Nonlinear Constraints with Gradients

Ordinarily, minimization routines use numerical gradients calculated by finite-difference approximation. This procedure systematically perturbs each of the variables in order to calculate function and constraint partial derivatives. Alternatively, you can provide a function to compute partial derivatives analytically. Typically, the problem is solved more accurately and efficiently if such a function is provided.

Consider how to solve

$\min_{x} f (x) = e^{x_{1}} (4 x_{1}^{2} + 2 x_{2}^{2} + 4 x_{1} x_{2} + 2 x_{2} + 1) .$

subject to the constraints

x₁x₂ – x₁ – x₂ ≤ –1.5,
x₁x₂ ≥ –10.

To solve the problem using analytically determined gradients, do the following.

Step 1: Write a file for the objective function and gradient.

function [f,gradf] = objfungrad(x)
f = exp(x(1))*(4*x(1)^2+2*x(2)^2+4*x(1)*x(2)+2*x(2)+1);
% Gradient of the objective function:
if nargout  > 1
    gradf = [ f + exp(x(1)) * (8*x(1) + 4*x(2)), 
    exp(x(1))*(4*x(1)+4*x(2)+2)];
end

Step 2: Write a file for the nonlinear constraints and the gradients of the nonlinear constraints.

function [c,ceq,DC,DCeq] = confungrad(x)
c(1) = 1.5 + x(1) * x(2) - x(1) - x(2); % Inequality constraints
c(2) = -x(1) * x(2)-10; 
% No nonlinear equality constraints
ceq=[];
% Gradient of the constraints:
if nargout > 2
    DC= [x(2)-1, -x(2);
        x(1)-1, -x(1)];
    DCeq = [];
end

gradf contains the partial derivatives of the objective function, f, returned by objfungrad(x), with respect to each of the elements in x:

\nabla f = [\begin{matrix} e^{x_{1}} (4 x_{1}^{2} + 2 x_{2}^{2} + 4 x_{1} x_{2} + 2 x_{2} + 1) + e^{x_{1}} (8 x_{1} + 4 x_{2}) \\ e^{x_{1}} (4 x_{1} + 4 x_{2} + 2) \end{matrix}] .

(6-58)

The columns of DC contain the partial derivatives for each respective constraint (i.e., the ith column of DC is the partial derivative of the ith constraint with respect to x). So in the above example, DC is

[\begin{matrix} \frac{\partial c_{1}}{\partial x_{1}} & \frac{\partial c_{2}}{\partial x_{1}} \\ \frac{\partial c_{1}}{\partial x_{2}} & \frac{\partial c_{2}}{\partial x_{2}} \end{matrix}] = [\begin{matrix} x_{2} - 1 & - x_{2} \\ x_{1} - 1 & - x_{1} \end{matrix}] .

(6-59)

Since you are providing the gradient of the objective in objfungrad.m and the gradient of the constraints in confungrad.m, you must tell fmincon that these files contain this additional information. Use optimoptions to turn the options GradObj and GradConstr to 'on' in the example's existing options:

options = optimoptions(options,'GradObj','on','GradConstr','on');

If you do not set these options to 'on', fmincon does not use the analytic gradients.

The arguments lb and ub place lower and upper bounds on the independent variables in x. In this example, there are no bound constraints, so set both to [].

Step 3: Invoke the constrained optimization routine.

x0 = [-1,1];            % Starting guess 
options = optimoptions(@fmincon,'Algorithm','sqp');
options = optimoptions(options,'GradObj','on','GradConstr','on');
lb = [ ]; ub = [ ];   % No upper or lower bounds
[x,fval] = fmincon(@objfungrad,x0,[],[],[],[],lb,ub,... 
   @confungrad,options);

The results:

x,fval
x =
    -9.5474    1.0474
fval =
     0.0236

[c,ceq] = confungrad(x) % Check the constraint values at x
c =
   1.0e-13 *
   -0.1066
    0.1066

ceq =
     []

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