I use fmincon to solve it.
Here is my code for the optimization with fmincon:
% main code
fun = @(x) x(9) ;
nonlcon = @unitdisk50;
lb = [0 0 0 0 0 0 0 0 0];
ub = [inf inf inf inf inf inf inf inf inf];
x0 = [0.02 0.64 0.95 1.38 1.72 2.12 2.52 2.9 3.1];
A = [];
b = [];
Aeq = [];
beq = [];
x = fmincon(fun,x0,[],[],[],[],lb,ub,nonlcon)
and the code of function
function [c,ceq] =unitdisk50(x)
J = 2;
w1 = 2.3026;
sigma1 = -0.3364;
w2 = 6.5298;
sigma2 = -3.4136;
c = [x(1)-x(2)+x(3)-x(4)+x(5)-x(6)+x(7)-x(8)+x(9)];
ceq = [1-exp(-sigma1*x(1))*cos(w1*x(1)) + exp(-sigma1*x(2))*cos(w1*x(2)) - exp(-sigma1*x(3))*cos(w1*x(3)) + exp(-sigma1*x(4))*cos(w1*x(4)) - exp(-sigma1*x(5))*cos(w1*x(5)) + exp(-sigma1*x(6))*cos(w1*x(6)) - exp(-sigma1*x(7))*cos(w1*x(7)) + exp(-sigma1*x(8))*cos(w1*x(8)) - exp(-sigma1*x(9))*cos(w1*x(9))
-exp(-sigma1*x(1))*sin(w1*x(1)) + exp(-sigma1*x(2))*sin(w1*x(2)) - exp(-sigma1*x(3))*sin(w1*x(3)) + exp(-sigma1*x(4))*sin(w1*x(4)) - exp(-sigma1*x(5))*sin(w1*x(5)) + exp(-sigma1*x(6))*sin(w1*x(6)) - exp(-sigma1*x(7))*sin(w1*x(7)) + exp(-sigma1*x(8))*sin(w1*x(8)) - exp(-sigma1*x(9))*sin(w1*x(9))
-x(1)*exp(-sigma1*x(1))*cos(w1*x(1)) + x(2)*exp(-sigma1*x(2))*cos(w1*x(2)) - x(3)*exp(-sigma1*x(3))*cos(w1*x(3)) + x(4)*exp(-sigma1*x(4))*cos(w1*x(4)) - x(5)*exp(-sigma1*x(5))*cos(w1*x(5)) + x(6)*exp(-sigma1*x(6))*cos(w1*x(6)) - x(7)*exp(-sigma1*x(7))*cos(w1*x(7)) + x(8)*exp(-sigma1*x(8))*cos(w1*x(8)) - x(9)*exp(-sigma1*x(9))*cos(w1*x(9))
-x(1)*exp(-sigma1*x(1))*sin(w1*x(1)) + x(2)*exp(-sigma1*x(2))*sin(w1*x(2)) - x(3)*exp(-sigma1*x(3))*sin(w1*x(3)) + x(4)*exp(-sigma1*x(4))*sin(w1*x(4)) - x(5)*exp(-sigma1*x(5))*sin(w1*x(5)) + x(6)*exp(-sigma1*x(6))*sin(w1*x(6)) - x(7)*exp(-sigma1*x(7))*sin(w1*x(7)) + x(8)*exp(-sigma1*x(8))*sin(w1*x(8)) - x(9)*exp(-sigma1*x(9))*sin(w1*x(9))
1-exp(-sigma2*x(1))*cos(w2*x(1)) + exp(-sigma2*x(2))*cos(w2*x(2)) - exp(-sigma2*x(3))*cos(w2*x(3)) + exp(-sigma2*x(4))*cos(w2*x(4)) - exp(-sigma2*x(5))*cos(w2*x(5)) + exp(-sigma2*x(6))*cos(w2*x(6)) - exp(-sigma2*x(7))*cos(w2*x(7)) + exp(-sigma2*x(8))*cos(w2*x(8)) - exp(-sigma2*x(9))*cos(w2*x(9))
-exp(-sigma2*x(1))*sin(w2*x(1)) + exp(-sigma2*x(2))*sin(w2*x(2)) - exp(-sigma2*x(3))*sin(w2*x(3)) + exp(-sigma2*x(4))*sin(w2*x(4)) - exp(-sigma2*x(5))*sin(w2*x(5)) + exp(-sigma2*x(6))*sin(w2*x(6)) - exp(-sigma2*x(7))*sin(w2*x(7)) + exp(-sigma2*x(8))*sin(w2*x(8)) - exp(-sigma2*x(9))*sin(w2*x(9))
-x(1)*exp(-sigma2*x(1))*cos(w2*x(1)) + x(2)*exp(-sigma2*x(2))*cos(w2*x(2)) - x(3)*exp(-sigma2*x(3))*cos(w2*x(3)) + x(4)*exp(-sigma2*x(4))*cos(w2*x(4)) - x(5)*exp(-sigma2*x(5))*cos(w2*x(5)) + x(6)*exp(-sigma2*x(6))*cos(w2*x(6)) - x(7)*exp(-sigma2*x(7))*cos(w2*x(7)) + x(8)*exp(-sigma2*x(8))*cos(w2*x(8)) - x(9)*exp(-sigma2*x(9))*cos(w2*x(9))
-x(1)*exp(-sigma2*x(1))*sin(w2*x(1)) + x(2)*exp(-sigma2*x(2))*sin(w2*x(2)) - x(3)*exp(-sigma2*x(3))*sin(w2*x(3)) + x(4)*exp(-sigma2*x(4))*sin(w2*x(4)) - x(5)*exp(-sigma2*x(5))*sin(w2*x(5)) + x(6)*exp(-sigma2*x(6))*sin(w2*x(6)) - x(7)*exp(-sigma2*x(7))*sin(w2*x(7)) + x(8)*exp(-sigma2*x(8))*sin(w2*x(8)) - x(9)*exp(-sigma2*x(9))*sin(w2*x(9))
J*(x(3)-x(2)+x(1)-x(4)+x(5)-x(6)+x(7)-x(8)+x(9))-1
];
Although it has found the solution
x
0.1333 1.0305 1.4458 1.7052 1.8199 2.2065 2.5552 2.8824 2.9252
but it didn't satisfiy the constraint '' J*(x(3)-x(2)+x(1)-x(4)+x(5)-x(6)+x(7)-x(8)+x(9))-1 '')
and it showed that :
'' Converged to an infeasible point.
fmincon stopped because the size of the current step is less than
the default value of the step size tolerance but constraints are not
satisfied to within the default value of the constraint tolerance. ''
Does anyone know how to solve this problem ? This is the first time I use optimal function.
Thanks.
Ray.

