## Documentation Center |

Ordinarily the medium-scale 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.

To solve

using analytically determined gradients, do the following.

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

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`:

(6-59) |

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

(6-60) |

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'` in
the options structure, `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 and so they are both
set to `[]`.

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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