533 lines
19 KiB
Python
533 lines
19 KiB
Python
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"""
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This module implements the Sequential Least SQuares Programming optimization
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algorithm (SLSQP), originally developed by Dieter Kraft.
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See http://www.netlib.org/toms/733
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Functions
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---------
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.. autosummary::
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:toctree: generated/
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approx_jacobian
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fmin_slsqp
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"""
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from __future__ import division, print_function, absolute_import
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__all__ = ['approx_jacobian', 'fmin_slsqp']
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import numpy as np
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from scipy.optimize._slsqp import slsqp
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from numpy import (zeros, array, linalg, append, asfarray, concatenate, finfo,
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sqrt, vstack, exp, inf, isfinite, atleast_1d)
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from .optimize import wrap_function, OptimizeResult, _check_unknown_options
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__docformat__ = "restructuredtext en"
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_epsilon = sqrt(finfo(float).eps)
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def approx_jacobian(x, func, epsilon, *args):
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"""
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Approximate the Jacobian matrix of a callable function.
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Parameters
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----------
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x : array_like
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The state vector at which to compute the Jacobian matrix.
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func : callable f(x,*args)
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The vector-valued function.
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epsilon : float
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The perturbation used to determine the partial derivatives.
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args : sequence
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Additional arguments passed to func.
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Returns
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-------
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An array of dimensions ``(lenf, lenx)`` where ``lenf`` is the length
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of the outputs of `func`, and ``lenx`` is the number of elements in
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`x`.
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Notes
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-----
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The approximation is done using forward differences.
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"""
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x0 = asfarray(x)
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f0 = atleast_1d(func(*((x0,)+args)))
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jac = zeros([len(x0), len(f0)])
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dx = zeros(len(x0))
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for i in range(len(x0)):
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dx[i] = epsilon
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jac[i] = (func(*((x0+dx,)+args)) - f0)/epsilon
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dx[i] = 0.0
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return jac.transpose()
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def fmin_slsqp(func, x0, eqcons=(), f_eqcons=None, ieqcons=(), f_ieqcons=None,
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bounds=(), fprime=None, fprime_eqcons=None,
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fprime_ieqcons=None, args=(), iter=100, acc=1.0E-6,
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iprint=1, disp=None, full_output=0, epsilon=_epsilon,
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callback=None):
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"""
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Minimize a function using Sequential Least SQuares Programming
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Python interface function for the SLSQP Optimization subroutine
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originally implemented by Dieter Kraft.
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Parameters
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----------
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func : callable f(x,*args)
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Objective function. Must return a scalar.
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x0 : 1-D ndarray of float
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Initial guess for the independent variable(s).
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eqcons : list, optional
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A list of functions of length n such that
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eqcons[j](x,*args) == 0.0 in a successfully optimized
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problem.
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f_eqcons : callable f(x,*args), optional
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Returns a 1-D array in which each element must equal 0.0 in a
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successfully optimized problem. If f_eqcons is specified,
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eqcons is ignored.
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ieqcons : list, optional
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A list of functions of length n such that
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ieqcons[j](x,*args) >= 0.0 in a successfully optimized
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problem.
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f_ieqcons : callable f(x,*args), optional
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Returns a 1-D ndarray in which each element must be greater or
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equal to 0.0 in a successfully optimized problem. If
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f_ieqcons is specified, ieqcons is ignored.
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bounds : list, optional
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A list of tuples specifying the lower and upper bound
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for each independent variable [(xl0, xu0),(xl1, xu1),...]
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Infinite values will be interpreted as large floating values.
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fprime : callable `f(x,*args)`, optional
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A function that evaluates the partial derivatives of func.
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fprime_eqcons : callable `f(x,*args)`, optional
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A function of the form `f(x, *args)` that returns the m by n
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array of equality constraint normals. If not provided,
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the normals will be approximated. The array returned by
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fprime_eqcons should be sized as ( len(eqcons), len(x0) ).
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fprime_ieqcons : callable `f(x,*args)`, optional
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A function of the form `f(x, *args)` that returns the m by n
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array of inequality constraint normals. If not provided,
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the normals will be approximated. The array returned by
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fprime_ieqcons should be sized as ( len(ieqcons), len(x0) ).
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args : sequence, optional
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Additional arguments passed to func and fprime.
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iter : int, optional
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The maximum number of iterations.
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acc : float, optional
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Requested accuracy.
