fsolve solves a system of nonlinear equations. However, it
does not allow you to include any constraints, even bound constraints. So how can
you solve a system of nonlinear equations when you have constraints?
A solution that satisfies your constraints is not guaranteed to exist. In fact, the problem might not have any solution, even one that does not satisfy your constraints. However techniques exist to help you search for solutions that satisfy your constraints.
To illustrate the techniques, consider how to solve the equations
where the components of x must be nonnegative. The equations have four solutions:
x = (–1,–2)
x = (10,–2),
x = (–1,20),
x = (10,20).
Only one solution satisfies the constraints, namely x = (10,20).
To solve the equations numerically, first enter code to calculate F(x).
function F = fbnd(x) F(1) = (x(1)+1)*(10-x(1))*(1+x(2)^2)/(1+x(2)^2+x(2)); F(2) = (x(2)+2)*(20-x(2))*(1+x(1)^2)/(1+x(1)^2+x(1));
Save this code as the file
fbnd.m on your MATLAB® path.
Generally, a system of N equations in N
variables has isolated solutions, meaning each solution has no nearby neighbors that
are also solutions. So, one way to search for a solution that satisfies some
constraints is to generate a number of initial points
fsolve starting at each
For this example, to look for a solution to Equation 1, take 10 random points that are normally distributed with mean 0 and standard deviation 100.
rng default % For reproducibility N = 10; % Try 10 random start points pts = 100*randn(N,2); % Initial points are rows in pts soln = zeros(N,2); % Allocate solution opts = optimoptions('fsolve','Display','off'); for k = 1:N soln(k,:) = fsolve(@fbnd,pts(k,:),opts); % Find solutions end
Examine the solutions in
soln, and note that several satisfy
fsolve has three algorithms. Each can lead to different
For this example, take
x0 = [1,9] and examine the
solution each algorithm returns.
x0 = [1,9]; opts = optimoptions(@fsolve,'Display','off',... 'Algorithm','trust-region-dogleg'); x1 = fsolve(@fbnd,x0,opts)
x1 = -1.0000 -2.0000
opts.Algorithm = 'trust-region'; x2 = fsolve(@fbnd,x0,opts)
x2 = -1.0000 20.0000
opts.Algorithm = 'levenberg-marquardt'; x3 = fsolve(@fbnd,x0,opts)
x3 = 0.9523 8.9941
Here, all three algorithms find different solutions for the same initial point. In
x3 is not even a solution, but is simply a locally
lsqnonlin tries to minimize the sum of squares of the
components in a vector function F(x).
Therefore, it attempts to solve the equation F(x) = 0. Also,
lsqnonlin accepts bound
Formulate the example problem for
lsqnonlin and solve
lb = [0,0]; rng default x0 = 100*randn(2,1); [x,res] = lsqnonlin(@fbnd,x0,lb)
x = 10.0000 20.0000 res = 2.4783e-25
You can use
lsqnonlin with the Global Optimization
MultiStart solver to search over many initial
points automatically. See MultiStart Using lsqcurvefit or lsqnonlin (Global Optimization Toolbox).
You can reformulate the problem and use
Give a constant objective function, such as
which evaluates to
0 for each
fsolve objective function as the nonlinear
equality constraints in
Give any other constraints in the usual
For this example, write a function file for the nonlinear constraints.
function [c,ceq] = fminconstr(x) c = ; % No nonlinear inequality ceq = fbnd(x); % fsolve objective is fmincon nonlinear equality constraints
Save this code as the file
fminconstr.m on your MATLAB path.
Solve the constrained problem.
lb = [0,0]; % Lower bound constraint rng default % Reproducible initial point x0 = 100*randn(2,1); opts = optimoptions(@fmincon,'Algorithm','interior-point','Display','off'); x = fmincon(@(x)0,x0,,,,,lb,,@fminconstr,opts)
x = 10.0000 20.0000