Why the lsqnonlin solver not able to solve the problem when used with MultiStart?

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Siva Rama Krishna Ayaluru
Spostato: Torsten il 9 Giu 2023
Ran in:
I want to solve two non-linear equations using lsqnonlin solver. For this I created a problem using CreateOptimProblem and solving it using MultiStart since my optimization problem is not convex rather having multiple local minima. I used the below code:
%The code below solves two equations. The initial part of the code generates 10000 values of random data U,V,x_0,y_0 which are seed values for fsolve
clc
clear all
close all
c=4;
d=10^-3;
alpha=1;
n=10;
N1=35;
%Generating 35 values of Gaussian distributed random delay in the range
%5*10^-3 and 6*10^-3
a=5*10^-3;
b=6*10^-3;
X = (b-a).*randn(1,35) + a;
X1 = X;
%Generating T_1_i_A
T_1_i_A=0.0001:0.0001:0.0035;
%Generating T_2_i_B and T_2_i_P from T_1_i_A,d,X
T_2_i_B=zeros(N1);
for j=1:N1
T_2_i_B(j)=T_1_i_A(j)+10*10^-3+2.02*(T_1_i_A(j)-T_1_i_A(1))+15*10^-4+X(j);
end
T_2_i_P=zeros(N1);
for j=1:N1
T_2_i_P(j)=T_1_i_A(j)+20*10^-3+1.01*(T_1_i_A(j)-T_1_i_A(1))+5*10^-4+X1(j);
end
%finding U and V
U=T_2_i_P-T_2_i_B;
V=T_1_i_A-T_1_i_A(1);
fun = @root2d;
%Generating 10000 random values of pre-defined clock offset in the range
%5*10^-6 and 30*10^-6
a=-10*10^-3;
b=10*10^-3;
%a=5*10^-6;
%b=30*10^-6;
x_0 = (b-a).*rand(1,n) + a;
%Generating 10000 random values of pre-defined clock skew in the range
%1.01 and 1.20
a1=0.99;
b1=1.01;
y_0=(b1-a1).*rand(1,n)+a1;
x0=zeros(2,n);
for i=1:n
x0(:,i)=[x_0(i); y_0(i)];
end
maxN=35;
x = zeros(size(x0,1), n, maxN);
for N=5:maxN
for i=1:n
problem=createOptimProblem('lsqnonlin','objective',@(x) root2d(x, U(N),V(N),N,c,d,alpha), 'x0', x0(:,i),'lb', [-10*10^-3;0.99], 'ub', [10*10^-3;1.01],'options',optimoptions(@lsqnonlin,'Algorithm','trust-region-reflective'));
ms = MultiStart('StartPointsToRun','bounds');
x(:,i,N) = run(ms,problem,20);
end
mse_theta_offset_hat_final(N)=mse(x(1,:,N),x_0);
mse_theta_skew_hat_final(N)=mse(x(2,:,N),y_0);
end
MultiStart completed the runs from all start points. All 20 local solver runs converged with a positive local solver exit flag. MultiStart completed the runs from all start points. All 20 local solver runs converged with a positive local solver exit flag. MultiStart completed the runs from all start points. All 20 local solver runs converged with a positive local solver exit flag. MultiStart completed the runs from all start points. All 20 local solver runs converged with a positive local solver exit flag. MultiStart completed the runs from all start points. All 20 local solver runs converged with a positive local solver exit flag. MultiStart completed the runs from all start points. All 20 local solver runs converged with a positive local solver exit flag. MultiStart completed the runs from all start points. All 20 local solver runs converged with a positive local solver exit flag. MultiStart completed the runs from all start points. 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All 20 local solver runs converged with a positive local solver exit flag. MultiStart completed the runs from all start points. All 20 local solver runs converged with a positive local solver exit flag. MultiStart completed the runs from all start points. All 20 local solver runs converged with a positive local solver exit flag. MultiStart completed the runs from all start points. All 20 local solver runs converged with a positive local solver exit flag. MultiStart completed the runs from all start points. All 20 local solver runs converged with a positive local solver exit flag. MultiStart completed the runs from all start points. All 20 local solver runs converged with a positive local solver exit flag. MultiStart completed the runs from all start points. All 20 local solver runs converged with a positive local solver exit flag. MultiStart completed the runs from all start points. All 20 local solver runs converged with a positive local solver exit flag. MultiStart completed the runs from all start points. All 20 local solver runs converged with a positive local solver exit flag. MultiStart completed the runs from all start points. All 20 local solver runs converged with a positive local solver exit flag. MultiStart completed the runs from all start points. All 20 local solver runs converged with a positive local solver exit flag. MultiStart completed the runs from all start points. All 20 local solver runs converged with a positive local solver exit flag. MultiStart completed the runs from all start points. All 20 local solver runs converged with a positive local solver exit flag. MultiStart completed the runs from all start points. All 20 local solver runs converged with a positive local solver exit flag. MultiStart completed the runs from all start points. All 20 local solver runs converged with a positive local solver exit flag. MultiStart completed the runs from all start points. All 20 local solver runs converged with a positive local solver exit flag. MultiStart completed the runs from all start points. All 20 