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iprint : int, optional
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The verbosity of fmin_slsqp :
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* iprint <= 0 : Silent operation
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* iprint == 1 : Print summary upon completion (default)
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* iprint >= 2 : Print status of each iterate and summary
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disp : int, optional
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Over-rides the iprint interface (preferred).
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full_output : bool, optional
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If False, return only the minimizer of func (default).
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Otherwise, output final objective function and summary
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information.
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epsilon : float, optional
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The step size for finite-difference derivative estimates.
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callback : callable, optional
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Called after each iteration, as ``callback(x)``, where ``x`` is the
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current parameter vector.
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Returns
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-------
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out : ndarray of float
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The final minimizer of func.
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fx : ndarray of float, if full_output is true
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The final value of the objective function.
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its : int, if full_output is true
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The number of iterations.
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imode : int, if full_output is true
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The exit mode from the optimizer (see below).
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smode : string, if full_output is true
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Message describing the exit mode from the optimizer.
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See also
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--------
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minimize: Interface to minimization algorithms for multivariate
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functions. See the 'SLSQP' `method` in particular.
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Notes
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-----
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Exit modes are defined as follows ::
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-1 : Gradient evaluation required (g & a)
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0 : Optimization terminated successfully.
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1 : Function evaluation required (f & c)
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2 : More equality constraints than independent variables
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3 : More than 3*n iterations in LSQ subproblem
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4 : Inequality constraints incompatible
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5 : Singular matrix E in LSQ subproblem
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6 : Singular matrix C in LSQ subproblem
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7 : Rank-deficient equality constraint subproblem HFTI
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8 : Positive directional derivative for linesearch
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9 : Iteration limit exceeded
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Examples
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--------
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Examples are given :ref:`in the tutorial <tutorial-sqlsp>`.
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"""
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if disp is not None:
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iprint = disp
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opts = {'maxiter': iter,
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'ftol': acc,
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'iprint': iprint,
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'disp': iprint != 0,
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'eps': epsilon,
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'callback': callback}
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# Build the constraints as a tuple of dictionaries
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cons = ()
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# 1. constraints of the 1st kind (eqcons, ieqcons); no Jacobian; take
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# the same extra arguments as the objective function.
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cons += tuple({'type': 'eq', 'fun': c, 'args': args} for c in eqcons)
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cons += tuple({'type': 'ineq', 'fun': c, 'args': args} for c in ieqcons)
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# 2. constraints of the 2nd kind (f_eqcons, f_ieqcons) and their Jacobian
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# (fprime_eqcons, fprime_ieqcons); also take the same extra arguments
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# as the objective function.
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if f_eqcons:
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cons += ({'type': 'eq', 'fun': f_eqcons, 'jac': fprime_eqcons,
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'args': args}, )
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if f_ieqcons:
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cons += ({'type': 'ineq', 'fun': f_ieqcons, 'jac': fprime_ieqcons,
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'args': args}, )
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res = _minimize_slsqp(func, x0, args, jac=fprime, bounds=bounds,
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constraints=cons, **opts)
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if full_output:
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return res['x'], res['fun'], res['nit'], res['status'], res['message']
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else:
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return res['x']
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def _minimize_slsqp(func, x0, args=(), jac=None, bounds=None,
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constraints=(),
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maxiter=100, ftol=1.0E-6, iprint=1, disp=False,
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eps=_epsilon, callback=None,
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**unknown_options):
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"""
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Minimize a scalar function of one or more variables using Sequential
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Least SQuares Programming (SLSQP).
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Options
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-------
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ftol : float
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Precision goal for the value of f in the stopping criterion.
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eps : float
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Step size used for numerical approximation of the Jacobian.
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disp : bool
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Set to True to print convergence messages. If False,
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`verbosity` is ignored and set to 0.
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maxiter : int
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Maximum number of iterations.
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"""
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_check_unknown_options(unknown_options)
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fprime = jac
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iter = maxiter
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acc = ftol
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epsilon = eps
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if not disp:
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iprint = 0
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# Constraints are triaged per type into a dictionary of tuples
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if isinstance(constraints, dict):
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constraints = (constraints, )
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cons = {'eq': (), 'ineq': ()}
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for ic, con in enumerate(constraints):
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# check type
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try:
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ctype = con['type'].lower()
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except KeyError:
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raise KeyError('Constraint %d has no type defined.' % ic)
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except TypeError:
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raise TypeError('Constraints must be defined using a '
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'dictionary.')
