This example shows how to solve a scalar minimization problem
with nonlinear inequality constraints. The problem is to find *x* that
solves

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

subject to the constraints

*x*_{1}*x*_{2} – *x*_{1} – *x*_{2} ≤
–1.5,

*x*_{1}*x*_{2} ≥
–10.

Because neither of the constraints is linear, you cannot pass
the constraints to `fmincon`

at
the command line. Instead you can create a second file, `confun.m`

,
that returns the value at both constraints at the current `x`

in
a vector `c`

. The constrained optimizer, `fmincon`

,
is then invoked. Because `fmincon`

expects the constraints
to be written in the form *c*(*x*) ≤ 0, you
must rewrite your constraints in the form

x_{1}x_{2} – x_{1} – x_{2} +
1.5 ≤ 0,– x_{1}x_{2} –10
≤ 0. | (2) |

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);

function [c, ceq] = confun(x) % Nonlinear inequality constraints c = [1.5 + x(1)*x(2) - x(1) - x(2); -x(1)*x(2) - 10]; % Nonlinear equality constraints ceq = [];

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

`fmincon`

produces the solution `x`

with
function value `fval`

:

x,fval x = -9.5474 1.0474 fval = 0.0236

You can evaluate the constraints at the solution by entering

[c,ceq] = confun(x)

This returns numbers close to zero, such as

c = 1.0e-14 * 0.5107 -0.5329 ceq = []

Note that both constraint values are, to within a small tolerance,
less than or equal to 0; that is, `x`

satisfies *c*(*x*) ≤ 0.