Many solvers allow you to supply a function that calculates first derivatives (gradients or Jacobians) of objective or constraint functions. You can check whether the derivatives calculated by your function match finite-difference approximations. This check can help you diagnose whether your derivative function is correct.
If a component of the gradient function is less than 1
,
"match" means the absolute difference of the gradient
function and the finite-difference approximation of that component
is less than 1e-6
.
Otherwise, "match" means that the relative
difference is less than 1e-6
.
The CheckGradients
option causes the solver
to check the supplied derivative against a finite-difference approximation
at just one point. If the finite-difference and supplied derivatives
do not match, the solver errors. If the derivatives match to within 1e-6
,
the solver reports the calculated differences, and continues iterating
without further derivative checks. Solvers check the match at a point
that is a small random perturbation of the initial point x0
,
modified to be within any bounds. Solvers do not include the computations
for CheckGradients
in the function count; see Iterations and Function Counts.
At the MATLAB^{®} command line:
Set the SpecifyObjectiveGradient
or SpecifyConstraintGradient
options
to true
using optimoptions
.
Make sure your objective or constraint functions supply the appropriate
derivatives.
Set the CheckGradients
option to true
.
Using the Optimization app:
In the Problem Setup and Results pane,
choose Derivatives: Objective function: Gradient
supplied
or Nonlinear constraint function: Derivatives: Gradient
supplied
. Make sure your objective or constraint functions
supply the appropriate derivatives.
In the Options pane, check User-supplied derivatives > Validate user-supplied derivatives
Central finite differences are more accurate than the default forward finite differences. To use central finite differences:
At the MATLAB command line, set FiniteDifferenceType
option
to 'central'
using optimoptions
.
Using the Optimization app, in the Approximated
derivatives pane, set Type to central
differences
.
Consider the problem of minimizing the Rosenbrock function within
the unit disk as described in Solve a Constrained Nonlinear Problem. The rosenboth
function
calculates the objective function and its gradient:
function [f g H] = rosenboth(x) f = 100*(x(2) - x(1)^2)^2 + (1-x(1))^2; if nargout > 1 g = [-400*(x(2)-x(1)^2)*x(1)-2*(1-x(1)); 200*(x(2)-x(1)^2)]; if nargout > 2 H = [1200*x(1)^2-400*x(2)+2, -400*x(1); -400*x(1), 200]; end end
rosenboth
calculates the Hessian, too, but
this example does not use the Hessian.
The unitdisk2
function correctly calculates
the constraint function and its gradient:
function [c,ceq,gc,gceq] = unitdisk2(x) c = x(1)^2 + x(2)^2 - 1; ceq = [ ]; if nargout > 2 gc = [2*x(1);2*x(2)]; gceq = []; end
The unitdiskb
function incorrectly calculates
gradient of the constraint function:
function [c ceq gc gceq] = unitdiskb(x) c = x(1)^2 + x(2)^2 - 1; ceq = [ ]; if nargout > 2 gc = [x(1);x(2)]; % Gradient incorrect: off by a factor of 2 gceq = []; end
Set the options to use the interior-point algorithm, gradient
of objective and constraint functions, and the CheckGradients
option:
% For reproducibility--CheckGradients randomly perturbs the initial point rng(0,'twister'); options = optimoptions(@fmincon,'Algorithm','interior-point',... 'CheckGradients',true,'SpecifyObjectiveGradient',true,'SpecifyConstraintGradient',true);
Solve the minimization with fmincon
using
the erroneous unitdiskb
constraint function:
[x fval exitflag output] = fmincon(@rosenboth,... [-1;2],[],[],[],[],[],[],@unitdiskb,options); ____________________________________________________________ Derivative Check Information Objective function derivatives: Maximum relative difference between user-supplied and finite-difference derivatives = 1.84768e-008. Nonlinear inequality constraint derivatives: Maximum relative difference between user-supplied and finite-difference derivatives = 1. User-supplied constraint derivative element (2,1): 1.99838 Finite-difference constraint derivative element (2,1): 3.99675 ____________________________________________________________ Error using validateFirstDerivatives Derivative Check failed: User-supplied and forward finite-difference derivatives do not match within 1e-006 relative tolerance. Error in fmincon at 805 validateFirstDerivatives(funfcn,confcn,X, ...
The constraint function does not match the calculated gradient, encouraging you to check the function for an error.
Replace the unitdiskb
constraint function
with unitdisk2
and run the minimization again:
[x fval exitflag output] = fmincon(@rosenboth,... [-1;2],[],[],[],[],[],[],@unitdisk2,options); ____________________________________________________________ Derivative Check Information Objective function derivatives: Maximum relative difference between user-supplied and finite-difference derivatives = 1.28553e-008. Nonlinear inequality constraint derivatives: Maximum relative difference between user-supplied and finite-difference derivatives = 1.46443e-008. Derivative Check successfully passed. ____________________________________________________________ Local minimum found that satisfies the constraints...
Note: The Optimization app warns that it will be removed in a future release. |
To set up the example using correct derivative functions, but
starting from [0 0]
, using the Optimization app:
Launch the Optimization app by entering optimtool
at
the command line.
Set the Problem Setup and Results pane to match the following figure:
Set the Options pane to match the following figure:
Press the Start button under Run solver and view results.
The output screen displays
The forward finite difference approximation is inaccurate enough
near [0 0]
that the derivative check fails.
To use the more accurate central differences, select central
differences
in the Approximated derivatives
> Type pane:
Click Run solver and view results > Clear Results, then Start. This time the derivative check is successful:
The derivative check also succeeds when you select the initial
point [-1 2]
, or most random points.