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except AttributeError:
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raise TypeError("Constraint's type must be a string.")
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else:
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if ctype not in ['eq', 'ineq']:
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raise ValueError("Unknown constraint type '%s'." % con['type'])
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# check function
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if 'fun' not in con:
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raise ValueError('Constraint %d has no function defined.' % ic)
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# check Jacobian
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cjac = con.get('jac')
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if cjac is None:
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# approximate Jacobian function. The factory function is needed
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# to keep a reference to `fun`, see gh-4240.
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def cjac_factory(fun):
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def cjac(x, *args):
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return approx_jacobian(x, fun, epsilon, *args)
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return cjac
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cjac = cjac_factory(con['fun'])
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# update constraints' dictionary
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cons[ctype] += ({'fun': con['fun'],
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'jac': cjac,
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'args': con.get('args', ())}, )
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exit_modes = {-1: "Gradient evaluation required (g & a)",
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0: "Optimization terminated successfully.",
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1: "Function evaluation required (f & c)",
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2: "More equality constraints than independent variables",
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3: "More than 3*n iterations in LSQ subproblem",
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4: "Inequality constraints incompatible",
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5: "Singular matrix E in LSQ subproblem",
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6: "Singular matrix C in LSQ subproblem",
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7: "Rank-deficient equality constraint subproblem HFTI",
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8: "Positive directional derivative for linesearch",
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9: "Iteration limit exceeded"}
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# Wrap func
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feval, func = wrap_function(func, args)
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# Wrap fprime, if provided, or approx_jacobian if not
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if fprime:
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geval, fprime = wrap_function(fprime, args)
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else:
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geval, fprime = wrap_function(approx_jacobian, (func, epsilon))
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# Transform x0 into an array.
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x = asfarray(x0).flatten()
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# Set the parameters that SLSQP will need
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# meq, mieq: number of equality and inequality constraints
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meq = sum(map(len, [atleast_1d(c['fun'](x, *c['args']))
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for c in cons['eq']]))
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mieq = sum(map(len, [atleast_1d(c['fun'](x, *c['args']))
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for c in cons['ineq']]))
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# m = The total number of constraints
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m = meq + mieq
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# la = The number of constraints, or 1 if there are no constraints
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la = array([1, m]).max()
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# n = The number of independent variables
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n = len(x)
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# Define the workspaces for SLSQP
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n1 = n + 1
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mineq = m - meq + n1 + n1
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len_w = (3*n1+m)*(n1+1)+(n1-meq+1)*(mineq+2) + 2*mineq+(n1+mineq)*(n1-meq) \
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+ 2*meq + n1 + ((n+1)*n)//2 + 2*m + 3*n + 3*n1 + 1
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len_jw = mineq
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w = zeros(len_w)
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jw = zeros(len_jw)
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# Decompose bounds into xl and xu
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if bounds is None or len(bounds) == 0:
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xl = np.empty(n, dtype=float)
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xu = np.empty(n, dtype=float)
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xl.fill(np.nan)
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xu.fill(np.nan)
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else:
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bnds = array(bounds, float)
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if bnds.shape[0] != n:
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raise IndexError('SLSQP Error: the length of bounds is not '
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'compatible with that of x0.')
