Minimize a function using the Constrained Optimization BY Linear Approximation (COBYLA) method. This method wraps a FORTRAN implentation of the algorithm.
Parameters :  func : callable
x0 : ndarray
cons : sequence
args : tuple
consargs : tuple
rhobeg : :
rhoend : :
iprint : {0, 1, 2, 3}
disp : {0, 1, 2, 3}
maxfun : int


Returns :  x : ndarray

See also
Notes
This algorithm is based on linear approximations to the objective function and each constraint. We briefly describe the algorithm.
Suppose the function is being minimized over k variables. At the jth iteration the algorithm has k+1 points v_1, ..., v_(k+1), an approximate solution x_j, and a radius RHO_j. (i.e. linear plus a constant) approximations to the objective function and constraint functions such that their function values agree with the linear approximation on the k+1 points v_1,.., v_(k+1). This gives a linear program to solve (where the linear approximations of the constraint functions are constrained to be nonnegative).
However the linear approximations are likely only good approximations near the current simplex, so the linear program is given the further requirement that the solution, which will become x_(j+1), must be within RHO_j from x_j. RHO_j only decreases, never increases. The initial RHO_j is rhobeg and the final RHO_j is rhoend. In this way COBYLA’s iterations behave like a trust region algorithm.
Additionally, the linear program may be inconsistent, or the approximation may give poor improvement. For details about how these issues are resolved, as well as how the points v_i are updated, refer to the source code or the references below.
References
Powell M.J.D. (1994), “A direct search optimization method that models the objective and constraint functions by linear interpolation.”, in Advances in Optimization and Numerical Analysis, eds. S. Gomez and JP Hennart, Kluwer Academic (Dordrecht), pp. 5167
Powell M.J.D. (1998), “Direct search algorithms for optimization calculations”, Acta Numerica 7, 287336
Powell M.J.D. (2007), “A view of algorithms for optimization without derivatives”, Cambridge University Technical Report DAMTP 2007/NA03
Examples
Minimize the objective function f(x,y) = x*y subject to the constraints x**2 + y**2 < 1 and y > 0:
>>> def objective(x):
... return x[0]*x[1]
...
>>> def constr1(x):
... return 1  (x[0]**2 + x[1]**2)
...
>>> def constr2(x):
... return x[1]
...
>>> fmin_cobyla(objective, [0.0, 0.1], [constr1, constr2], rhoend=1e7)
Normal return from subroutine COBYLA
NFVALS = 64 F =5.000000E01 MAXCV = 1.998401E14
X =7.071069E01 7.071067E01
array([0.70710685, 0.70710671])
The exact solution is (sqrt(2)/2, sqrt(2)/2).