U Gv fg @sdZddlmZddlmZddlZddddd d gZGd dde Z d*ddZ d+ddZ e Z d,ddZd-dd ZddZddZddZd.dd Zd/dd Zd0d!d"Zd1d%d&Zd2d(d)ZdS)3z Functions --------- .. autosummary:: :toctree: generated/ line_search_armijo line_search_wolfe1 line_search_wolfe2 scalar_search_wolfe1 scalar_search_wolfe2 )warn) _minpack2NLineSearchWarningline_search_wolfe1line_search_wolfe2scalar_search_wolfe1scalar_search_wolfe2line_search_armijoc@s eZdZdS)rN)__name__ __module__ __qualname__r r >/tmp/pip-unpacked-wheel-96ln3f52/scipy/optimize/_linesearch.pyrsr -C6??2:0yE>+=c  s|dkrf}|gdgdgfdd} fdd}t|}t| |||||| | | | d \}}}|dd||dfS)a As `scalar_search_wolfe1` but do a line search to direction `pk` Parameters ---------- f : callable Function `f(x)` fprime : callable Gradient of `f` xk : array_like Current point pk : array_like Search direction gfk : array_like, optional Gradient of `f` at point `xk` old_fval : float, optional Value of `f` at point `xk` old_old_fval : float, optional Value of `f` at point preceding `xk` The rest of the parameters are the same as for `scalar_search_wolfe1`. Returns ------- stp, f_count, g_count, fval, old_fval As in `line_search_wolfe1` gval : array Gradient of `f` at the final point Nrcs&dd7<|fSNrr sargsffcpkxkr rphiIszline_search_wolfe1..phics:|fd<dd7<tdSrnpdotr)rfprimegcgvalrrr rderphiMsz"line_search_wolfe1..derphi)c1c2amaxaminxtol)r r!r)rr"rrgfkold_fval old_old_fvalrr&r'r(r)r*rr%derphi0stpZfvalr )rrrr"r#r$rrrrs*#  c Cs|dkr|d}|dkr |d}|dk rT|dkrTtdd|||} | dkrXd} nd} |} |} tdtj} tdt}d}d }t|D]T}t| | | ||| |||| | \}} } }|dd d kr|} ||} ||} qqqd}|dd d ks|dddkr d}|| |fS)a, Scalar function search for alpha that satisfies strong Wolfe conditions alpha > 0 is assumed to be a descent direction. Parameters ---------- phi : callable phi(alpha) Function at point `alpha` derphi : callable phi'(alpha) Objective function derivative. Returns a scalar. phi0 : float, optional Value of phi at 0 old_phi0 : float, optional Value of phi at previous point derphi0 : float, optional Value derphi at 0 c1 : float, optional Parameter for Armijo condition rule. c2 : float, optional Parameter for curvature condition rule. amax, amin : float, optional Maximum and minimum step size xtol : float, optional Relative tolerance for an acceptable step. Returns ------- alpha : float Step size, or None if no suitable step was found phi : float Value of `phi` at the new point `alpha` phi0 : float Value of `phi` at `alpha=0` Notes ----- Uses routine DCSRCH from MINPACK. Nr?)\(@)) sSTARTdr3sFGsERRORsWARN)minr zerosZintcfloatrangeminpack2Zdcsrch)rr%phi0old_phi0r.r&r'r(r)r*alpha1phi1Zderphi1ZisaveZdsaveZtaskmaxiterir/r r rr[sF,     $ c  sdgdgdgdg fdd} | fdd|dkr^ f}t| }dk r fdd}nd}t| ||||| | || d \}}}}|dkrtd tnd}|dd|||fS) a Find alpha that satisfies strong Wolfe conditions. Parameters ---------- f : callable f(x,*args) Objective function. myfprime : callable f'(x,*args) Objective function gradient. xk : ndarray Starting point. pk : ndarray Search direction. gfk : ndarray, optional Gradient value for x=xk (xk being the current parameter estimate). Will be recomputed if omitted. old_fval : float, optional Function value for x=xk. Will be recomputed if omitted. old_old_fval : float, optional Function value for the point preceding x=xk. args : tuple, optional Additional arguments passed to objective function. c1 : float, optional Parameter for Armijo condition rule. c2 : float, optional Parameter for curvature condition rule. amax : float, optional Maximum step size extra_condition : callable, optional A callable of the form ``extra_condition(alpha, x, f, g)`` returning a boolean. Arguments are the proposed step ``alpha`` and the corresponding ``x``, ``f`` and ``g`` values. The line search accepts the value of ``alpha`` only if this callable returns ``True``. If the callable returns ``False`` for the step length, the algorithm will continue with new iterates. The callable is only called for iterates satisfying the strong Wolfe conditions. maxiter : int, optional Maximum number of iterations to perform. Returns ------- alpha : float or None Alpha for which ``x_new = x0 + alpha * pk``, or None if the line search algorithm did not converge. fc : int Number of function evaluations made. gc : int Number of gradient evaluations made. new_fval : float or None New function value ``f(x_new)=f(x0+alpha*pk)``, or None if the line search algorithm did not converge. old_fval : float Old function value ``f(x0)``. new_slope : float or None The local slope along the search direction at the new value ````, or None if the line search algorithm did not converge. Notes ----- Uses the line search algorithm to enforce strong Wolfe conditions. See Wright and Nocedal, 'Numerical Optimization', 1999, pp. 59-61. Examples -------- >>> import numpy as np >>> from scipy.optimize import line_search A objective function and its gradient are defined. >>> def obj_func(x): ... return (x[0])**2+(x[1])**2 >>> def obj_grad(x): ... return [2*x[0], 2*x[1]] We can find alpha that satisfies strong Wolfe conditions. >>> start_point = np.array([1.8, 1.7]) >>> search_gradient = np.array([-1.0, -1.0]) >>> line_search(obj_func, obj_grad, start_point, search_gradient) (1.0, 2, 1, 1.1300000000000001, 6.13, [1.6, 1.4]) rNcs&dd7<|fSrr alpharr rrszline_search_wolfe2..phicsBdd7<|fd<|d<tdSrrrD)rr"r#r$ gval_alpharrr rr%sz"line_search_wolfe2..derphics2d|kr||}|||dS)Nrr )rErx)r%extra_conditionr$rFrrr rextra_condition2%s  z,line_search_wolfe2..extra_condition2)rA*The line search algorithm did not converge)r r!rrr)rZmyfprimerrr+r,r-rr&r'r(rHrArr.rI alpha_starphi_star derphi_starr ) rr%rHrrr"r#r$rFrrrrs:X  c Cs |dkr|d}|dkr |d}d} |dk rL|dkrLtdd|||} nd} | dkr\d} |dk rnt| |} || } |} |}|dkrdd}t| D]P}| dks|dk r| |krd}|}|}d}| dkrd}n d d |}t|tq|dk}| ||| |ks| | krF|rFt| | | | |||||||| \}}}q|| }t|| |kr|| | r| }| }|}q|dkrt| | | | |||||||| \}}}qd | }|dk rt||}| } |} | } || } |}q| }| }d}td t||||fS) aFind alpha that satisfies strong Wolfe conditions. alpha > 0 is assumed to be a descent direction. Parameters ---------- phi : callable phi(alpha) Objective scalar function. derphi : callable phi'(alpha) Objective function derivative. Returns a scalar. phi0 : float, optional Value of phi at 0. old_phi0 : float, optional Value of phi at previous point. derphi0 : float, optional Value of derphi at 0 c1 : float, optional Parameter for Armijo condition rule. c2 : float, optional Parameter for curvature condition rule. amax : float, optional Maximum step size. extra_condition : callable, optional A callable of