Enforce condition in lsqnoneg

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sanjay 2024-7-5，8:20

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编辑： Matt J 2024-7-7，20:29

I working on inverse problem which is rank deficient and for that i am using tikhonov regularization , for minization i am using lsqnoneg to resolve it which is giving me good result but now i have to enforce a condition in each iteration of minimzation of lsqnoneg, however lsqnoneg oterations are automatic i cant control it manually althohg i used fmincon but it is not giving me the same results can some one help what algorithm should i use so that i can enforce my conditon during optimization iterations and it gives results like lsqnoneg,

Tikh_Output.IA_recovered_line_1 = lsqnonneg(Combined_Coeff_Matrix,Combined_projection_line_1);

Tikh_Output.IA_recovered_line_2 = lsqnonneg(Combined_Coeff_Matrix,Combined_projection_line_2);

here is my code

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Torsten 2024-7-6，9:30

编辑：Torsten 2024-7-6，9:33

As I said: You can't change solution variables during the optimization process in MATLAB optimizers. The changes must be initiated by your definition of the objective function or by the definition of your constraints. So neither "lsqlin" or "fmincon" can help you in this respect.

But if you want the optimal solution that satisfies x(1) >= x(1601), ... , x(1600) >= x(1601), then the constraint-based solution with "lsqlin" should be correct in my opinion (althogh the result you get might not be as expected).

Do you get the same results with lsqnonneg and lsqlin if you work without the A*x <= b constraint and only set the lower bounds vector lb to zeros(1601,1) ?

sanjay 2024-7-6，9:44

Yes i tried without constraint then the reconstruction results are coming same in lsqin and lsqnoneg,

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Matt J 2024-7-6，11:32

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编辑：Matt J 2024-7-6，11:35

在 MATLAB Online 中打开

Suppose your original problem is,

Rewrite the problem by making the change of variables x=Q*z where z>=0 and

Q=eye(1601); Q(:,end)=1;

Then the problem becomes,

which you can solve with lsqnonneg. After solving for z, you can retrieve x with x=Q*z.

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Matt J 2024-7-7，12:14

编辑：Matt J 2024-7-7，12:38

@sanjay, please indicate whether your comments are directed to me or to Torsten (or somebody else).

Regarding what is is happening during optimization: when lsqnonneg is used to optimize over z, it generates a sequence of iteration vectors z_i, i=0,1,2,... which, because of the particulars of lsqnonneg's algorithm, all satisfy z_i>=0 throughout the optimization. lsqnonneg does not know about x at all or do any manipulations in the space of x.

However, as we have discussed, since the sequence generated by lsqnnnoneg satisfies z_i>=0, then you could create a corresponding sequence x_i=Q*z_i which would satisfy your desired boundary conditions for all x_i.

Also, for the purposes of optimization, it is irrelevant whether your constraints are satisfied at every iteration of the optimization (even if in this case they are). As long as the z_i converges to an optimal solution for z (which it does), then the x_i will converge to an optimal solution for x, because the two problems are in a 1-1 relationship.

Torsten 2024-7-7，20:13

编辑：Torsten 2024-7-7，20:20

在 MATLAB Online 中打开

Both approaches (using lsqlin with lb = 0 and A*x <= b or using lsqnonneg for optimization in z where x is determined after the optimization as x = Q*z) are correct and should give the same result (at least ||C*x-d|| and ||C*Q*z-d|| should be the same)). Test it.

If you don't trust in the appoach with the coordinate transformation

z(i) = x(i) - x(1601) (1 <= i <= 1600)
z(1601) = x(1601)

or explicitly rewritten in x

x(i) = z(i) + z(1601) (1 < = i <= 1600)
x(1601) = z(1601)

or rewritten in matrix form

x = Q*z

use the approach in x.

I don't know what you mean by "recover my X and then apply this condition".

Matt J 2024-7-7，20:24

编辑：Matt J 2024-7-7，20:29

i am enforcing the condition which X(1)>=X(1601),X(2)>=X(1601),,,,,,,,X(1600)>=X(1601) where this condition is satisfied using Q and z approach?

Yes. Torsten already said it, but just for emphasis, the condition X(1)>=X(1601),X(2)>=X(1601),,,,,,,,X(1600)>=X(1601) means the very same thing as the condition z>=0. Therefore, if you have optimized over z subject to z>=0, you have done the very same thing as when you optimize over X subject to the constraints X(1)>=X(1601),X(2)>=X(1601),,,,,,,,X(1600)>=X(1601).

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