fminunc works, fmincon does not

I hope someone can help explaining why fminunc works (it does!), and why fmincon does not, in the code below. The estimates returned by fminunc are within the boundaries given to fmincon. The motivation for getting fmincon to work is the need to exchange the data, which causes fminunc to vomit (the parameters gets outside the boundaries and the objective function explodes).

Code

clear all

MyDataSet=importdata('MyDataSet.mat');

% Y and RK

RK=cell2mat(MyDataSet(2:end,5)); RK(RK==0)=100000*eps;

Y=cell2mat(MyDataSet(2:end,35));

%%Unconstrained Optimization

par=[0.5;0.1;4;0.3;0.3;0.01;0.7;0.05;0.05;0.05;0.05]; 

fminuncDNF = true;

while fminuncDNF

      try
          [x,fval] = fminunc(@(P) -fLogLik(P,Y,RK),par,options);
          fminuncDNF = false;                       
      catch
          disp('fminunc failed, changing start guess');
          a=rand(1,1);
          par=[rand(3,1);a;0.95-a;randn(6,1)];
      end
  end
%%Constrained optimization

par=[0.5;0.1;4;0.3;0.3;0.01;0.7;0.05;0.05;0.05;0.05];

A=[zeros(4,2) [-eye(3);0 1 1] zeros(4,6)];

b=[zeros(3,1);1]; 

b=b-[zeros(3,1);]*2*optimget(options,'TolCon',1e-6);

fminconDNF = true;
while fminconDNF
    try
        [x,fval] = fmincon('-fLogLik(P)',par,A,b,[],[],[],[],[],options,Y,RK);
        fminconDNF = false;                
    catch
        disp('fmincon failed, changing start guess');
        a=rand(1,1);
        par=[rand(3,1);a;0.95-a;randn(6,1)];
    end
end
----

function S=fLogLik(P,Y,RK)

h_u=fFilter_realLogGARCH(P,Y,RK);

mu0=P(2);

sig2_u=mean(h_u(2:end,2).^2);

tmp1=log(h_u(:,1));

tmp2=(Y-mu0).^2;

tmp2=tmp2./h_u(:,1);

tmp3=log(sig2_u);

l=-1*(tmp1+tmp2+tmp3+1);

S=sum(l(2:end,1));

end

function h_u=fFilter_realLogGARCH(P,Y,RK)

T=length(Y); h=zeros(T,1); u=h; h(1,1)=exp(P(1));

for t=2:T

      z_tm1=(Y(t-1,1)-P(2))/sqrt(h(t-1,1));
      tmp1=P(4)*log(h(t-1,1));
      tmp2=P(5)*log(RK(t-1,1));
      tmp3=P(6)*z_tm1;
      tmp4=P(7)*(z_tm1^2-1);
      h(t,1)=exp(P(3)+tmp1+tmp2+tmp3+tmp4);
      z_t=(Y(t,1)-P(2))/sqrt(h(t,1));
      tmp1=log(RK(t,1))-P(8);
      tmp2=P(9)*log(h(t,1));
      tmp3=P(10)*z_t;
      tmp4=P(11)*(z_t^2-1);
      u(t,1)=tmp1-tmp2-tmp3-tmp4;
end

h_u=[h,u];

end

Best, Kasper

Answer 1

[Steve Grikschat]

Hi Kasper,

It's hard to tell what's going on from the code, so I'll just offer some generalities.

Each solver and algorithm is different and will take different paths to (hopefully) reach the solution. Perhaps, putting constraints on the problem will guide the constrained algorithm into a difficult spot from which it can't recover. It's hard to say from this vantage point. You are starting some of the parameters close to the bounds.

On a related note, you should restate the first 3 linear inequalities as bounds. This has some distinct advantages: - the finite-difference derivative estimates will stay within the bounds - the interior-point and sqp algorithms of fmincon will honor the bounds at each step

Since you've mentioned that veering outside the constraints causes the objective to blow up, this seems important.

[Kasper]

First of all, thank you very much.

I have now implemented your suggestion but as expected it does not solve the problem. fmincon never gets started - it jumps to "catch" immediately. Do you have any suggestion to avoid this? I can't "demand it" to start as fminunc does? (I didn't find much help in the documentation on the latter point...)

[Steve Grikschat]

Jumping to catch means there is an error from fmincon or from the problem set up. You need to debug this before you can say whether or not fmincon "works".

My initial reaction is to the specification of the objective:

fmincon('-fLogLik(P)',...

is probably not gonna work. Try making this like your call to fminunc:

fmincon(@(P)-fLogLik(P,Y,RK),...

If that doesn't work, use the debugger to find out the problem.

Regards