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

Question

Siva Rama Krishna Ayaluru il 9 Giu 2023

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https://it.mathworks.com/matlabcentral/answers/1980529-why-the-lsqnonlin-solver-not-able-to-solve-the-problem-when-used-with-multistart

Spostato: Torsten il 9 Giu 2023

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