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This example is a nonlinear minimization problem with all possible types of constraints. The example does not use gradients.

The problem has five variables, `x(1)`

through `x(5)`

. The objective function is a polynomial in the variables.

$$f(x)=6{x}_{2}{x}_{5}+7{x}_{1}{x}_{3}+3{x}_{2}^{2}$$.

The objective function is in the local function `myobj(x)`

, which is nested inside the function `fullexample`

. The code for `fullexample`

appears at the end of this example.

The nonlinear constraints are likewise polynomial expressions.

$${x}_{1}-0.2{x}_{2}{x}_{5}\le 71$$

$$0.9{x}_{3}-{x}_{4}^{2}\le 67$$

$$3{x}_{2}^{2}{x}_{5}+3{x}_{1}^{2}{x}_{3}=20.875$$.

The nonlinear constraints are in the local function `myconstr(x)`

, which is nested inside the function `fullexample`

.

The problem has bounds on $${x}_{3}$$ and $${x}_{5}$$.

$$0\le {x}_{3}\le 20$$, $${x}_{5}\ge 1$$.

The problem has a linear equality constraint $${x}_{1}=0.3{x}_{2}$$, which you can write as $${x}_{1}-0.3{x}_{2}=0$$.

The problem also has three linear inequality constraints:

$$\begin{array}{c}0.1{x}_{5}\le {x}_{4}\\ {x}_{4}\le 0.5{x}_{5}\\ 0.9{x}_{5}\le {x}_{3}.\end{array}$$

Represent the bounds and linear constraints as matrices and vectors. The code that creates these arrays is in the `fullexample`

function. As described in the `fmincon`

Input Arguments section, the `lb`

and `ub`

vectors represent the constraints

`lb`

$$\le x\le $$ `ub`

.

The matrix `A`

and vector `b`

represent the linear inequality constraints

`A*x`

$$\le $$ `b`

,

and the matrix `Aeq`

and vector `beq`

represent the linear equality constraints

`Aeq*x = b`

.

Call `fullexample`

to solve the minimization problem subject to all types of constraints.

[x,fval,exitflag] = fullexample

Local minimum found that satisfies the constraints. Optimization completed because the objective function is non-decreasing in feasible directions, to within the value of the optimality tolerance, and constraints are satisfied to within the value of the constraint tolerance.

`x = `*5×1*
0.6114
2.0380
1.3948
0.1572
1.5498

fval = 37.3806

exitflag = 1

The exit flag value of `1`

indicates that `fmincon`

converges to a local minimum that satisfies all of the constraints.

This code creates the `fullexample`

function, which contains the nested functions `myobj`

and `myconstr`

.

function [x,fval,exitflag] = fullexample x0 = [1; 4; 5; 2; 5]; lb = [-Inf; -Inf; 0; -Inf; 1]; ub = [ Inf; Inf; 20; Inf; Inf]; Aeq = [1 -0.3 0 0 0]; beq = 0; A = [0 0 0 -1 0.1 0 0 0 1 -0.5 0 0 -1 0 0.9]; b = [0; 0; 0]; opts = optimoptions(@fmincon,'Algorithm','sqp'); [x,fval,exitflag] = fmincon(@myobj,x0,A,b,Aeq,beq,lb,ub,... @myconstr,opts); %--------------------------------------------------------- function f = myobj(x) f = 6*x(2)*x(5) + 7*x(1)*x(3) + 3*x(2)^2; end %--------------------------------------------------------- function [c, ceq] = myconstr(x) c = [x(1) - 0.2*x(2)*x(5) - 71 0.9*x(3) - x(4)^2 - 67]; ceq = 3*x(2)^2*x(5) + 3*x(1)^2*x(3) - 20.875; end end