Non-linear constrained optimization in Matlab - matlab

Question

Given a function f(k',w') = ((k'-k)^2+(w'-w)^2)^(1/2), where k and w are known real parameters. The objective is to find a couple (k',w') such that f(k',w') is minimal under the following constraints
b(v,s,w') < 10s <=> w'< 10s
b(v,s,w')< a(v,s,k')^2 <=> (w'-10s)-(k'-3)^2/s < 0
q( a(v,s,k'),b(v,s,w')) < s[v^(1/2)]
where b(v,s,w') = (v/s)(w'-10s) and a(v,s,k') = (1/s)(k'-3)v^(1/2). Above that, v (=vari) > 0 and s (=skew) < 0 are known parameters. Furthermore q(a,b) is a root of the quartic polynomial:
(48a^2 + 16b)x^4 + (-40a^3 - 168ab)x^3 + (-45a^4 + 225a^2b + 72b^2)x^2 + (27a^3b - 162ab^2)x + 27b^3.
To be more precise, whenever the quartic has four real roots then q is the second greatest root. If the quartic has two real and two complex roots then q is the greatest real root. The problem is that the algebraic expressions for q are quite monstrous. Ideally, I would like an analytical solution for the above non-linear constrained optimization problem. However, I think that might turn out quite ugly. Therefore, I thought it was better to do it numerically with Matlab by using constrained optimizers such as fmincon.
f = #(x) sqrt((x(1)-hyp_skew)^2+(x(2)-kurt)^2);
A = [1,0];
d = 10*skew;
Aeq = [];
beq = [];
ub = [];
lb = [];
[x,fval,exitflag] = fmincon(f,[w,k],A,d,Aeq,beq,lb,ub,#(x)quarticcondition2(x,skew,vari),options);
where the (non-linear) optimization function is given by
function [c, ceq] = quarticcondition2(x,skew,vari)
av = ((x(2)-3)*sqrt(vari))/skew;
bv = (vari/skew)*(x(1)-10*skew);
A = (-40*av^3-168*av*bv)/(48*av^2+16*bv);
B = (-45*av^4+225*av^2*bv+72*bv^2)/(48*av^2+16*bv);
C = (27*av^3*bv-162*av*bv^2)/(48*av^2+16*bv);
D = (27*bv^3)/(48*av^2+16*bv);
roots_quartic = roots([1,A,B,C,D]);
z = imag(roots_quartic);
in = find(z ~= 0);
if isempty(in)
r = sort(roots_quartic);
c2 = r(2)-skew*sqrt(vari);
else
index = find(z == 0);
c2 = max(roots_quartic(index))-skew*sqrt(vari);
end
c1 = ((x(2)-3)^2/skew)-(x(1)-10*skew);
c = [c1 c2];
ceq = [];
end
My code works for some initial parameter sets [w,k]. However, finding this initial parameter set turns out to be quite difficult (since the constraints are hard to handle). I need to run the program for quite some possible scenarios, hence it would be nice to have some logic in choosing my starting values. I know this is a well known issue when using optimization solvers. However, is there a good/proper way to find start values?
Thanks!
Cheers