Using 'fminunc', I receive Optimization completed because the size of the gradient is less than the value of the optimality tolerance.

Question

Ahmed Nasr 2020-5-21

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此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/528963-using-fminunc-i-receive-optimization-completed-because-the-size-of-the-gradient-is-less-than-the

编辑： Matt J 2020-5-24

I use 'fminunc' to find the value of B (2*4 matrix) that minimzes the difference between the corresponding elements in two vectors as indicated in the attached code. In other words, I want to find the B (2*4 matrix) that makes the elements of beta_d (1*4 vector) which is a function of B matrix, equal to the corresponding ones of a "given" beta_u (1*4 vector), i.e. beta_d(1,1) = beta_u(1,1) && beta_d(1,2) = beta_u(1,2) && beta_d(1,3) = beta_u(1,3) && beta_d(1,4) = beta_u(1,4).

However, I usually receive the following message without getting any result and the program seems to go on an infinite loop!

Local minimum found.
Optimization completed because the size of the gradient is less than
the value of the optimality tolerance.
<stopping criteria details>

the code:

clc;
clear;
% System paramters:
N = 2;
K = 4;
C_l = 4;
H = [-0.3208  -0.9784; -1.5994   -1.4689; -1.5197   -0.4568; -0.0993   -0.7667]; % 4*2 matrix
A = [-1 1; 0 1]; % 2*2 matrix
C = [-0.20  0.4  0.6 -0.2; -0.2  0.4  0.6 -0.2; 0.4  0.2 -0.2  0.4; 0.4  0.2 -0.2  0.4]; % 4*4 matrix
P = [250000   0   0   0; 0  250000  0   0; 0   0  250000  0; 0  0  0  250000]; % 4*4 matrix
beta_u = [50.2207   50.2207   20.3433   20.3433]; % 1*4 vector
beta_d = zeros(1,4); % intial value
B = zeros(2,4); %intial value
%store inputs to a struct for shorter syntax
s=struct;[s.H,s.A,s.C,s.P,s.C_l,s.N,s.K]=deal(H,A,C,P,C_l,N,K);
%fminunc optimization
while (sum(abs(beta_u-beta_d))>=0.1)
    initial_guess = randn(2,4);
    OLS = @(B_d,input_vars)sum((beta_u-myfun(B_d,input_vars)).^2);%ordinary least squares cost function
    opts = optimoptions(@fminunc,'MaxIterations',10000,'MaxFunctionEvaluations',50000,'CheckGradients',true);
    B = fminunc(OLS, initial_guess, opts,s);
    input_vars = s;
    [beta_d, D_d]=myfun(B,input_vars);      
end
%calculate beta_d from B and the other inputs
function [beta_d, D_d]=myfun(B,input_vars)
%load parameters
s=input_vars;[H,A,C,P,C_l,N,K]=deal(s.H,s.A,s.C,s.P,s.C_l,s.N,s.K);
for j=1:1:N
    d(j) = (B(j,:)*P*B(j,:)')/((2^(2*C_l))-(norm(A(:,j))^2));
end
D_d = diag(d);
for i=1:1:K   
    V_d(i)=C(i,:)*P*B'*H(i,:)'*inv(1+H(i,:)*(A'*D_d*A+B*P*B')*H(i,:)');
    sigma_d(i)=norm((V_d(i)*H(i,:)*B-C(i,:))*(P^(1/2)))^2+(V_d(i)^2)*(1+H(i,:)*A'*D_d*A*H(i,:)');
    beta_d(i)=((P(i,i))/sigma_d(:,i));   
end
end

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Answer 1

Matt J 2020-5-21

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链接

此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/528963-using-fminunc-i-receive-optimization-completed-because-the-size-of-the-gradient-is-less-than-the#answer_435178

编辑：Matt J 2020-5-21

在 MATLAB Online 中打开

If the problem doesn't have a solution satisfying the tolerance condition in your while loop

(sum(abs(beta_u-beta_d))<0.1)

then the loop has no way of stopping.

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Ahmed Nasr 2020-5-21

在 MATLAB Online 中打开

But, when I change the values of the given H, A, C, P matrices and beta_u vector, the fminunc sometimes gives me a result matched with the while function! .. I think this means that the math problem is solvable such as:

clc;
clear;
% System paramters:
N = 2;
K = 4;
C_l = 4;
% >>> New values 
H = [-0.5119 -0.8390; -0.0719  0.6697; 1.4303  2.0519; 1.4391 0.6676]
A = [-1 1; 1 0];
C = [-0.20  0.4  0.2 -0.6; -0.2  0.4  0.2 -0.6; -0.0154 0.1077 -0.0615 -0.2769; -0.0154 0.1077 -0.0615 -0.2769];
P = [2500   0   0   0; 0  2500  0   0; 0   0  2500  0; 0  0  0  2500];
beta_u = [22.6467   22.6467   17.5767   17.5767];
beta_d = zeros(1,4); % intial value
%store inputs to a struct for shorter syntax
s=struct;[s.H,s.A,s.C,s.P,s.C_l,s.N,s.K]=deal(H,A,C,P,C_l,N,K);
while (sum(abs(beta_u-beta_d))>=0.1)
    initial_guess = randn(2,4);
    OLS = @(B_d,input_vars)sum((beta_u-myfun(B_d,input_vars)).^2);%ordinary least squares cost function
    opts = optimoptions(@fminunc,'MaxIterations',10000,'MaxFunctionEvaluations',50000,'CheckGradients',true);
    B = fminunc(OLS, initial_guess, opts,s);
    input_vars = s;
    [beta_d, D_d]=myfun(B,input_vars);      
end
%calculate beta_d from B and the other inputs
function [beta_d, D_d]=myfun(B,input_vars)
%load parameters
s=input_vars;[H,A,C,P,C_l,N,K]=deal(s.H,s.A,s.C,s.P,s.C_l,s.N,s.K);
for j=1:1:N
    d(j) = (B(j,:)*P*B(j,:)')/((2^(2*C_l))-(norm(A(:,j))^2));
end
D_d = diag(d);
for i=1:1:K   
    V_d(i)=C(i,:)*P*B'*H(i,:)'*inv(1+H(i,:)*(A'*D_d*A+B*P*B')*H(i,:)');
    sigma_d(i)=norm((V_d(i)*H(i,:)*B-C(i,:))*(P^(1/2)))^2+(V_d(i)^2)*(1+H(i,:)*A'*D_d*A*H(i,:)');
    beta_d(i)=((P(i,i))/sigma_d(:,i));   
end
end

The optimization is solved correctly, and the obtained beta_d is equal to beta_u as required as:

beta_d =
   22.6467   22.6467   17.5767   17.5767

I want to know why when I use the values of H, P, A, C, and beta_u indicated in the question, the optimization is not solved?!

Ahmed Nasr 2020-5-22

在 MATLAB Online 中打开

This code sometimes gives me a result, and sometimes gives me the previous mentioned error!

clc;
clear;
% System paramters:
N = 2;
K = 4;
C_l = 4;
counter = 0;
H = [-0.3208  -0.9784; -1.5994   -1.4689; -1.5197   -0.4568; -0.0993   -0.7667]; % 4*2 matrix
A = [-1 1; 0 1]; % 2*2 matrix
C = [-0.20  0.4  0.6 -0.2; -0.2  0.4  0.6 -0.2; 0.4  0.2 -0.2  0.4; 0.4  0.2 -0.2  0.4]; % 4*4 matrix
P = [250000   0   0   0; 0  250000  0   0; 0   0  250000  0; 0  0  0  250000]; % 4*4 matrix
beta_u = [50.2207   50.2207   20.3433   20.3433]; % 1*4 vector
beta_d = zeros(1,4); % intial value
%store inputs to a struct for shorter syntax
s=struct;[s.H,s.A,s.C,s.P,s.C_l,s.N,s.K]=deal(H,A,C,P,C_l,N,K);
while (sum(abs(beta_u-beta_d))>0.1 && counter< 500)
    initial_guess = randn(2,4);
    OLS = @(B_d,input_vars)sum(abs(beta_u-myfun(B_d,input_vars)).^2);%ordinary least squares cost function
    opts = optimoptions(@fminunc,'MaxIterations',10000,'MaxFunctionEvaluations',50000);
    [B,FVAL,grad] = fminunc(OLS, initial_guess, opts,s);
    input_vars = s;
    [beta_d, D_d]=myfun(B,input_vars);
    counter = counter+1;
end
%calculate beta_d from B and the other inputs
function [beta_d, D_d]=myfun(B,input_vars)
%load parameters
s=input_vars;[H,A,C,P,C_l,N,K]=deal(s.H,s.A,s.C,s.P,s.C_l,s.N,s.K);
for j=1:1:N
    d(j) = (B(j,:)*P*B(j,:)')/((2^(2*C_l))-(norm(A(:,j))^2));
end
D_d = diag(d);
for i=1:1:K   
    V_d(i)=C(i,:)*P*B'*H(i,:)'*inv(1+H(i,:)*(A'*D_d*A+B*P*B')*H(i,:)');
    sigma_d(i)=norm((V_d(i)*H(i,:)*B-C(i,:))*(P^(1/2)))^2+(V_d(i)^2)*(1+H(i,:)*A'*D_d*A*H(i,:)');
    beta_d(i)=((P(i,i))/sigma_d(:,i));   
end
end

Matt J 2020-5-24

编辑：Matt J 2020-5-24

fminunc is not succeeding in approximating the gradient of your objective function accurately using its default finite difference method. You could try supplying your own gradient calculation using the SpecifyObjectiveGradient option,

https://www.mathworks.com/help/optim/ug/writing-scalar-objective-functions.html#bu2w6h1

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