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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 that your nonlinear equality constraint is

x12+x2=1

and your nonlinear inequality constraint is

x1x2-10.

Rewrite these constraints as

x12+x2-1=0-x1x2-100.

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

Objective Function

Solve the problem

minxf(x)=ex1(4x12+2x22+4x1x2+2x2+1)

subject to the constraints. The objfun 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];

There are no bounds or linear constraints in the problem, 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 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 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

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