fmincon 以及 其他优化

不同软件的优化器有所区别，MATLAB中的fmincon 是较为简单好用的， Python中的Scipy Minimize，以及Ox中可以选择 BFGS， 以及BFGS-BOUND。

MATLAB 输入 和 输出：

• MATLAB中的varargin用法; nargout

一个优化器cvxpy, 如何安装 Python：安装cvxpy，进行凸优化； 解凸优化的工具箱或者包

scipy.optimize.minimize； Matlab fmincon algorithm

返回message信息：

ineq_constraints = ({'type': 'ineq', 'fun': lambda x: 1-x[0]-x[1]})
unrest = Bounds([1e-5, 0.01], [1, 1], keep_feasible=True)  # create bounds objects
options = {'gtol': 1e-6, 'disp': True}
tol = 1e-6

SLSQP

或者 fmin_slsqp

opt_ = minimize(
    lik_fn,
    sv,
    args=(stdData, unconcorr, backcast),
    jac='2-point',
    # hess=SR1(),
    method="SLSQP",
    bounds=bounds,
    constraints=ineq_constraints,
    tol=tol,
    options=options,
    # callback=llf._callback
)

 fun: 9571.083803555019
 jac: array([0.00292969, 0.00012207])
message: 'Optimization terminated successfully'
nfev: 36
 nit: 10
njev: 10
status: 0
success: True
   x: array([0.02523875, 0.94423303])

BFGS

【python】scipy包中的BFGS算法

res_ = minimize(lik_fn, sv, method='BFGS',
               jac='2-point',
               args=(stdData, unconcorr, backcast),
               options={'gtol': 1e-6, 'disp': True})

    fun: 9571.083803555195
hess_inv: array([[ 0.0001261 , -0.00031329],
     [-0.00031329,  0.00078031]])
    jac: array([0.00146484, 0.00012207])
message: 'Desired error not necessarily achieved due to precision loss.'
   nfev: 281
    nit: 10
   njev: 87
 status: 2
success: False
      x: array([0.02523853, 0.94423353])

Nelder-Mead

res_NM = minimize(lik_fn, sv, method='Nelder-Mead',
                # jac='2-point',
                bounds=bounds,
                args=(stdData, unconcorr, backcast),
                options={'gtol': 1e-6, 'disp': True})

final_simplex: (array([[0.0252493 , 0.94420818],
     [0.02522809, 0.94425249],
     [0.02522306, 0.9442968 ]]), array([9571.08380417, 9571.08380451, 9571.08380711]))
         fun: 9571.083804171478
     message: 'Optimization terminated successfully.'
        nfev: 72
         nit: 38
      status: 0
     success: True
           x: array([0.0252493 , 0.94420818])

L-BFGS-B

• scipy.optimize.LbfgsInvHessProduct； 参数设置为 scipy.optimize.minimize 设置收敛容差

res_BFGS_B = minimize(lik_fn, sv, method='L-BFGS-B',
                # jac='2-point',
                bounds=bounds,
                args=(stdData, unconcorr, backcast),
                options={'gtol': 1e-6, 'disp': True})
res_BFGS_B.hess_inv.todense()  # 返回矩阵

res_BFGS_B
      fun: 9571.083803563144
 hess_inv: <2x2 LbfgsInvHessProduct with dtype=float64>
      jac: array([0.00400178, 0.00800355])
  message: 'CONVERGENCE: REL_REDUCTION_OF_F_<=_FACTR*EPSMCH'
     nfev: 78
      nit: 10
     njev: 26
   status: 0
  success: True
        x: array([0.02523756, 0.94423662])

COBYLA

opt_C = minimize(
    lik_fn,
    sv,
    args=(stdData, unconcorr, backcast),
    # jac='2-point',
    # hess=SR1(),
    method="COBYLA",
    # bounds=bounds,
    constraints=ineq_constraints,
    tol=tol,
    options=options,
    # callback=llf._callback
    )

opt_C
     fun: 9571.083804272734
   maxcv: 0.0
 message: 'Optimization terminated successfully.'
    nfev: 376
  status: 1
 success: True
       x: array([0.02524922, 0.94419908])

Basinhopping

basinhopping, 默认方法是 L-BFGS-B;

niter = 80
step_r = bs.StepGM2013(rest=True, max=2.5)
step_u = bs.StepGM2013(rest=False, max=2.5)

rest_reg = basinhopping(gm_llf.simpleGM, x0, T=0,
                        minimizer_kwargs=bs.min_arg((args,),  rest),
                        niter=niter,
                        take_step=step_r,
                        # callback=bs.print_fun
                        )