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with np.errstate(invalid='ignore'):
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bnderr = bnds[:, 0] > bnds[:, 1]
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if bnderr.any():
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raise ValueError('SLSQP Error: lb > ub in bounds %s.' %
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', '.join(str(b) for b in bnderr))
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xl, xu = bnds[:, 0], bnds[:, 1]
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# Mark infinite bounds with nans; the Fortran code understands this
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infbnd = ~isfinite(bnds)
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xl[infbnd[:, 0]] = np.nan
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xu[infbnd[:, 1]] = np.nan
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# Clip initial guess to bounds (SLSQP may fail with bounds-infeasible
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# initial point)
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have_bound = np.isfinite(xl)
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x[have_bound] = np.clip(x[have_bound], xl[have_bound], np.inf)
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have_bound = np.isfinite(xu)
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x[have_bound] = np.clip(x[have_bound], -np.inf, xu[have_bound])
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||
|
|
|
||
|
|
# Initialize the iteration counter and the mode value
|
||
|
|
mode = array(0, int)
|
||
|
|
acc = array(acc, float)
|
||
|
|
majiter = array(iter, int)
|
||
|
|
majiter_prev = 0
|
||
|
|
|
||
|
|
# Initialize internal SLSQP state variables
|
||
|
|
alpha = array(0, float)
|
||
|
|
f0 = array(0, float)
|
||
|
|
gs = array(0, float)
|
||
|
|
h1 = array(0, float)
|
||
|
|
h2 = array(0, float)
|
||
|
|
h3 = array(0, float)
|
||
|
|
h4 = array(0, float)
|
||
|
|
t = array(0, float)
|
||
|
|
t0 = array(0, float)
|
||
|
|
tol = array(0, float)
|
||
|
|
iexact = array(0, int)
|
||
|
|
incons = array(0, int)
|
||
|
|
ireset = array(0, int)
|
||
|
|
itermx = array(0, int)
|
||
|
|
line = array(0, int)
|
||
|
|
n1 = array(0, int)
|
||
|
|
n2 = array(0, int)
|
||
|
|
n3 = array(0, int)
|
||
|
|
|
||
|
|
# Print the header if iprint >= 2
|
||
|
|
if iprint >= 2:
|
||
|
|
print("%5s %5s %16s %16s" % ("NIT", "FC", "OBJFUN", "GNORM"))
|
||
|
|
|
||
|
|
while 1:
|
||
|
|
|
||
|
|
if mode == 0 or mode == 1: # objective and constraint evaluation required
|
||
|
|
|
||
|
|
# Compute objective function
|
||
|
|
fx = func(x)
|
||
|
|
try:
|
||
|
|
fx = float(np.asarray(fx))
|
||
|
|
except (TypeError, ValueError):
|
||
|
|
raise ValueError("Objective function must return a scalar")
|
||
|
|
# Compute the constraints
|
||
|
|
if cons['eq']:
|
||
|
|
c_eq = concatenate([atleast_1d(con['fun'](x, *con['args']))
|
||
|
|
for con in cons['eq']])
|
||
|
|
else:
|
||
|
|
c_eq = zeros(0)
|
||
|
|
if cons['ineq']:
|
||
|
|
c_ieq = concatenate([atleast_1d(con['fun'](x, *con['args']))
|
||
|
|
for con in cons['ineq']])
|
||
|
|
else:
|
||
|
|
c_ieq = zeros(0)
|
||
|
|
|
||
|
|
# Now combine c_eq and c_ieq into a single matrix
|
||
|
|
c = concatenate((c_eq, c_ieq))
|
||
|
|
|
||
|
|
if mode == 0 or mode == -1: # gradient evaluation required
|
||
|
|
|
||
|
|
# Compute the derivatives of the objective function
|
||
|
|
# For some reason SLSQP wants g dimensioned to n+1
|
||
|
|
g = append(fprime(x), 0.0)
|
||
|
|
|
||
|
|
# Compute the normals of the constraints
|
||
|
|
if cons['eq']:
|
||
|
|
a_eq = vstack([con['jac'](x, *con['args'])
|
||
|
|
for con in cons['eq']])
|
||
|
|
else: # no equality constraint
|
||
|
|