local solver runs converged with a positive local solver exit flag. MultiStart completed the runs from all start points. All 20 local solver runs converged with a positive local solver exit flag. MultiStart completed the runs from all start points. All 20 local solver runs converged with a positive local solver exit flag. MultiStart completed the runs from all start points. All 20 local solver runs converged with a positive local solver exit flag. MultiStart completed the runs from all start points. All 20 local solver runs converged with a positive local solver exit flag. MultiStart completed the runs from all start points. All 20 local solver runs converged with a positive local solver exit flag. MultiStart completed the runs from all start points. All 20 local solver runs converged with a positive local solver exit flag. MultiStart completed the runs from all start points. All 20 local solver runs converged with a positive local solver exit flag. MultiStart completed the runs from all start points. All 20 local solver runs converged with a positive local solver exit flag. MultiStart completed the runs from all start points. All 20 local solver runs converged with a positive local solver exit flag. MultiStart completed the runs from all start points. All 20 local solver runs converged with a positive local solver exit flag. MultiStart completed the runs from all start points. All 20 local solver runs converged with a positive local solver exit flag. MultiStart completed the runs from all start points. All 20 local solver runs converged with a positive local solver exit flag. MultiStart completed the runs from all start points. All 20 local solver runs converged with a positive local solver exit flag. MultiStart completed the runs from all start points. All 20 local solver runs converged with a positive local solver exit flag. MultiStart completed the runs from all start points. All 20 local solver runs converged with a positive local solver exit flag. MultiStart completed the runs from all start points. All 20 local solver runs converged with a positive local solver exit flag. MultiStart completed the runs from all start points. All 20 local solver runs converged with a positive local solver exit flag. MultiStart completed the runs from all start points. All 20 local solver runs converged with a positive local solver exit flag. MultiStart completed the runs from all start points. All 20 local solver runs converged with a positive local solver exit flag. MultiStart completed the runs from all start points. All 20 local solver runs converged with a positive local solver exit flag. MultiStart completed the runs from all start points. All 20 local solver runs converged with a positive local solver exit flag. MultiStart completed the runs from all start points. All 20 local solver runs converged with a positive local solver exit flag. MultiStart completed the runs from all start points. All 20 local solver runs converged with a positive local solver exit flag. MultiStart completed the runs from all start points. All 20 local solver runs converged with a positive local solver exit flag. MultiStart completed the runs from all start points. All 20 local solver runs converged with a positive local solver exit flag. MultiStart completed the runs from all start points. All 20 local solver runs converged with a positive local solver exit flag. MultiStart completed the runs from all start points. All 20 local solver runs converged with a positive local solver exit flag. MultiStart completed the runs from all start points. All 20 local solver runs converged with a positive local solver exit flag. MultiStart completed the runs from all start points. All 20 local solver runs converged with a positive local solver exit flag. MultiStart completed the runs from all start points. All 20 local solver runs converged with a positive local solver exit flag. MultiStart completed the runs from all start points. All 20 local solver runs converged with a positive local solver exit flag.
%Plots
k=5:N1;
subplot(2,1,1),plot(k,mse_theta_offset_hat_final(5:N1),'r+-')
title('clock offset')
xlabel('No. of Observations')
ylabel('MSE')
grid on
subplot(2,1,2),plot(k,mse_theta_skew_hat_final(5:N1),'-*')
title('clock skew')
xlabel('No. of Observations')
ylabel('MSE')
grid on
function f = root2d(x, U, V, N,c,d,alpha)
f(1)=(1.5./(U-x(1)-x(2)*V+d)-(c/2)./(U-x(1)-x(2)*V+d).^2)*N+alpha*sum(1:N);
f(2)=((1.5*V)./(U-x(1)-x(2)*V+d)-(c/2)*V./(U-x(1)-x(2)*V+d).^2+alpha*V)*N;
end

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Risposte (1)

Torsten
Torsten il 9 Giu 2023
Spostato: Torsten il 9 Giu 2023
If you substitute
y = 1/(U-x(1)-x(2)*V+d)
you will see that you have two equations in one unknown, namely y.
So most probably, you won't get a solution that satisfies both equations exactly.
And if you aim at the least-squares solution y*, there will most probably be infinitly many combinations for x(1) and x(2) such that
y* = 1/(U-x(1)-x(2)*V+d)
In other words: Your code will produce garbage.

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