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## Nonlinear Systems with Constraints

### Solve Equations with Inequality Constraints

`fsolve` solves systems of nonlinear equations. However, it does not allow you to include any constraints, even bound constraints. The question is, how can you solve systems of nonlinear equations when you have constraints?

The short answer is, there are no guarantees that a solution exists that satisfies your constraints. There is no guarantee that any solution exists, even one that does not satisfy your constraints. Nevertheless, there are techniques that can help you search for solutions that satisfy your constraints.

To illustrate the techniques, consider how to solve the equations

 $\begin{array}{c}{F}_{1}\left(x\right)=\left({x}_{1}+1\right)\left(10-{x}_{1}\right)\frac{1+{x}_{2}^{2}}{1+{x}_{2}^{2}+{x}_{2}}\\ {F}_{2}\left(x\right)=\left({x}_{2}+2\right)\left(20-{x}_{2}\right)\frac{1+{x}_{1}^{2}}{1+{x}_{1}^{2}+{x}_{1}},\end{array}$ (1)

where the components of x must be nonnegative. Clearly, there are four solutions to the equations:

x = (–1,–2)
x = (10,–2),
x = (–1,20),
x = (10,20).

There is only one solution that 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.

### Use Different Start Points

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 `x0`, and run `fsolve` starting at each `x0`.

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 you find several that satisfy the constraints.

### Use Different Algorithms

There are three `fsolve` algorithms. Each can lead to different solutions.

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 fact, `x3` is not even a solution, but is simply a locally stationary point.

### Use lsqnonlin with Bounds

`lsqnonlin` tries to minimize the sum of squares of the components of a vector function F(x). Therefore, it attempts to solve the equation F(x) = 0. Furthermore, `lsqnonlin` accepts bound constraints.

Formulate the example problem for `lsqnonlin` and solve it.

```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 Toolbox `MultiStart` solver to search over many initial points automatically. See MultiStart Using lsqcurvefit or lsqnonlin (Global Optimization Toolbox).

### Set Equations and Inequalities as fmincon Constraints

You can reformulate the problem and use `fmincon` as follows:

• Give a constant objective function, such as `@(x)0`, which evaluates to `0` for each `x`.

• Set the `fsolve` objective function as the nonlinear equality constraints in `fmincon`.

• Give any other constraints in the usual `fmincon` syntax.

For this example, write a function file for the nonlinear inequality constraint.

```function [c,ceq] = fminconstr(x) c = []; % no nonlinear inequality ceq = fbnd(x); % the fsolve objective is fmincon 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```