Nonlinear Equality and Inequality Constraints

This example shows how to solve an optimization problem containing nonlinear constraints. 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.

Nonlinear Constraints

Suppose you have the nonlinear equality constraint

$$x_1^2+x_2=1$$

and the nonlinear inequality constraint

$$x_1{x}_2\ge -10$$.

Rewrite these constraints as

$$\begin{array}{c}{x}_1^2+x_2-1=0\\ -x_1{x}_2-10\le 0.\end{array}$$

The confuneq helper function at the end of this example implements these inequalities in the correct syntax.

Objective Function

Solve the problem

$$\underset{x}{\min }f(x)=e^{x_1}\left(4x_1^2+2x_2^2+4x_1{x}_2+2x_2+1\right)$$

subject to the constraints. The objfun helper function at the end of this example implements this objective function.

Solve Problem

Solve the problem by calling the fmincon solver. This solver requires an initial point; use the point x0 = [-1,-1].

x0 = [-1,-1];

The problem has no bounds or linear constraints, so set those inputs to [].

A = [];
b = [];
Aeq = [];
beq = [];
lb = [];
ub = [];

Call the solver.

[x,fval] = fmincon(@objfun,x0,A,b,Aeq,beq,lb,ub,@confuneq)

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 = 1×2

   -0.7529    0.4332

fval = 
1.5093

The solver reports that the constraints are satisfied at the solution. Check the nonlinear constraints at the solution.

[c,ceq] = confuneq(x)

c = 
-9.6739

ceq = 
2.0668e-12

c is less than 0, as required. ceq is equal to 0 within the default constraint tolerance of 1e-6.

Helper Functions

The following code creates the confuneq helper function.

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

The following code creates the objfun helper function.

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);
end

Related Topics

• Nonlinear Inequality Constraints
• Optimize Live Editor Task with fmincon Solver