|GNU Octave Manual Version 3|
by John W. Eaton, David Bateman, Søren Hauberg
Paperback (6"x9"), 568 pages
RRP £24.95 ($39.95)
23.3 Nonlinear Programming
Octave can also perform general nonlinear minimization using a successive quadratic programming solver.
- Function File: [x, obj, info, iter, nf, lambda] = sqp (x, phi, g, h)
- Solve the nonlinear program
min phi (x) x
g(x) = 0 h(x) >= 0
using a successive quadratic programming method.
The first argument is the initial guess for the vector x.
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 to compute 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.
If supplied, the gradient function must be of the form
g = gradient (x)
in which x is a vector and g is a vector.
If 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 your problem does not have equality (or inequality) constraints, you may pass an empty matrix for cef (or cif).
If 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 ]
Here is an example of calling
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
The value returned in info may be one of the following:
The algorithm terminated because the norm of the last step was less
tol * norm (x))(the value of tol is currently fixed at
sqrt (eps)in ‘sqp.m’).
- The BFGS update failed.
- The maximum number of iterations was reached (the maximum number of allowed iterations is currently fixed at 100 in ‘sqp.m’).
See also qp
|ISBN 095461206X||GNU Octave Manual Version 3||See the print edition|