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## Nonlinear Equality and Inequality Constraints

You can include nonlinear constraints by writing a function that computes both equality and inequality constraint values. A nonlinear constraint function has the syntax

`[c,ceq] = nonlinconstr(x)`

The function `c(x)` represents the constraint ```c(x) <= 0```. The function `ceq(x)` represents the constraint `ceq(x) = 0`.

### Note

You must have the nonlinear constraint function return both `c(x)` and `ceq(x)`, even if you have only one type of nonlinear constraint. If a constraint does not exist, have the function return `[]` for that constraint.

For example, if you have the nonlinear equality constraint ${x}_{1}^{2}+{x}_{2}=1$ and the nonlinear inequality constraint x1x2 ≥ –10, rewrite them as

`$\begin{array}{c}{x}_{1}^{2}+{x}_{2}-1=0,\\ -{x}_{1}{x}_{2}-10\le 0,\end{array}$`

and then solve the problem using the following steps.

For this example, solve the problem

`$\underset{x}{\mathrm{min}}f\left(x\right)={e}^{{x}_{1}}\left(4{x}_{1}^{2}+2{x}_{2}^{2}+4{x}_{1}{x}_{2}+2{x}_{2}+1\right).$`

subject to these nonlinear constraints.

### Step 1: Write a file objfun.m.

```function f = objfun(x) f = exp(x(1))*(4*x(1)^2+2*x(2)^2+4*x(1)*x(2)+2*x(2)+1);```

### Step 2: Write a file confuneq.m for the nonlinear constraints.

```function [c,ceq] = confuneq(x) % Nonlinear inequality constraints c = -x(1)*x(2) - 10; % Nonlinear equality constraints ceq = x(1)^2 + x(2) - 1;```

### Step 3: Invoke constrained optimization routine.

```x0 = [-1,1]; % Make a starting guess at the solution options = optimoptions(@fmincon,'Algorithm','sqp'); [x,fval] = fmincon(@objfun,x0,[],[],[],[],[],[],... @confuneq,options);```

After 21 function evaluations, the solution produced is

```x,fval x = -0.7529 0.4332 fval = 1.5093 [c,ceq] = confuneq(x) % Check the constraint values at x c = -9.6739 ceq = -2.2204e-16```

Note that `ceq` is equal to 0 within the default tolerance on the constraints of `1.0e-006` and that `c` is less than or equal to 0, as desired.

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