Optimization problem with a function of 2 parameters

I have a function with h, t parameters. And i want to find the values of the parameters that minimise the function. My function is quadratic and i have inequalities for the 2 parameters. h=[1.1;inf]; and t=[1000;10000];

I am trying to use the fmincon function as i have constrained nonlinear multivariable function but i dont find an example of a function with 2 parameters. Can you write an example of fmincon function for a function with 2 parameters? Can you tell me how can i find the 'x0' value?

[Torsten]

All the examples given here

https://de.mathworks.com/help/optim/ug/fmincon.html

are examples with two parameters.

[Stathis Kritikos]

my function is f(h,t). all the examples are f(x). I want to minimize f(h,t) when h=[1.1;inf]; and t=[1000;10000]. is there an example for my problem?

Thank you for the answer.

Stathis

[Torsten]

Set h = x(1) and t = x(2).

[Stathis Kritikos]

TC=@(h,t) (QC(h)+FC(t)+RC)./E(t)

I have this function. and i want to find h,t when TC is minimized. when h=(1.5*10^-5:2.5*10^-5) and t=(1000 :100000). Which function of the optimization toolbox can i use and how. TC is a big quadratic function.

The paper i am reading says that i have to implement an SQP. But as i said the fynction is too big to find all the matrix in order to formulate sqp.

Thank you

Answer 1

[Torsten]

TC = @(x) (QC(x(1))+FC(x(2))+ RC)/E(x(2));
lb = [1.5e-5,1000];
ub = [2.5e-5,100000];
sol = fmincon(TC,[2e-5,10000],[],[],[],[],lb,ub)

[Stathis Kritikos]

in this way i have to change all the parameters in all the functionsincludes in TC in X(1) and X(2)?

I run it and i have this problem

Index exceeds the number of array elements (1).?What is the problem?

Thank you so much for your help

[Torsten]

We need to see your code. If QC,FC and E are normal functions accepting a scalar and returning a scalar, all should work fine.

[Stathis Kritikos]

QC=@(y) -k.*DX0.*(MX0+y(1)).*normcdf(((y(1)-MX0)./DX0)+DX0.*MX0.*k.*normcdf(-MX0./DX0)-k.*(DX0.^2+MX0.^2).*normcdf(-MX0./DX0)+k.*(DX0.^2+MX0.^2).*normcdf(((y(1)-MX0)./DX0)+(1-normcdf((X(1)-MX0)./DX0)+normcdf(-MX0./DX0)).*s+CI))

FC=@(y) fC.*(1-(normcdf((H-2.*pi.*c.*mr.*MF.*y(2)))))./(2.*pi.*c.*y(2).*sqrt(dr.^2.*DF.^2+dr.^2.*MF.^2+mr.^2.*DF.^2))./R(1000)

RC=50

E= @(y)(1./R(1000)).*(y(2).*R(y(2))-1000.*R(1000)+integral(FT,1000,y(2)))

TC=@(y) (QC(y(1))+FC(y(2))+RC)/E(y(2))

All the other letters are constant or values been found.

FT is @(y)t.*fT(y) and

fT = @(y)(H./(sqrt(2.*pi).*b.*y(2).^2)).*exp((-1).*((H-a.*y(2)).^2)./(2.*b.^2.*y(2).^2)).

In the beggining i used h,t till you told me to use y, y(1)=h and y(2)=t. the problem is in the optimization and i dont know how to fix it.

[Torsten]

You don't need to rewrite QC, FC and E in y. Just leave them as they were before.

E.g.

QC=@(h) -k.*DX0.*(MX0+h).*normcdf(((h-MX0)./DX0)+DX0.*MX0.*k.*normcdf(-MX0./DX0)-k.*(DX0.^2+MX0.^2).*normcdf(-MX0./DX0)+k.*(DX0.^2+MX0.^2).*normcdf(((h-MX0)./DX0)+(1-normcdf((X(1)-MX0)./DX0)+normcdf(-MX0./DX0)).*s+CI))

FC = @(t) fC.*(1-(normcdf((H-2.*pi.*c.*mr.*MF.*t))))./(2.*pi.*c.*t.*sqrt(dr.^2.*DF.^2+dr.^2.*MF.^2+mr.^2.*DF.^2))./R(1000)

etc.

[Stathis Kritikos]

i run it succesfully. Thank you so much for your help.

A few answers would be useful for me instead.

Looking the code you send me.

TC = @(x) (QC(x(1))+FC(x(2))+ RC)/E(x(2));

lb = [1.5e-5,1000];

ub = [2.5e-5,100000];

sol = fmincon(TC,[2e-5,10000],[],[],[],[],lb,ub)

The [2e-5,10000] is the x0? x0 means the starting values of h and t?

Thank you

[Torsten]

Two times yes.

[Stathis Kritikos]

ΟΚ. Finally the code you rote me is an SQP method right?

[Torsten]

You can choose the algorithm under "options" :

https://de.mathworks.com/help/optim/ug/fmincon.html#busow0u-1_1

[Stathis Kritikos]

If i choose the code you wrote me before,i have a problem?fmincon is not an sqp method?

[Stathis Kritikos]

Matlab shows to me a result of this format at the end of optimization. what does it mean?

sol =

1.0e+04 *

0.0000 9.9927

Which is the x1 and which is the x2?