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### 24.3 Nonlinear Programming

Octave can also perform general nonlinear minimization using a successive quadratic programming solver.

— Function File: [x, obj, info, iter, nf, lambda] = sqp (x0, phi)
— Function File: [...] = sqp (x0, phi, g)
— Function File: [...] = sqp (x0, phi, g, h)
— Function File: [...] = sqp (x0, phi, g, h, lb, ub)
— Function File: [...] = sqp (x0, phi, g, h, lb, ub, maxiter)
— Function File: [...] = sqp (x0, phi, g, h, lb, ub, maxiter, tolerance)

> Solve the nonlinear program

>                    min phi (x)
>                     x

> subject to

>                    g(x)  = 0
>                    h(x) >= 0
>                    lb <= x <= ub

> using a sequential quadratic programming method.

> The first argument is the initial guess for the vector x0.

> The second argument is a function handle pointing to the objective function. The objective function must be of the form

>                    y = phi (x)

> in which x is a vector and y is a scalar.

> The second argument may also be a 2- or 3-element cell array of function handles. The first element should point to the objective function, the second should point to a function that computes the gradient of the objective function, and the third should point to a function that computes the Hessian of the objective function. If the gradient function is not supplied, the gradient is computed by finite differences. If the Hessian function is not supplied, a BFGS update formula is used to approximate the Hessian.

> When supplied, the gradient function must be of the form

>               g = gradient (x)

> in which x is a vector and g is a vector.

> When supplied, the Hessian function must be of the form

>               h = hessian (x)

> in which x is a vector and h is a matrix.

> The third and fourth arguments are function handles pointing to functions that compute the equality constraints and the inequality constraints, respectively.

> If the problem does not have equality (or inequality) constraints, then use an empty matrix ([]) for cef (or cif).

> When supplied, the equality and inequality constraint functions must be of the form

>               r = f (x)

> in which x is a vector and r is a vector.

> The third and fourth arguments may also be 2-element cell arrays of function handles. The first element should point to the constraint function and the second should point to a function that computes the gradient of the constraint function:

>                               [ d f(x)   d f(x)        d f(x) ]
>                   transpose ( [ ------   -----   ...   ------ ] )
>                               [  dx_1     dx_2          dx_N  ]

> The fifth and sixth arguments contain lower and upper bounds on x. These must be consistent with the equality and inequality constraints g and h. If the arguments are vectors then x(i) is bound by lb(i) and ub(i). A bound can also be a scalar in which case all elements of x will share the same bound. If only one bound (lb, ub) is specified then the other will default to (-realmax, +realmax).

> The seventh argument specifies the maximum number of iterations. The default value is 100.

> The eighth argument specifies the tolerance for the stopping criteria. The default value is `sqrt(eps)`.

> The value returned in info may be one of the following:

> 101
>   The algorithm terminated normally. Either all constraints meet the requested tolerance, or the stepsize, delta x, is less than `tol * norm (x)`.
> 102
>   The BFGS update failed.
> 103
>   The maximum number of iterations was reached.

> An example of calling `sqp`:

>               function r = g (x)
>                 r = [ sumsq(x)-10;
>                       x(2)*x(3)-5*x(4)*x(5);
>                       x(1)^3+x(2)^3+1 ];
>               endfunction

>               function obj = phi (x)
>                 obj = exp(prod(x)) - 0.5*(x(1)^3+x(2)^3+1)^2;
>               endfunction

>               x0 = [-1.8; 1.7; 1.9; -0.8; -0.8];

>               [x, obj, info, iter, nf, lambda] = sqp (x0, @phi, @g, [])

>               x =

>                 -1.71714
>                  1.59571
>                  1.82725
>                 -0.76364
>                 -0.76364

>               obj = 0.053950
>               info = 101
>               iter = 8
>               nf = 10
>               lambda =

>                 -0.0401627
>                  0.0379578
>                 -0.0052227

> See also: qp.