Minimize a function using the Constrained Optimization BY Linear Approximation (COBYLA) method. This method wraps a FORTRAN implementation of the algorithm.
Parameters ---------- func : callable Function to minimize. In the form func(x, \*args). x0 : ndarray Initial guess. cons : sequence Constraint functions; must all be ``>=0`` (a single function if only 1 constraint). Each function takes the parameters `x` as its first argument, and it can return either a single number or an array or list of numbers. args : tuple, optional Extra arguments to pass to function. consargs : tuple, optional Extra arguments to pass to constraint functions (default of None means use same extra arguments as those passed to func). Use ``()`` for no extra arguments. rhobeg : float, optional Reasonable initial changes to the variables. rhoend : float, optional Final accuracy in the optimization (not precisely guaranteed). This is a lower bound on the size of the trust region. disp :
, 1, 2, 3
, optional Controls the frequency of output; 0 implies no output. maxfun : int, optional Maximum number of function evaluations. catol : float, optional Absolute tolerance for constraint violations.
Returns ------- x : ndarray The argument that minimises `f`.
See also -------- minimize: Interface to minimization algorithms for multivariate functions. See the 'COBYLA' `method` in particular.
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 non-negative).
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 J-P Hennart, Kluwer Academic (Dordrecht), pp. 51-67
Powell M.J.D. (1998), 'Direct search algorithms for optimization calculations', Acta Numerica 7, 287-336
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 x0*x1 ... >>> def constr1(x): ... return 1 - (x0**2 + x1**2) ... >>> def constr2(x): ... return x1 ... >>> from scipy.optimize import fmin_cobyla >>> fmin_cobyla(objective, 0.0, 0.1, constr1, constr2, rhoend=1e-7) array(-0.70710685, 0.70710671)
The exact solution is (-sqrt(2)/2, sqrt(2)/2).