the form ``extra_condition(alpha, phi_value)`` returning a boolean. The line search accepts the value of ``alpha`` only if this callable returns ``True``. If the callable returns ``False`` for the step length, the algorithm will continue with new iterates. The callable is only called for iterates satisfying the strong Wolfe conditions. maxiter : int, optional Maximum number of iterations to perform. Returns ------- alpha_star : float or None Best alpha, or None if the line search algorithm did not converge. phi_star : float phi at alpha_star. phi0 : float phi at 0. derphi_star : float or None derphi at alpha_star, or None if the line search algorithm did not converge. Notes ----- Uses the line search algorithm to enforce strong Wolfe conditions. See Wright and Nocedal, 'Numerical Optimization', 1999, pp. 59-61. Nr0rr1r2cSsdS)NTr )rErr r rz&scalar_search_wolfe2..z7Rounding errors prevent the line search from convergingz4The line search algorithm could not find a solution zless than or equal to amax: %sr3rJ)r8r;rr_zoomabs)rr%r=r>r.r&r'r(rHrAalpha0r?phi_a1phi_a0Z derphi_a0rBrKrLrMmsgZnot_first_iterationZ derphi_a1alpha2r r rr=s9       c Cs6tjdddd z|}||}||} || d|| } td} | d| d<|d | d<| d | d<|d| d <t| t|||||||| g\} } | | } | | } | | d| |}|| t|d| }Wn"tk rYW5QRd SXW5QRXt|s2d S|S) z Finds the minimizer for a cubic polynomial that goes through the points (a,fa), (b,fb), and (c,fc) with derivative at a of fpa. If no minimizer can be found, return None. raisedivideZoverinvalidr3)r3r3)rr)rr)rr)rrN) r errstateemptyr!ZasarrayflattensqrtArithmeticErrorisfinite)afafpabfbcrCdbdcZdenomd1ABradicalxminr r r _cubicmins.      rpc Cstjddddhz@|}|}||d}||||||}||d|} Wn tk rrYW5QRdSXW5QRXt| sdS| S)z Finds the minimizer for a quadratic polynomial that goes through the points (a,fa), (b,fb) with derivative at a of fpa. rWrXr1@N)r r\r`ra) rbrcrdrerfDrhrirmror r r_quadmins  rsc Csd} d} d}d}|}d}||}|dkr4||}}n ||}}| dkrb||}t|||||||}| dks|dks|||ks|||kr||}t|||||}|dks|||ks|||kr|d|}||}||| ||ks||kr|}|}|}|}np||}t|| |kr>| ||r>|}|}|}q|||dkrb|}|}|}|}n|}|}|}|}|}| d7} | | krd}d}d}qq|||fS)aZoom stage of approximate linesearch satisfying strong Wolfe conditions. Part of the optimization algorithm in `scalar_search_wolfe2`. Notes ----- Implements Algorithm 3.6 (zoom) in Wright and Nocedal, 'Numerical Optimization', 1999, pp. 61. rCrg?皙?N?r)rprsrQ)Za_loZa_hiZphi_loZphi_hiZ derphi_lorr%r=r.r&r'rHrArBZdelta1Zdelta2Zphi_recZa_recZdalpharbreZcchkZa_jZqchkZphi_ajZ derphi_ajZa_starZval_starZ valprime_starr r rrPsd     (   rPrc sjtdgfdd}|dkr6|d} n|} t|} t|| | ||d\} } | d| fS)aMinimize over alpha, the function ``f(xk+alpha pk)``. Parameters ---------- f : callable Function to be minimized. xk : array_like Current point. pk : array_like Search direction. gfk : array_like Gradient of `f` at point `xk`. old_fval : float Value of `f` at point `xk`. args : tuple, optional Optional arguments. c1 : float, optional Value to control stopping criterion. alpha0 : scalar, optional Value of `alpha` at start of the optimization. Returns ------- alpha f_count f_val_at_alpha Notes ----- Uses the interpolation algorithm (Armijo backtracking) as suggested by Wright and Nocedal in 'Numerical Optimization', 1999, pp. 56-57 rcs&dd7<|fSrr )r?rr rrszline_search_armijo..phiNr0)r&rR)r