U Gv f=@s|dZddlZddlmZddlmZddlmZdddgZ Gd ddZ Gd d d e Z Gd dde Z Gd dde Z dS)z@Hessian update strategies for quasi-Newton optimization methods.N)norm)get_blas_funcs)warnHessianUpdateStrategyBFGSSR1c@s0eZdZdZddZddZddZdd Zd S) ra]Interface for implementing Hessian update strategies. Many optimization methods make use of Hessian (or inverse Hessian) approximations, such as the quasi-Newton methods BFGS, SR1, L-BFGS. Some of these approximations, however, do not actually need to store the entire matrix or can compute the internal matrix product with a given vector in a very efficiently manner. This class serves as an abstract interface between the optimization algorithm and the quasi-Newton update strategies, giving freedom of implementation to store and update the internal matrix as efficiently as possible. Different choices of initialization and update procedure will result in different quasi-Newton strategies. Four methods should be implemented in derived classes: ``initialize``, ``update``, ``dot`` and ``get_matrix``. Notes ----- Any instance of a class that implements this interface, can be accepted by the method ``minimize`` and used by the compatible solvers to approximate the Hessian (or inverse Hessian) used by the optimization algorithms. cCs tddS)Initialize internal matrix. Allocate internal memory for storing and updating the Hessian or its inverse. Parameters ---------- n : int Problem dimension. approx_type : {'hess', 'inv_hess'} Selects either the Hessian or the inverse Hessian. When set to 'hess' the Hessian will be stored and updated. When set to 'inv_hess' its inverse will be used instead. z=The method ``initialize(n, approx_type)`` is not implemented.NNotImplementedErrorselfn approx_typerK/tmp/pip-unpacked-wheel-96ln3f52/scipy/optimize/_hessian_update_strategy.py initialize$sz HessianUpdateStrategy.initializecCs tddS)Update internal matrix. Update Hessian matrix or its inverse (depending on how 'approx_type' is defined) using information about the last evaluated points. Parameters ---------- delta_x : ndarray The difference between two points the gradient function have been evaluated at: ``delta_x = x2 - x1``. delta_grad : ndarray The difference between the gradients: ``delta_grad = grad(x2) - grad(x1)``. z>The method ``update(delta_x, delta_grad)`` is not implemented.Nr r delta_x delta_gradrrrupdate6szHessianUpdateStrategy.updatecCs tddS)PCompute the product of the internal matrix with the given vector. Parameters ---------- p : array_like 1-D array representing a vector. Returns ------- Hp : array 1-D represents the result of multiplying the approximation matrix by vector p. z)The method ``dot(p)`` is not implemented.Nr r prrrdotHszHessianUpdateStrategy.dotcCs tddS)zReturn current internal matrix. Returns ------- H : ndarray, shape (n, n) Dense matrix containing either the Hessian or its inverse (depending on how 'approx_type' is defined). z0The method ``get_matrix(p)`` is not implemented.Nr )r rrr get_matrixYs z HessianUpdateStrategy.get_matrixN)__name__ __module__ __qualname____doc__rrrrrrrrr s c@sneZdZdZedddZedddZedddZddd Zd d Z d d Z ddZ ddZ ddZ ddZdS)FullHessianUpdateStrategyzKHessian update strategy with full dimensional internal representation. ZsyrdZdtypeZsyr2ZsymvautocCs"||_d|_d|_d|_d|_dSN) init_scalefirst_iterationrBH)r r%rrr__init__os z"FullHessianUpdateStrategy.__init__cCsRd|_||_||_|dkr"td|jdkr>tj|td|_ntj|td|_dS)rT)hessZinv_hessz+`approx_type` must be 'hess' or 'inv_hess'.r*r"N) r&r r ValueErrornpeyefloatr'r(r rrrrxs z$FullHessianUpdateStrategy.initializecCsdt||}t||}tt||}|dksB|dksB|dkrFdS|jdkrX||S||SdS)Nrr*)r,rabsr)r rrZs_norm2Zy_norm2ysrrr _auto_scales   z%FullHessianUpdateStrategy._auto_scalecCs tddS)Nz9The method ``_update_implementation`` is not implemented.r rrrr_update_implementationsz0FullHessianUpdateStrategy._update_implementationcCst|dkrdSt|dkr.tdtdS|jr|jdkrL|||}n t|j}|jdkrp|j |9_ n|j |9_ d|_| ||dS)rr/Nzdelta_grad == 0.0. Check if the approximated function is linear. If the function is linear better results can be obtained by defining the Hessian as zero instead of using quasi-Newton approximations.r#r*F) r,allr UserWarningr&r%r3r.rr'r(r4)r rrscalerrrrs    z FullHessianUpdateStrategy.updatecCs.