Yes, [0 0 0 0] is the correct solution to your problem.
For all other c vectors >= 0, min g(x,c) would be negative.
Since your problem is linear in the c's, here is an easier way to solve your problem:
poly1=@(x)x.^2-x-1;
poly2=@(x)x.^4-x.^3-3*x.^2+x+1;
poly3=@(x)x.^8-x.^7-7*x.^6+4*x.^5+13*x.^4-4*x.^3-7*x.^2+x+1;
poly4=@(x)x.^3+x.^2-2*x-1;
X = [0.1037;0.0259;0.2288;0.0938;0.0917;0.2386;0.2003];
A = [log(abs(poly1(X))),log(abs(poly2(X))),log(abs(poly3(X))),log(abs(poly4(X))),ones(numel(X),1)] b = zeros(numel(X),1);
f = [0 0 0 0 -1].';
lb = [0 0 0 0 -Inf].';
ub = [1000 1000 1000 1000 Inf].';
c = linprog(f,A,b,[],[],lb,ub)
