Objective function value stuck at a point for fminsearch optimization

Hi,

I have the following code that I am trying to optimize using fminsearch but the objective function value stuck at 3 and doesn't go below that even though I have tried hundreds of different initial guesses. Code is given below:

function [ h ] = newk( param )

sum=0;
a    = param(1);
b    = param(2);
pnot = param(3);
sm    = param(4);
st = param(5);

if sm<st
    sum=sum+((-0.0607)-pnot+a*s^2+b*s)^2;

      sum=sum+(1.1429-((a*st^3)/3 + (b*st^2)/2 + (pnot*st) - (0.0001*a)/3 - (0.001*b)/2 -0.1*pnot))^2;

      sum=sum+(0-((a*sm^3)/3 + (b*sm^2)/2 + (pnot*sm) - (0.0001*a)/3 - (0.001*b)/2 -0.1*pnot))^2;

      sum=sum+(1-(-1*cos(180/pi)*((a*st^3)/3+(b*st^2)/2+pnot*st - (0.001*a)/3 - (0.001*b)/2 - 0.1*pnot)/(a*st^2 + b*st+pnot))  -(-1*cos(180/pi)*(0)/(a*st^2 + b*st+pnot)))^2;

      sum=sum+(9-(-1*cos(180/pi)*((a*sm^3)/3+(b*sm^2)/2+pnot*sm - (0.001*a)/3 - (0.001*b)/2 - 0.1*pnot)/(a*sm^2 + b*sm+pnot))  -(-1*cos(180/pi)*(0)/(a*sm^2 + b*sm+pnot)))^2;

      sum=sum+(33-((sin(180/pi)*(thetanot+(a*st^3)/3+(b*st^2)/2+pnot*st)-(0.0001*a)/3-(0.001*b)/2-(0.1*pnot))/a*st^2 + b*st + pnot))^2;

      sum=sum+(20-((sin(180/pi)*(thetanot+(a*sm^3)/3+(b*sm^2)/2+pnot*sm)-(0.0001*a)/3-(0.001*b)/2-(0.1*pnot))/a*sm^2 + b*sm + pnot))^2;

      h=sum;

else
    h = NaN;
end
end

Does anyone know how to improve the objective function value or any other better way to solve this problem?

Thanks in advance

[Geoff Hayes]

One comment - don't use sum as variable name as that is the name of a MATLAB built-in function.

Also, can you describe what the above code is doing and how you are using it with fminsearch?

[Walter Roberson]

We need to know what your function s or your shared variable s is. Also thetanot.

You will get further if you replace the NaN with realmax

[Research Guy]

I am trying to solve for the values of a,b,sm,st and pnot using optimization toolbox. Therefore I am using fminsearch. I have five equations with five unknowns so I am trying to solve the equations for the unknowns.

[Research Guy]

Here s is the variable in polynomial equation while so sm and st are variables while a,b and pnot are coefficients of a polynomial equation.

[Walter Roberson]

There is no s in your inputs. If you are constructing a polynomial in s, then you need to declare s as symbolic and you need to apply the polynomial to some value, as fminsearch cannot minimize symbolic functions. And you still have not said what thetanot is.

[Research Guy]

thetanot is 0. And I want to find the value of s too. s is also unknown.

[Walter Roberson]

Are there any constraints other than sm<st and thetanot = 0? Are any of the values constrained to be positive? Are any of the values constrained to be real?

Could I ask you to confirm that the sin() and cos() only go as far as including the 180/pi ? sin(180/pi) and nothing more? Not sin(180/pi * the next expression) ?

[Research Guy]

No there are no any other constraints.

Answer 1

[Walter Roberson]

My tests indicate

With parameter order a, b, pnot, s, sm, st

Overall best: approximately 0.653438069904368368 @ [-498.85377613580755, 221.483885941090421, -24.4119283679524557, 0.535194263358808153, 0.200512886217885877, 0.259221258479518379]

Best zone [-498.853793637251272, -498.853749303693007; 221.483880618247383, 221.483889651746154; -24.4119284892117108, -24.4119283114555756; -0.0912086804123130035, 0.535194279291770658; 0.200512882607536191, 0.200512891548387973; 0.259221254021537173, 0.259221265398527967]

The zone shows the search area containing the minima -- for example, the best for the first parameter, a, is -498.853793637251272 to -498.853749303693007

Notice the wide range for the 4th parameter, s: there are two s values that are very close in producing minima; my tests suggest that if the minima from the two are different then it is not by much.

The function minimized was

@(a,b,pnot,s,sm,st)(-607/10000-pnot+a.*s.^2+b.*s).^2+(1/900000000).*(10000.*a.*st.^3+15000.*b.*st.^2+30000.*pnot.*st-a-15.*b-3000.*pnot-34287).^2+(-(1/3).*a.*sm.^3-(1/2).*b.*sm.^2-pnot.*sm+(1/30000).*a+(1/2000).*b+(1/10).*pnot).^2+(1+cos(180/pi).*((1/3).*a.*st.^3+(1/2).*b.*st.^2+pnot.*st-(1/3000).*a-(1/2000).*b-(1/10).*pnot)./(a.*st.^2+b.*st+pnot)).^2+(9+cos(180/pi).*((1/3).*a.*sm.^3+(1/2).*b.*sm.^2+pnot.*sm-(1/3000).*a-(1/2000).*b-(1/10).*pnot)./(a.*sm.^2+b.*sm+pnot)).^2+(33-(sin(180/pi).*((1/3).*a.*st.^3+(1/2).*b.*st.^2+pnot.*st)-(1/30000).*a-(1/2000).*b-(1/10).*pnot).*st.^2./a-b.*st-pnot).^2+(20-(sin(180/pi).*((1/3).*a.*sm.^3+(1/2).*b.*sm.^2+pnot.*sm)-(1/30000).*a-(1/2000).*b-(1/10).*pnot).*sm.^2./a-b.*sm-pnot).^2

[Research Guy]

Thank you for answer. But why you include s? sm and st are the only ones we need to find. I am sorry for the mistake because I used s in the first equation while there is st instead of s. Sorry about that.

[Walter Roberson]

Earlier I asked about s and you said,

"And I want to find the value of s too. s is also unknown."

[Walter Roberson]

With the correction of s to st, the best point I find so far is

a, b, pnot, sm, st as -18880.3302350366, 800.005971699948, -6.37050123940534, 0.0330024744393139,0.0491711511781789

[Research Guy]

Thank you, This works but can you tell me how did you find the best point? Also is there any better way to solve this porblem?

[Walter Roberson]

A few months ago I wrote my own hybrid minimizer, and I have been tuning it since then. It uses a mix of brute force and fminsearch.

Unfortunately there are a number of local minima, which make it difficult to find good values using fmincon or ga. I have not had an opportunity yet to try simulated annealing.

[Walter Roberson]

I noticed that your present posted code is not exactly the same as what I had optimized for (because I used the previous version of what you posted), so I re-optimized. I find two distinct minima groups.

[0.000555266562406788 -0.00389896399517703 0.00542068020598675 2.10243357365545 3.63610626474992]

The above is an area near which the residue is 1.34 -ish

[-527.652989562935 8869.56269660075 -1709.20884144656 0.194959774473371 0.196426461653078]

The above is an area near which the residue is 1.73 -ish

The difference between 1.34-ish and 1.73-ish is significant, so you might prefer one to the other, but on the other hand both of those are much smaller than "typical" values of the surface. The best I was able to do with simulanneabnd starting from random points was in the 81 to 83 range, way out near

[-887.209391792006 14913.4480950303 -2874.95731224742 0.194120083848097 0.194986847835464]