|jdkr|d|j|S|d|j|SdS)rr*r0N)r_symvr'r(rrrrrs zFullHessianUpdateStrategy.dotcCsD|jdkrt|j}n t|j}tj|dd}|j|||<|S)zReturn the current internal matrix. Returns ------- M : ndarray, shape (n, n) Dense matrix containing either the Hessian or its inverse (depending on how `approx_type` was defined). r*)k)rr,copyr'r(Ztril_indices_fromT)r Mlirrrrs  z$FullHessianUpdateStrategy.get_matrixN)r#)rrrrr_syr_syr2r8r)rr3r4rrrrrrrr gs    &r cs:eZdZdZd fdd ZddZd d Zd d ZZS)raBroyden-Fletcher-Goldfarb-Shanno (BFGS) Hessian update strategy. Parameters ---------- exception_strategy : {'skip_update', 'damp_update'}, optional Define how to proceed when the curvature condition is violated. Set it to 'skip_update' to just skip the update. Or, alternatively, set it to 'damp_update' to interpolate between the actual BFGS result and the unmodified matrix. Both exceptions strategies are explained in [1]_, p.536-537. min_curvature : float This number, scaled by a normalization factor, defines the minimum curvature ``dot(delta_grad, delta_x)`` allowed to go unaffected by the exception strategy. By default is equal to 1e-8 when ``exception_strategy = 'skip_update'`` and equal to 0.2 when ``exception_strategy = 'damp_update'``. init_scale : {float, 'auto'} Matrix scale at first iteration. At the first iteration the Hessian matrix or its inverse will be initialized with ``init_scale*np.eye(n)``, where ``n`` is the problem dimension. Set it to 'auto' in order to use an automatic heuristic for choosing the initial scale. The heuristic is described in [1]_, p.143. By default uses 'auto'. Notes ----- The update is based on the description in [1]_, p.140. References ---------- .. [1] Nocedal, Jorge, and Stephen J. Wright. "Numerical optimization" Second Edition (2006). skip_updateNr#cs^|dkr |dk r||_qHd|_n(|dkr@|dk r8||_qHd|_ntdt|||_dS)NrA:0yE> damp_updateg?z<`exception_strategy` must be 'skip_update' or 'damp_update'.) min_curvaturer+superr)exception_strategy)r rFrDr% __class__rrr)s z BFGS.__init__cCs>|jd||||jd|_|j|||d||jd|_dS)aUpdate the inverse Hessian matrix. BFGS update using the formula: ``H <- H + ((H*y).T*y + s.T*y)/(s.T*y)^2 * (s*s.T) - 1/(s.T*y) * ((H*y)*s.T + s*(H*y).T)`` where ``s = delta_x`` and ``y = delta_grad``. This formula is equivalent to (6.17) in [1]_ written in a more efficient way for implementation. References ---------- .. [1] Nocedal, Jorge, and Stephen J. Wright. "Numerical optimization" Second Edition (2006). aN)r@r(r?)r r2ZHyZyHysrrr_update_inverse_hessian%szBFGS._update_inverse_hessiancCs4|jd|||jd|_|jd|||jd|_dS)aUpdate the Hessian matrix. BFGS update using the formula: ``B <- B - (B*s)*(B*s).T/s.T*(B*s) + y*y^T/s.T*y`` where ``s`` is short for ``delta_x`` and ``y`` is short for ``delta_grad``. Formula (6.19) in [1]_. References ---------- .. [1] Nocedal, Jorge, and Stephen J. Wright. "Numerical optimization" Second Edition (2006). g?rJrIN)r?r')r r2ZBsZsBsyrrr_update_hessian9szBFGS._update_hessianc Cs*|jdkr|}|}n|}|}t||}||}||}|dkr|||}|jdkrr|tj|jtd|_n|tj|jtd|_||}||}||j |kr|j dkrdS|j dkrd|j d||} | |d| |}t||}|jdkr| ||||n| ||||dS)Nr*r/r"rArCr0) rr,rr3r-r r.r'r(rDrFrPrN) r rrwzZwzMwZwMwr7Z update_factorrrrr4Ks2            zBFGS._update_implementation)rANr#) rrrrr)rNrPr4 __classcell__rrrGrrs"cs*eZdZdZdfdd ZddZZS) raSymmetric-rank-1 Hessian update strategy. Parameters ---------- min_denominator : float This number, scaled by a normalization factor, defines the minimum denominator magnitude allowed in the update. When the condition is violated we skip the update. By default uses ``1e-8``. init_scale : {float, 'auto'}, optional Matrix scale at first iteration. At the first iteration the Hessian matrix or its inverse will be initialized with ``init_scale*np.eye(n)``, where ``n`` is the problem dimension. Set it to 'auto' in order to use an automatic heuristic for choosing the initial scale. The heuristic is described in [1]_, p.143. By default uses 'auto'. Notes ----- The update is based on the description in [1]_, p.144-146. References ---------- .. [1] Nocedal, Jorge, and Stephen J. Wright. "Numerical optimization" Second Edition (2006). rBr#cs||_t|dSr$)min_denominatorrEr))r rUr%rGrrr)sz SR1.__init__cCs|jdkr|}|}n|}|}||}||}t||}t||jt|t|kr^dS|jdkr|jd|||jd|_n|jd|||jd|_dS)Nr*r0rJ) rrr,r1rUrr?r'r()r rrrQrRrSZ z_minus_Mw denominatorrrrr4s     zSR1._update_implementation)rBr#)rrrrr)r4rTrrrGrrys)rZnumpyr,Z numpy.linalgrZ scipy.linalgrwarningsr__all__rr rrrrrrs    \