Z atleast_1dr!scalar_search_armijo) rrrr+r,rr&rRrr=r.rEr@r rrr `s"     c Cs0t||||||||d}|d|dd|dfS)z8 Compatibility wrapper for `line_search_armijo` )rr&rRrrr3)r ) rrrr+r,rr&rRrr r rline_search_BFGSsrxcCs||}|||||kr$||fS| |dd||||}||}|||||krj||fS||kr|d|d||} |d|||||d||||} | | } |d |||||d||||} | | } | tt| dd| |d| } || } | ||| |krP| | fS|| |dkstd| |dkr||d} |}| }|}| }qjd|fS)a(Minimize over alpha, the function ``phi(alpha)``. Uses the interpolation algorithm (Armijo backtracking) as suggested by Wright and Nocedal in 'Numerical Optimization', 1999, pp. 56-57 alpha > 0 is assumed to be a descent direction. Returns ------- alpha phi1 r3rqr[g@rgQ?N)r r_rQ)rr=r.r&rRr)rTr?rSZfactorrbrerVZphi_a2r r rrvs8" ,$rvrtrucCs|d}t|} d} d} d} || |} || \}}|| ||| d|krX| } q| d||d| d|}|| |} || \}}|| ||| d|kr| } q| d||d| d|}t||| || } t||| || } q| | ||fS)a@ Nonmonotone backtracking line search as described in [1]_ Parameters ---------- f : callable Function returning a tuple ``(f, F)`` where ``f`` is the value of a merit function and ``F`` the residual. x_k : ndarray Initial position. d : ndarray Search direction. prev_fs : float List of previous merit function values. Should have ``len(prev_fs) <= M`` where ``M`` is the nonmonotonicity window parameter. eta : float Allowed merit function increase, see [1]_ gamma, tau_min, tau_max : float, optional Search parameters, see [1]_ Returns ------- alpha : float Step length xp : ndarray Next position fp : float Merit function value at next position Fp : ndarray Residual at next position References ---------- [1] "Spectral residual method without gradient information for solving large-scale nonlinear systems of equations." W. La Cruz, J.M. Martinez, M. Raydan. Math. Comp. **75**, 1429 (2006). rr3)maxr clip)rx_kdZprev_fsetagammatau_mintau_maxf_kZf_baralpha_palpha_mrExpfpFpalpha_tpalpha_tmr r r_nonmonotone_line_search_cruzs((      r333333?c Cs(d} d} d} || |}||\}}||||| d|krF| } q| d||d| d|}|| |}||\}}||||| d|kr| } q| d||d| d|}t||| | | } t||| | | } q | |d}| |||||}|}| |||||fS)a Nonmonotone line search from [1] Parameters ---------- f : callable Function returning a tuple ``(f, F)`` where ``f`` is the value of a merit function and ``F`` the residual. x_k : ndarray Initial position. d : ndarray Search direction. f_k : float Initial merit function value. C, Q : float Control parameters. On the first iteration, give values Q=1.0, C=f_k eta : float Allowed merit function increase, see [1]_ nu, gamma, tau_min, tau_max : float, optional Search parameters, see [1]_ Returns ------- alpha : float Step length xp : ndarray Next position fp : float Merit function value at next position Fp : ndarray Residual at next position C : float New value for the control parameter C Q : float New value for the control parameter Q References ---------- .. [1] W. Cheng & D.-H. Li, ''A derivative-free nonmonotone line search and its application to the spectral residual method'', IMA J. Numer. Anal. 29, 814 (2009). rr3)r r{)rr|r}rrhQr~rrrnurrrErrrrrZQ_nextr r r_nonmonotone_line_search_cheng#s*/       r) NNNr rrrrr)NNNrrrrr) NNNr rrNNrC)NNNrrNNrC)r rr)r rr)rrr)rrtru)rrtrur)__doc__warningsrZscipy.optimizerr<Znumpyr __all__RuntimeWarningrrrZ line_searchrrrprsrPr rxrvrrr r r rs|   < S   "[ 4 ? I