Matlab Frank Wolfe algorithm for non-linear constraints - matlab

Question

I have a non-linear objective function with non-linear constraints as below:
fun = #(x)x(11)^2 + x(12)^2 + x(13)^2 + x(8)^2 + x(15)^2 + x(16)^2 + x(17)^2 + x(18)^2 +
x(19)^2 + x(20)^2 + x(21)^2 + x(14)^2;
A = [];
b = [];
Aeq = [];
beq = [];
lb = [0,0,0,0,0,0,0,-Inf,0,0,-Inf,-Inf,-Inf,-Inf,-Inf,-Inf,-Inf,-Inf,-Inf,-Inf,-Inf];
ub = [Inf,Inf,Inf,Inf,Inf,Inf,Inf,Inf,Inf,Inf,Inf,Inf,Inf,Inf,Inf,Inf,Inf,Inf,Inf,Inf,Inf];
x0(1:21) = 0;
nonlcon = #consts;
options = optimoptions('fmincon','Display','iter','Algorithm','interior-point');
x = fmincon(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon,options);
function [c,ceq] = consts(x)
c = x(1) + x(2) + x(3) + x(4) + x(5) + x(6) - 1;
ceq(1) = x(7) + x(8) - 0;
ceq(2) = x(5)*x(9) + x(3)*x(10) + x(7) + x(11) - 0;
ceq(3) = x(7) + x(5)*x(9) + x(12) - 0;
ceq(4) = x(7) + x(3)*x(10) + x(13) - 0;
ceq(5) = x(10) + x(14) - 9.666666666666666;
ceq(6) = x(6)*x(9) + x(1)*x(7) + x(10) + x(15) - 17;
ceq(7) = x(10) + x(6)*x(9) + x(16) - 0;
ceq(8) = x(10) + x(1)*x(7) + x(17) - 2.211764705882353;
ceq(9) = x(9) + x(18) - 0;
ceq(10) = x(4)*x(10) + x(2)*x(7) + x(9) + x(19) - 0;
ceq(11) = x(9) + x(4)*x(10) + x(20) - 0;
ceq(12) = x(9) + x(2)*x(7) + x(21) - 0;
end
'fun' is the objective function and 'consts' is the constraints.
I solve my problem with 'interior-point' however, I want to solve this problem with Frank_Wolfe and I cannot find where can I define the constraints.
Based on the documentation:
function [X,fval,i]=frank_wolfe(f,X0,e,A,b,Aeq,beq,lb,ub)
%
% Yinxiao Li
%
% function function [X,fval,i]=frank_wolfe(f,X0,e,A,b,Aeq,beq,lb,ub)
%
% order = 3;
%
% Input: f cost function
% X0 starting feasible point
% e stopping criteria
% A defined in linprog
% b
% Aeq
% beq
% lb
% ub defined in linprog
%
% Output: X optimal point
% fval cost at the optimal point
% i iterations
X=X0;
syms x1 x2 x3 lamida;
df_dx1=diff(f,x1);
df_dx2=diff(f,x2);
df_dx3=diff(f,x3);
x=[100;100;100];
g_f=[100;100;100];
i=0;
while(abs(g_f'*(x-X))>e)%stopping criteria
g_f1=subs(df_dx1,{x1,x2,x3},X);
g_f2=subs(df_dx2,{x1,x2,x3},X);
g_f3=subs(df_dx3,{x1,x2,x3},X);
g_f=[double(g_f1);double(g_f2);double(g_f3)];
[x,feval]=linprog(g_f,A,b,Aeq,beq,lb,ub);
f0=subs(f,{x1,x2,x3},X+lamida*(x-X));
f1=inline(char(f0));
[lamida,fval]=fminbnd(f1,0,1);
X1=X;
X=X+lamida*(x-X);
clear lamida;
syms lamida;
i=i+1;
end
fval=subs(f,{x1,x2,x3},X);
I am not sure:
1- where to define constraints?
2- what should I provide in "linprog" cause as far as I know it is for linear programming.
3- Does Frank wolfe find a global optima or local optima?