Optimization of vectors with optimization toolbox

hello, new to optimization toolbox here :)

i wolud like to optimize 2 vectors:

omega (100X1)

T (100X1)

i have 2 constrains on this vectors

-200<omega>200

-60<T<60

H it's a matrix equal to :H = [omega 1 T]

and M = transpose(H)* H;

i wrote my M matrix result alredy.

my code look like this:

omega = optimvar('omega',100,'LowerBound',-200,'UpperBound',200);
T     = optimvar('Temperature',100,'LowerBound',-60,'UpperBound',60);
t     = transpose(0:1:100-1); 
M = [sum(omega.*omega) sum(omega) sum(omega.*T) ; ...
     sum(omega) sum(ones(length(omega),1)) sum(T) ;
     sum(omega.*T) sum(T) sum(T.*T) ];

i want to optimize omega and T with this two objective functions

1.trace(M)

2.Det(M)

J1 = trace(M);
J2 = det(M);
prob  = optimproblem("ObjectiveSense",'maximize');
prob.Objective =J1;
sol = solve(prob)

my questions:

1. i would like to add 2 more constrains, about domega/dt and for dT/dt. any one have idea how to do that?
2. when i try to solve this problem, i get :The problem is non-convex. how sould i continue?
3. when i try to calculate det(M) i get this error : Check for missing argument or incorrect argument data type in call to function 'det'. can i even use this function with optimization varible ?

[Walter Roberson]

No, det() is not defined for optimvar .

Fortunately you have a 3 x 3 system so you can construct the determinant

M1_1 = sum(omega.^2)
M1_2 = sum(omega)
M1_3 = sum(omega.*T)
and so on
detM = M1_1*M2_2*M3_3 - M1_1*M2_3*M3_2 - M1_2*M2_1*M3_3 + M1_2*M2_3*M3_1 + M1_3*M2_1*M3_2 - M1_3*M2_2*M3_1

[eden meirovich]

is there another way? i'm working with 3X3 system now, but later on i will have 7X7 or 8X8 systems even.

is there a way to use the symbolic tool box maybe?

[Bruno Luong]

Use solver based rather than problem based.

BTW I can't see t is used anywhere.

You might revise how you formulate the optimization problem.

[eden meirovich]

t is just a vector steps time.

do you have an idea which solver shoud i use for this kind of problem?

[Matt J]

i want to optimize omega and T with this two objective functions...

The solutions are trivially omega=T=zeros(100,1) for both objective functions, unless the constraints on the derivatives that you intend to impose bound at least some of the derivatves strictly away from zero.

[Bruno Luong]

FMINCON?

[eden meirovich]

i know the solution (i try to max the trace and the det, not minimize).

i just want to get this "easy" one, before i continue to one i don't know

Answer 1

[Matt J]

i would like to add 2 more constrains, about domega/dt and for dT/dt. any one have idea how to do that?

You can use diff() to implement constraints on numerical derivatives, e.g.,

omega = optimvar('omega',100,'LowerBound',-200,'UpperBound',200);
T     = optimvar('Temperature',100,'LowerBound',-60,'UpperBound',60);

domega=diff(omega); %derivatives
dT=diff(T);

C.con_domega=lower_domega<=domega & domega<=upper_domega; %bounds on derivatives
C.con_dT=lower_dT<=dT & dT<=upper_dT

As long as all these constraints are linear, you can then use prob2matrices,

https://www.mathworks.com/matlabcentral/fileexchange/74481-prob2matrices-a-selective-version-of-prob2struct

to assist in the conversion from the problem-based to the solver-based framework. E.g.,

sol0.omega=omega0; %initial guesses
sol0.T=T0;

prob=prob2matrices({omega,T},'Constraints',C,'sol0',sol0);
  prob.x0=reshape(prob.x0,[],2);
  prob.solver='fmincon';

 prob.fun=@(x) sum(x.^2,'all');  %equivalent to trace M
x1=fmincon(prob);

  e = ones(length(omega),1);
  prob.fun=@(x) det([x,e].' * [x,e]);
x2=fmincon(prob);