# Documentation

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## 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

`$\underset{x}{\mathrm{min}}f\left(x\right)={e}^{{x}_{1}}\left(4{x}_{1}^{2}+2{x}_{2}^{2}+4{x}_{1}{x}_{2}+2{x}_{2}+1\right).$`

subject to the constraints

x1x2x1x2 ≤ –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`:

 $\nabla f=\left[\begin{array}{c}{e}^{{x}_{1}}\left(4{x}_{1}^{2}+2{x}_{2}^{2}+4{x}_{1}{x}_{2}+2{x}_{2}+1\right)+{e}^{{x}_{1}}\left(8{x}_{1}+4{x}_{2}\right)\\ {e}^{{x}_{1}}\left(4{x}_{1}+4{x}_{2}+2\right)\end{array}\right].$ (6-58)

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

 $\left[\begin{array}{cc}\frac{\partial {c}_{1}}{\partial {x}_{1}}& \frac{\partial {c}_{2}}{\partial {x}_{1}}\\ \frac{\partial {c}_{1}}{\partial {x}_{2}}& \frac{\partial {c}_{2}}{\partial {x}_{2}}\end{array}\right]=\left[\begin{array}{cc}{x}_{2}-1& -{x}_{2}\\ {x}_{1}-1& -{x}_{1}\end{array}\right].$ (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 `SpecifyObjectiveGradient` and `SpecifyConstraintGradient` to `true` in the example's existing `options`:

`options = optimoptions(options,'SpecifyObjectiveGradient',true,'SpecifyConstraintGradient',true);`

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,'SpecifyObjectiveGradient',true,'SpecifyConstraintGradient',true); 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 = []```