xfidZddlmZddlmZddlZgdZGdde Z ddZ dd Z e Z dd Z dd Zd Zd ZdZddZddZddZ ddZ ddZy)z Functions --------- .. autosummary:: :toctree: generated/ line_search_armijo line_search_wolfe1 line_search_wolfe2 scalar_search_wolfe1 scalar_search_wolfe2 )warn) _minpack2N)LineSearchWarningline_search_wolfe1line_search_wolfe2scalar_search_wolfe1scalar_search_wolfe2line_search_armijoc eZdZy)rN)__name__ __module__ __qualname__.phiIs( 1 ad"T""rct|zzgd<dxxdz cc<tjdSrnpdot)rrfprimegcgvalrrs rderphiz"line_search_wolfe1..derphiMs@ad*T*Q 1 vvd1gr""r)c1c2amaxaminxtol)rr r)rr!rrgfkold_fval old_old_fvalrr%r&r'r(r)rr$derphi0stpfvalrr"r#s```` ` @@@rrrsF {R$ 5D B B#### ffS"oG. <bt$T;Cx 1r!udHd1g 55rc ||d}||d}|"|dk7rtdd||z z|z } | dkrd} nd} |} |} tjdtj} tjdt}d}d }t |D]B}t j| | | ||| |||| | \}} } }|dd d k(r|} ||} ||} Bnd}|dd d k(s|dddk(rd}|| |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?)\(@)) sSTARTdr4sFGsERRORsWARN)minrzerosintcfloatrangeminpack2dcsrch)rr$phi0old_phi0r-r%r&r'r(r)alpha1phi1derphi1isavedsavetaskmaxiterir.s rrr[s0X |2w*1 S&$/27:; A:F DG HHT277 #E HHUE "E DG 7^ #+??6435r437ue$M T7D 8u Fs8DSkG   BQx8tBQx72 d?rc < dgdgdgdgfd} |fd| g}tj|}  fd}nd}t| ||||| | ||  \}}}}|tdtnd}|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. The search direction must be a descent direction for the algorithm to converge. 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. The search direction `pk` must be a descent direction (e.g. ``-myfprime(xk)``) to find a step length that satisfies the strong Wolfe conditions. If the search direction is not a descent direction (e.g. ``myfprime(xk)``), then `alpha`, `new_fval`, and `new_slope` will be None. 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]) rNc<dxxdz cc<|zzgSrr)alpharrrrrs rrzline_search_wolfe2..phis( 1 ebj(4((rc~dxxdz cc<|zzgd<|d<tjdSrr)rLrr!r"r# gval_alpharrs rr$z"line_search_wolfe2..derphi sI 1 ebj040Q 1 vvd1gr""rcPd|k7r||zz}|||dS)Nrr) rLrxr$extra_conditionr#rNrrs rextra_condition2z,line_search_wolfe2..extra_condition2-s8!}%u URZA"5!S$q': :r)rH*The line search algorithm did not converge)rr r rr)rmyfprimerrr*r+r,rr%r&r'rQrHrr-rR alpha_starphi_star derphi_starr$rr!r"r#rNs` `` ` ` @@@@@@rrrs| B B 6DJ))F##  {R$ffS"oG" ; ;  2F <"b$ g3//J(K 9;LM 1g r!ubeXx DDrc ||d}||d}d} ||dk7rtdd||z z|z } nd} | dkrd} | t| |} || } |} |}|d}t| D]}| dk(s|/| |k(r*d}|}|}d}| dk(rd}ndd |zz}t|tn|dkD}| ||| z|zzkDs| | k\r|rt | | | | |||||||| \}}}n|| }t || |zkr|| | r| }| }|}n\|dk\rt | | | | |||||||| \}}}n;d | z}| t||}| } |} | } || } |}| }| }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. Nr1rr2r3cy)NTr)rLrs rrQz-scalar_search_wolfe2..extra_conditionsrz7Rounding errors prevent the line search from convergingz4The line search algorithm could not find a solution zless than or equal to amax: %sr4rS)r9r=rr_zoomabs)rr$r@rAr-r%r&r'rQrHalpha0rBphi_a1phi_a0 derphi_a0rIrUrVrWmsgnot_first_iteration derphi_a1alpha2s rr r Es1r |2w* F1 S&$/27:; z VT" [FFI 7^8N Q;4+$JHDK{OL6=> ' ( !e TBK'11 1 v #6fff$if"GR_F .J+ 6N  Nrc'k )vv.# !'  Nfff$if"GR_F .J+ V  &FV c8Nj   9;LM x{ 22rc ptjddd5 |}||z }||z } || zdz|| z z} tjd} | dz| d<|dz | d<| dz | d<|dz| d <tj| tj||z ||zz ||z || zz gj \} } | | z} | | z} | | zd| z|zz }|| tj |zd| zz z} d d d tjsy |S#t$r Yd d d y wxYw#1swY8xYw) 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. raisedivideoverinvalidr4)r4r4)rr)rr)rr)rrN) rerrstateemptyr asarrayflattensqrtArithmeticErrorisfinite)afafpabfbcrCdbdcdenomd1ABradicalxmins r _cubicminrsk G'7 C AQBQB"WNb2g.E&!BQwBtHaxBtHaxBtHQwBtHVVB BGa"f,<,.Ga"f,<,>!??FwyJFQ JA JA!ea!eai'GRWWW--!a%88D!