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 (x, 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 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 ]

The fifth and sixth arguments are vectors containing lower and upper bounds on x. These must be consistent with equality and inequality constraints g and h. If the bounds are not specified, or are empty, they are set to -realmax and realmax by default.

The seventh argument is max. number of iterations. If not specified, the default value is 100.

The eighth argument is tolerance for stopping criteria. If not specified, the default value is eps.

Here is 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

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

101

The algorithm terminated because the norm of the last step was less than tol * norm (x)) (the value of tol is currently fixed at sqrt (eps)—edit ‘sqp.m’ to modify this value.

102

The BFGS update failed.

103

The maximum number of iterations was reached (the maximum number of allowed iterations is currently fixed at 100—edit ‘sqp.m’ to increase this value).

See also: qp.