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Checking Validity of Gradients or Jacobians

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 DerivativeCheck 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 DerivativeCheck in the function count; see Iterations and Function Counts.

How to Check Derivatives

  • At the MATLAB® command line:

    1. Set the GradObj, GradConstr, or Jacobian options to 'on' with optimoptions. Make sure your objective or constraint functions supply the appropriate derivatives.

    2. Set the DerivativeCheck option to 'on'.

  • Using the Optimization app:

    1. 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.

    2. 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 FinDiffType option to 'central' with optimoptions.

  • Using the Optimization app, in the Approximated derivatives pane, set Type to central differences.

Example: Checking Derivatives of Objective and Constraint Functions

Objective and Constraint Functions

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

Checking Derivatives at the Command Line

  1. Set the options to use the interior-point algorithm, gradient of objective and constraint functions, and the DerivativeCheck option:

    % For reproducibility--DerivativeCheck randomly perturbs the initial point
    rng(0,'twister'); 
    options = optimoptions(@fmincon,'Algorithm','interior-point',...
        'DerivativeCheck','on','GradObj','on','GradConstr','on');
  2. 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.

  3. 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...

Checking Derivatives with the Optimization App

To set up the example using correct derivative functions, but starting from [0 0], using the Optimization app:

  1. Launch the Optimization app by entering optimtool at the command line.

  2. Set the Problem Setup and Results pane to match the following figure:

  3. Set the Options pane to match the following figure:

  4. 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.

  5. To use the more accurate central differences, select central differences in the Approximated derivatives > Type pane:

  6. 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.

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