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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.