octave: Nonlinear Programming
25.3 Nonlinear Programming
==========================
Octave can also perform general nonlinear minimization using a
successive quadratic programming solver.
-- : [X, OBJ, INFO, ITER, NF, LAMBDA] = sqp (X0, PHI)
-- : [...] = sqp (X0, PHI, G)
-- : [...] = sqp (X0, PHI, G, H)
-- : [...] = sqp (X0, PHI, G, H, LB, UB)
-- : [...] = sqp (X0, PHI, G, H, LB, UB, MAXITER)
-- : [...] = sqp (X0, PHI, G, H, LB, UB, MAXITER, TOL)
Minimize an objective function using sequential quadratic
programming (SQP).
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 PHI. The objective function must accept one vector
argument and return 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 ‘PHI{2}’ must accept one
vector argument and return a vector. When supplied, the Hessian
function ‘PHI{3}’ must accept one vector argument and return a
matrix.
The third and fourth arguments G and H 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 G (or H). When supplied, these equality and inequality
constraint functions must accept one vector argument and return 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, LB and UB, 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 MAXITER specifies the maximum number of
iterations. The default value is 100.
The eighth argument TOL 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. All constraints meet the
specified tolerance.
102
The BFGS update failed.
103
The maximum number of iterations was reached.
104
The stepsize has become too small, i.e., delta X, is less than
‘TOL * norm (x)’.
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 XREFqp.
Info Catalog
octave: Quadratic Programming
octave: Optimization
octave: Linear Least Squares