a_eq = zeros((meq, n))
|
||
|
|
|
||
|
|
if cons['ineq']:
|
||
|
|
a_ieq = vstack([con['jac'](x, *con['args'])
|
||
|
|
for con in cons['ineq']])
|
||
|
|
else: # no inequality constraint
|
||
|
|
a_ieq = zeros((mieq, n))
|
||
|
|
|
||
|
|
# Now combine a_eq and a_ieq into a single a matrix
|
||
|
|
if m == 0: # no constraints
|
||
|
|
a = zeros((la, n))
|
||
|
|
else:
|
||
|
|
a = vstack((a_eq, a_ieq))
|
||
|
|
a = concatenate((a, zeros([la, 1])), 1)
|
||
|
|
|
||
|
|
# Call SLSQP
|
||
|
|
slsqp(m, meq, x, xl, xu, fx, c, g, a, acc, majiter, mode, w, jw,
|
||
|
|
alpha, f0, gs, h1, h2, h3, h4, t, t0, tol,
|
||
|
|
iexact, incons, ireset, itermx, line,
|
||
|
|
n1, n2, n3)
|
||
|
|
|
||
|
|
# call callback if major iteration has incremented
|
||
|
|
if callback is not None and majiter > majiter_prev:
|
||
|
|
callback(np.copy(x))
|
||
|
|
|
||
|
|
# Print the status of the current iterate if iprint > 2 and the
|
||
|
|
# major iteration has incremented
|
||
|
|
if iprint >= 2 and majiter > majiter_prev:
|
||
|
|
print("%5i %5i % 16.6E % 16.6E" % (majiter, feval[0],
|
||
|
|
fx, linalg.norm(g)))
|
||
|
|
|
||
|
|
# If exit mode is not -1 or 1, slsqp has completed
|
||
|
|
if abs(mode) != 1:
|
||
|
|
break
|
||
|
|
|
||
|
|
majiter_prev = int(majiter)
|
||
|
|
|
||
|
|
# Optimization loop complete. Print status if requested
|
||
|
|
if iprint >= 1:
|
||
|
|
print(exit_modes[int(mode)] + " (Exit mode " + str(mode) + ')')
|
||
|
|
print(" Current function value:", fx)
|
||
|
|
print(" Iterations:", majiter)
|
||
|
|
print(" Function evaluations:", feval[0])
|
||
|
|
print(" Gradient evaluations:", geval[0])
|
||
|
|
|
||
|
|
return OptimizeResult(x=x, fun=fx, jac=g[:-1], nit=int(majiter),
|
||
|
|
nfev=feval[0], njev=geval[0], status=int(mode),
|
||
|
|
message=exit_modes[int(mode)], success=(mode == 0))
|
||
|
|
|
||
|
|
|
||
|
|
if __name__ == '__main__':
|
||
|
|
|
||
|
|
# objective function
|
||
|
|
def fun(x, r=[4, 2, 4, 2, 1]):
|
||
|
|
""" Objective function """
|
||
|
|
return exp(x[0]) * (r[0] * x[0]**2 + r[1] * x[1]**2 +
|
||
|
|
r[2] * x[0] * x[1] + r[3] * x[1] +
|
||
|
|
r[4])
|
||
|
|
|
||
|
|
# bounds
|
||
|
|
bnds = array([[-inf]*2, [inf]*2]).T
|
||
|
|
bnds[:, 0] = [0.1, 0.2]
|
||
|
|
|
||
|
|
# constraints
|
||
|
|
def feqcon(x, b=1):
|
||
|
|
""" Equality constraint """
|
||
|
|
return array([x[0]**2 + x[1] - b])
|
||
|
|
|
||
|
|
def jeqcon(x, b=1):
|
||
|
|
""" Jacobian of equality constraint """
|
||
|
|
return array([[2*x[0], 1]])
|
||
|
|
|
||
|
|
def fieqcon(x, c=10):
|
||
|
|
""" Inequality constraint """
|
||
|
|
return array([x[0] * x[1] + c])
|
||
|
|
|
||
|
|
def jieqcon(x, c=10):
|
||
|
|
""" Jacobian of Inequality constraint """
|
||
|
|
return array([[1, 1]])
|
||
|
|
|
||
|
|
# constraints dictionaries
|
||
|
|
cons = ({'type': 'eq', 'fun': feqcon, 'jac': jeqcon, 'args': (1, )},
|
||
|
|
{'type': 'ineq', 'fun': fieqcon, 'jac': jieqcon, 'args': (10,)})
|
||
|
|
|
||
|
|
# Bounds constraint problem
|
||
|
|
print(' Bounds constraints '.center(72, '-'))
|
||
|
|
print(' * fmin_slsqp')
|
||
|
|
x, f = fmin_slsqp(fun, array([-1, 1]), bounds=bnds, disp=1,
|
||
|
|
full_output=True)[:2]
|
||
|
|
print(' * _minimize_slsqp')
|
||
|
|
res = _minimize_slsqp(fun, array([-1, 1]), bounds=bnds,
|
||
|
|
**{'disp': True})
|
||
|
|
|
||
|
|
# Equality and inequality constraints problem
|
||
|
|
print(' Equality and inequality constraints '.center(72, '-'))
|
||
|
|
print(' * fmin_slsqp')
|
||
|
|
x, f = fmin_slsqp(fun, array([-1, 1]),
|
||
|
|
f_eqcons=feqcon, fprime_eqcons=jeqcon,
|
||
|
|
f_ieqcons=fieqcon, fprime_ieqcons=jieqcon,
|
||
|
|
disp=1, full_output=True)[:2]
|
||
|
|
print(' * _minimize_slsqp')
|
||
|
|
res = _minimize_slsqp(fun, array([-1, 1]), constraints=cons,
|
||
|
|
**{'disp': True})
|