#### 1 Comment

Matt J on 3 Oct 2019
Note that these constraints are in fact linear and should not be in your nonlcon function.
J*(x(3)-x(2)+x(1)-x(4)+x(5)-x(6)+x(7)-x(8)+x(9))-1=0
x(1)-x(2)+x(3)-x(4)+x(5)-x(6)+x(7)-x(8)+x(9)<=0

Matt J on 3 Oct 2019
Edited: Matt J on 3 Oct 2019
It seems highly likely that your problem is over-constrained. With 9 equality constraints and 9 unknowns, you may have enough equations to solve for the unknown uniquely. Who knows if that unique point will also satisfy the remaining inequality constraints in your problem.

Alex Sha on 1 Mar 2020
one inequality constraint is: x(1)-x(2)+x(3)-x(4)+x(5)-x(6)+x(7)-x(8)+x(9)<=0 (1)
while one of equality constraint is: J*(x(3)-x(2)+x(1)-x(4)+x(5)-x(6)+x(7)-x(8)+x(9))-1=0
this same as: x(3)-x(2)+x(1)-x(4)+x(5)-x(6)+x(7)-x(8)+x(9)=1/J=1/2=0.5 (2)
note the left sides of (1) and (2) are exact same, so the inequality constraint and equality constraint are contradictory, therefor there is no solution.

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