Several optimization solvers accept nonlinear constraints, including
fminimax, and the Global Optimization
MultiStart. 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.
Nonlinear constraint functions must return both
the inequality and equality constraint functions, even if they do
not both exist. Return an empty entry
 for a
For example, suppose that you have the following inequalities as constraints:
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
ellipseparabolareturns an empty entry
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
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
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
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
gradci, j =
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
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 Symbolic Math Toolbox Calculates Gradients and Hessians.
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?
deal function distributes multiple
outputs. For example, suppose that you have the nonlinear inequalities
Suppose that you have the nonlinear equality
x2 = tanh(x1).
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(x1) + sinh(x2) subject
to the constraints in
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 default value of the function tolerance, and constraints are satisfied to within the default value of the constraint tolerance. z = -0.6530 -0.5737
To check how well the resulting point
the constraints, use
[cout,ceqout] = nonlinfcn(z) cout = -0.8704 0 ceqout = 0
z satisfies all the constraints to within the default value of the
For information on anonymous objective functions, see Anonymous Function Objectives.