Nonlinear Constraints

Several optimization solvers accept nonlinear constraints, including fmincon, fseminf, fgoalattain, fminimax, and the Global Optimization Toolbox solvers ga (Global Optimization Toolbox), gamultiobj (Global Optimization Toolbox), patternsearch (Global Optimization Toolbox), paretosearch (Global Optimization Toolbox), GlobalSearch (Global Optimization Toolbox), and MultiStart (Global Optimization Toolbox). Nonlinear constraints allow you to restrict the solution to any region that can be described in terms of smooth functions.

Nonlinear inequality constraints have the form c(x) ≤ 0, where c is a vector of constraints, one component for each constraint. Similarly, nonlinear equality constraints have the form ceq(x) = 0.

Note

Nonlinear constraint functions must return both c and ceq, the inequality and equality constraint functions, even if they do not both exist. Return an empty entry [] for a nonexistent constraint.

For example, suppose that you have the following inequalities as constraints:

$$\begin{array}{c}\frac{x_1^2}{9}+\frac{x_2^2}{4}\le 1,\\ x_2\ge x_1^2-1.\end{array}$$

Write these constraints in a function file as follows:

function [c,ceq] = ellipseparabola(x)
c(1) = (x(1)^2)/9 + (x(2)^2)/4 - 1;
c(2) = x(1)^2 - x(2) - 1;
ceq = [];
end

ellipseparabola returns an empty entry [] for ceq, the nonlinear equality constraint function. Also, the second inequality is rewritten to ≤ 0 form.

Minimize the function exp(x(1) + 2*x(2)) subject to the ellipseparabola constraints.

fun = @(x)exp(x(1) + 2*x(2));
nonlcon = @ellipseparabola;
x0 = [0 0];
A = []; % No other constraints
b = [];
Aeq = [];
beq = [];
lb = [];
ub = [];
x = fmincon(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon)

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 =

   -0.2500   -0.9375

Including Gradients in Constraint Functions

If you provide gradients for c and ceq, the solver can run faster and give more reliable results.

Providing a gradient has another advantage. A solver can reach a point x such that x is feasible, but finite differences around x always lead to an infeasible point. In this case, a solver can fail or halt prematurely. Providing a gradient allows a solver to proceed.

To include gradient information, write a conditionalized function as follows:

function [c,ceq,gradc,gradceq] = ellipseparabola(x)
c(1) = x(1)^2/9 + x(2)^2/4 - 1;
c(2) = x(1)^2 - x(2) - 1;
ceq = [];

if nargout > 2
    gradc = [2*x(1)/9, 2*x(1); ...
             x(2)/2, -1];
    gradceq = [];
end

See Writing Scalar Objective Functions for information on conditionalized functions. The gradient matrix has the form

gradc_{i, j} = [∂c(j)/∂x_i].

The first column of the gradient matrix is associated with c(1), and the second column is associated with c(2). This derivative form is the transpose of the form of Jacobians.

To have a solver use gradients of nonlinear constraints, indicate that they exist by using optimoptions:

options = optimoptions(@fmincon,'SpecifyConstraintGradient',true);

Make sure to pass the options structure to the solver:

[x,fval] = fmincon(@myobj,x0,A,b,Aeq,beq,lb,ub, ...
           @ellipseparabola,options)

If you have a Symbolic Math Toolbox™ license, you can calculate gradients and Hessians automatically, as described in Calculate Gradients and Hessians Using Symbolic Math Toolbox.

Anonymous Nonlinear Constraint Functions

Nonlinear constraint functions must return two outputs. The first output corresponds to nonlinear inequalities, and the second corresponds to nonlinear equalities.

Anonymous functions return just one output. So how can you write an anonymous function as a nonlinear constraint?

The deal function distributes multiple outputs. For example, suppose that you have the nonlinear inequalities

$$\begin{array}{c}\frac{x_1^2}{9}+\frac{x_2^2}{4}\le 1,\\ x_2\ge x_1^2-1.\end{array}$$

Suppose that you have the nonlinear equality

$$x_2=\tanh (x_1)$$.

Write a nonlinear constraint function as follows.

c = @(x)[x(1)^2/9 + x(2)^2/4 - 1;
        x(1)^2 - x(2) - 1];
ceq = @(x)tanh(x(1)) - x(2);
nonlinfcn = @(x)deal(c(x),ceq(x));

To minimize the function $$\cosh (x_1) + \sinh (x_2)$$ subject to the constraints in nonlinfcn, use fmincon.

obj = @(x)cosh(x(1))+sinh(x(2));
opts = optimoptions(@fmincon,'Algorithm','sqp');
z = fmincon(obj,[0;0],[],[],[],[],[],[],nonlinfcn,opts)

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.

z = 2×1

   -0.6530
   -0.5737

To check how well the resulting point z satisfies the constraints, use nonlinfcn.

[cout,ceqout] = nonlinfcn(z)

cout = 2×1

   -0.8704
         0

ceqout = 
0

z satisfies all the constraints to within the default value of the constraint tolerance ConstraintTolerance, 1e-6.

For information on anonymous objective functions, see Anonymous Function Objectives.