Optimization based line fitting

Question

I am using the following optimization script by which I am fitting the following curve y(t)=c_1+c_2 e^{-t} between two points y_1(t_1=0)=1 and y_2(t_2=10)=7

My question is, how can I solve the same optimization problem by adding the constraints y(t=5)>4?

clc;
clear;

tic

%The Data: Time and response data
t = [0 10]';
y = [1 7]';
%Look at the Data
subplot(2,1,1)
plot(t,y,'*','MarkerSize',10)
grid on
xlabel('Time')
ylabel('Response')
hold on
%Curve to Fit
E = [ones(size(t)) exp(-t)]
%Solving constrained linear least squares problem
% cNew = lsqlin(E,y,[],[],[1 1],y(1),[],[],[],opt) % Solver-based approach
p = optimproblem;
c = optimvar('c',2);
p.ObjectiveSense = 'minimize';
p.Objective = sum((E*c-y).^2);
% constraint example: p.Constraints.intercept = c(1) + c(2) == 0.82
sol = solve(p);
cNew = sol.c;
tf = (0:0.1:10)';
Ef = [ones(size(tf)) exp(-tf)];
yhatc = Ef*cNew;
%plot the curve\
subplot(2,1,2)
plot(t,y,'*','MarkerSize',10)
grid on
xlabel('Time')
ylabel('Response')
hold on
plot(tf,yhatc)
title('y(t)=c_1 + c_2e^{-t}')

toc

Answer 1

From MATLAB's documentation, you can add the constraint using

p.Constraints.cons1 = c(1) + c(2) * exp(-5) >= 4;

Also, note that the strict inequality > is not supported by the optimizer for numerical reasons. So if you still want the strict inequality, then add a tolerance such as

tol = 1e-10;
p.Constraints.cons1 = c(1) + c(2) * exp(-5) >= 4 + tol;

Then proceed by solving the optimization as you did in your code.