How can i solve this optimization problem?

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Hi,
My cost function is:
clear all
clc
syms delta_Uk
F = rand(4,5);
Xf = rand(5,1);
phi = rand(4,2);
Q = rand(4,4);
R_bar = rand(2,2);
Rs = rand(4,1);
YY = F * Xf + phi * delta_Uk;
J = (transpose(Rs - YY)) * Q * (Rs - YY) + (transpose(delta_Uk)) * R_bar * delta_Uk;
Which command / optimization technique may be used to find delta_Uk which make J minimum? There is no contraint on this problem.
Thanks,

Accepted Answer

Bjorn Gustavsson
Bjorn Gustavsson on 31 Aug 2021
Best route might be to properly differentiate J with respect to delta_Uk and solve the normal equations and find an analytical solution to that system of equations, and check/verify that it is the global minimum - this is after all some kind of 2-D quadratic equation. It should be possible to solve with the symbolic tools available too for general forms of the constant matrices if you define those variables as arrays:
syms F [4 5]
and so on. For a numerical solution you simply have to define YY and J as functions which you can do like this:
F = rand(4,5);
Xf = rand(5,1);
phi = rand(4,2);
Q = rand(4,4);
R_bar = rand(2,2);
Rs = rand(4,1);
YY = @(delta_Uk) F * Xf + phi * delta_Uk;
J = @(dUk) (transpose(Rs - YY(dUk))) * Q * (Rs - YY(dUk)) + (transpose(dUk)) * R_bar * dUk;
This gives you your cost-function J as a function-handle which you can treat pretty much identically like ordinary matlab-function. To minimize it you can simply call fminsearch:
best_dUk = fminsearch(J,[1;2])
Which should give you the optimal value for delta_Uk.
HTH
  3 Comments
Volkan Yangin
Volkan Yangin on 31 Aug 2021
Finally, i want to ask another question. If we have constraints, is there any alternative way instead of fminsearch?
Bjorn Gustavsson
Bjorn Gustavsson on 31 Aug 2021
My pleasure.
You have at least two FEX-submissions that helps you with that problem: fminsearchbnd and minimize. Then you have the optimisation-toolbox functions fmincon (and possibly lsqnonlin if you can differentiate your quadratic equation into normal-equations).

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