& ;;t  K  %" #s)D,CD D)D,(D))D,,D5ctjddd5 |}|}||dzz }||z ||zz ||zz }||d|zz z } dddtj sy| S#t$r YdddywxYw#1swY8xYw)z Finds the minimizer for a quadratic polynomial that goes through the points (a,fa), (b,fb) with derivative at a of fpa. rerfr2@N)rrkrprq) rrrsrtrurvDrxryr~rs r_quadminrs G'7 C AAQWBa!b&R"W-AqC!G}$D  ;;t  K   s(A;(A$$ A8-A;7A88A;;Bc d} d} d}d}|}d} ||z }|dkr||}}n||}}| dkDr||z}t|||||||}| dk(s||z kDs|||zkr.||z}t|||||}||||z kDs|||zkr|d|zz}||}||| |z|zzkDs||k\r |}|}|}|}nH||}t|| |zkr| ||r|}|}|}n0|||z zdk\r |}|}|}|}n|}|}|}|}|}| dz } | | kDrd}d}d}n|||fS)a Zoom 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. rg?皙?N?r)rrr[)a_loa_hiphi_lophi_hi derphi_lorr$r@r-r%r&rQrHrIdelta1delta2phi_reca_recdalpharrrucchka_jqchkphi_aj derphi_aja_starval_star valprime_stars rrZrZsG A F FG E  A:qAqA EF?DD&)T6!7,C F q4xS1t8^F?D4D&AC qv34<SZ'S TBsF7N* *&0@GEDFs I9~"W,f1M! ) $+&!+  DF!I Q KFH M A B 8] **rc tjdg  fd}| |d} n|} tj|} t|| | ||\} } | d| fS)aMinimize over alpha, the function ``f(xk+alpha pk)``. Parameters ---------- f : callable Function to be minimized. xk : array_like Current point. pk : array_like Search direction. gfk : array_like Gradient of `f` at point `xk`. old_fval : float Value of `f` at point `xk`. args : tuple, optional Optional arguments. c1 : float, optional Value to control stopping criterion. alpha0 : scalar, optional Value of `alpha` at start of the optimization. Returns ------- alpha f_count f_val_at_alpha Notes ----- Uses the interpolation algorithm (Armijo backtracking) as suggested by Wright and Nocedal in 'Numerical Optimization', 1999, pp. 56-57 rc<dxxdz cc<|zzgSrr)rBrrrrrs rrzline_search_armijo..phis( 1 fRi'$''rr1)r%r\)r atleast_1dr scalar_search_armijo)rrrr*r+rr%r\rr@r-rLrCrs``` ` @rr r istD r B B((2wffS"oG&sD'b.46KE4 "Q% rc Ft||||||||}|d|dd|dfS)z8 Compatibility wrapper for `line_search_armijo` )rr%r\rrr4)r ) rrrr*r+rr%r\rs rline_search_BFGSrs: 1b"c8$2"( *A Q41q!A$ rc^||}||||z|zzkr||fS| |dzzdz ||z ||zz z }||}||||z|zzkr||fS||kDr|dz|dzz||z z} |dz||z ||zz z|dz||z ||zz zz } | | z } |dz ||z ||zz z|dz||z ||zz zz} | | z } | tjt| dzd| z|zz zd| zz } || } | ||| z|zzkr| | fS|| z |dz kDs d| |z z dkr|dz } |}| }|}| }||kDrd|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 r4rrjg@rgQ?N)rror[)rr@r-r%r\r(r^rBr]factorrrrurcphi_a2s rrrs[F 6 ')))v~Z&!) #c )Vd]Wv=M-M NF [F$F7***v~ 4-VQY&&-8 AI$7 8 AI$7 8 9 J QYJ&4-'&.8 9 AI$7 8 9 J"rwws1a4!a%'/#9:;;AFV dRYw.. .6> ! VOv| +F6M0AT/Ic\F+ 4-0 <rc|d}t|} d} d} d} || |zz} || \}}|| |z|| dzz|zz kr| } n| dz|z|d| zdz |zzz }|| |zz } || \}}|| |z|| dzz|zz kr| } nR| dz|z|d| zdz |zzz }tj||| z|| z} tj||| z|| z} | | ||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). rr4)maxrclip)rx_kdprev_fsetagammatau_mintau_maxf_kf_baralpha_palpha_mrLxpfpFpalpha_tpalpha_tms r_nonmonotone_line_search_cruzrsGP "+C LEGG E  7Q; 2B uwz1C77 7E A:#rQwY]C,?'?@ 7Q; 2B uwz1C77 7HE A:#rQwY]C,?'?@''(Gg$5w7HI''(Gg$5w7HI) , "b" rc d} d} d} || |zz}||\}}|||z|| dzz|zz kr| } n| dz|z|d| zdz |zzz }|| |zz }||\}}|||z|| dzz|zz kr| } nR| dz|z|d| zdz |zzz }tj||| z| | z} tj||| z| | z} | |zdz}| |z||zz|z|z }|}| |||||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). rr4)rr)rrrrrxQrrrrnurrrLrrrrrQ_nexts r_nonmonotone_line_search_chengr,sg^GG E  7Q; 2B S57A:-33 3E A:#rQwY]C,?'?@ 7Q; 2B S57A:-33 3HE A:#rQwY]C,?'?@''(Gg$5w7HI''(Gg$5w7HI) .!VaZF a1s7 b F*AA "b"a ""r) NNNr-C6??2:0yE>+=)NNNrrrrr) NNNrrrNNr)NNNrrNNr)rrr)rrr)rrr)rrrg333333?)__doc__warningsrscipy.optimizerr>numpyr__all__RuntimeWarningrrr line_searchrr rrrZr rrrrrrrrs 0 !  /337?C!96xIM%(27Pf! @DIM57KE\,004/379O3dD*T+v1h7~DGEREH&*N#r