sg@XdZddlmZmZmZddlmZddlZddl m Z gdZ dZ dd Z dd Z dd Zd Zd ZdZej&ddddddfdZy)z3Equality-constrained quadratic programming solvers.)linalgbmat csc_matrix)copysignN)norm) eqp_kktfactsphere_intersectionsbox_intersectionsbox_sphere_intersectionsinside_box_boundariesmodified_dogleg projected_cgcHtj|\}tj|\}tt||jg|dgg}tj | | g}t j|}|j|} | d|} | |||z } | | fS)aSolve equality-constrained quadratic programming (EQP) problem. Solve ``min 1/2 x.T H x + x.t c`` subject to ``A x + b = 0`` using direct factorization of the KKT system. Parameters ---------- H : sparse matrix, shape (n, n) Hessian matrix of the EQP problem. c : array_like, shape (n,) Gradient of the quadratic objective function. A : sparse matrix Jacobian matrix of the EQP problem. b : array_like, shape (m,) Right-hand side of the constraint equation. Returns ------- x : array_like, shape (n,) Solution of the KKT problem. lagrange_multipliers : ndarray, shape (m,) Lagrange multipliers of the KKT problem. N) npshaperrThstackrsplusolve) HcAbnm kkt_matrixkkt_veclukkt_solxlagrange_multiplierss c/var/www/html/venv/lib/python3.12/site-packages/scipy/optimize/_trustregion_constr/qp_subproblem.pyrrs0 !BA !BA D1acc(QI!678Jii!aR!G Z BhhwG A#AacN? " ""Fcft|dk(rytj|r/|r"tj }tj}nd}d}d}|||fStj||}dtj||z}tj|||dzz } ||zd|z| zz } | dkrd}dd|fStj | } |t | |z} | d|zz }d| z| z }t||g\}}|rd}n+|dks|dkDrd}d}d}nd}td|}td|}|||fS) aHFind the intersection between segment (or line) and spherical constraints. Find the intersection between the segment (or line) defined by the parametric equation ``x(t) = z + t*d`` and the ball ``||x|| <= trust_radius``. Parameters ---------- z : array_like, shape (n,) Initial point. d : array_like, shape (n,) Direction. trust_radius : float Ball radius. entire_line : bool, optional When ``True``, the function returns the intersection between the line ``x(t) = z + t*d`` (``t`` can assume any value) and the ball ``||x|| <= trust_radius``. When ``False``, the function returns the intersection between the segment ``x(t) = z + t*d``, ``0 <= t <= 1``, and the ball. Returns ------- ta, tb : float The line/segment ``x(t) = z + t*d`` is inside the ball for for ``ta <= t <= tb``. intersect : bool When ``True``, there is a intersection between the line/segment and the sphere. On the other hand, when ``False``, there is no intersection. rrrFTF) rrisinfinfdotsqrtrsortedmaxmin) zd trust_radius entire_linetatb intersectarr discriminantsqrt_discriminantauxs r"r r As\B Aw!| xx  &&BBBB 2y   q! A BFF1aLA q! |Q&AQ31Q;La !Y - h(!, ,C 1B AB RH FB  6R!VIBBIQBQB r9 r#ctj|}tj|}tj|}tj|}t|dk(ry|dk(}||||kjs||||kDjrd}dd|fStj|}||}||}||}||}||z |z }||z |z } t tj || } ttj|| } | | krd}nd}|s)| dks| dkDrd}d} d} nt d| } td| } | | |fS)a5Find the intersection between segment (or line) and box constraints. Find the intersection between the segment (or line) defined by the parametric equation ``x(t) = z + t*d`` and the rectangular box ``lb <= x <= ub``. Parameters ---------- z : array_like, shape (n,) Initial point. d : array_like, shape (n,) Direction. lb : array_like, shape (n,) Lower bounds to each one of the components of ``x``. Used to delimit the rectangular box. ub : array_like, shape (n, ) Upper bounds to each one of the components of ``x``. Used to delimit the rectangular box. entire_line : bool, optional When ``True``, the function returns the intersection between the line ``x(t) = z + t*d`` (``t`` can assume any value) and the rectangular box. When ``False``, the function returns the intersection between the segment ``x(t) = z + t*d``, ``0 <= t <= 1``, and the rectangular box. Returns ------- ta, tb : float The line/segment ``x(t) = z + t*d`` is inside the box for for ``ta <= t <= tb``. intersect : bool When ``True``, there is a intersection between the line (or segment) and the rectangular box. On the other hand, when ``False``, there is no intersection. rr%FTr&) rasarrayrany logical_notr/minimumr0maximum) r1r2lbubr4zero_dr7 not_zero_dt_lbt_ubr5r6s r"r r seJ 1 A 1 A BB BB Aw!|1fF & BvJ##%!F)bj*@)E)E)G !Y'J * A * A JB JB qDA:D qDA:D RZZd # $B RZZd # $B Rx   6R!VIBBQBQB r9 r#ct|||||\}}} t||||\} } } tj|| } tj|| }| r | r| |krd}nd}|r| | | d}||| d}| ||||fS| ||fS)aFind the intersection between segment (or line) and box/sphere constraints. Find the intersection between the segment (or line) defined by the parametric equation ``x(t) = z + t*d``, the rectangular box ``lb <= x <= ub`` and the ball ``||x|| <= trust_radius``. Parameters ---------- z : array_like, shape (n,) Initial point. d : array_like, shape (n,) Direction. lb : array_like, shape (n,) Lower bounds to each one of the components of ``x``. Used to delimit the rectangular box. ub : array_like, shape (n, ) Upper bounds to each one of the components of ``x``. Used to delimit the rectangular box. trust_radius : float Ball radius. entire_line : bool, optional When ``True``, the function returns the intersection between the line ``x(t) = z + t*d`` (``t`` can assume any value) and the constraints. When ``False``, the function returns the intersection between the segment ``x(t) = z + t*d``, ``0 <= t <= 1`` and the constraints. extra_info : bool, optional When ``True``, the function returns ``intersect_sphere`` and ``intersect_box``. Returns ------- ta, tb : float The line/segment ``x(t) = z + t*d`` is inside the rectangular box and inside the ball for ``ta <= t <= tb``. intersect : bool When ``True``, there is a intersection between the line (or segment) and both constraints. On the other hand, when ``False``, there is no intersection. sphere_info : dict, optional Dictionary ``{ta, tb, intersect}`` containing the interval ``[ta, tb]`` for which the line intercepts the ball. And a boolean value indicating whether the sphere is intersected by the line. box_info : dict, optional Dictionary ``{ta, tb, intersect}`` containing the interval ``[ta, tb]`` for which the line intercepts the box. And a boolean value indicating whether the box is intersected by the line. TF)r5r6r7)r r rrAr@)r1r2rBrCr3r4 extra_infota_btb_b intersect_bta_stb_s intersect_sr5r6r7 sphere_infobox_infos r"r r sb01b"0;=D$ 21a3?3>@D$  D$ B D$ B{rRx  !KH dE2y+x772y  r#cR||kjxr||kjS)zCheck if lb <= x <= ub.)allr rBrCs r"r r 1s! !G==? .R}}.r#cVtjtj|||S)zReturn clipped value of x)rr@rArTs r"reinforce_box_boundariesrV6s ::bjjB' ,,r#c|j| }t|||rt||kr|}|S|jj|}|j|} t j|| t j| | z |z} t j | } | } || z } t | | |||\}}}|r | || zz}n| } | } t | | |||\}}}| || zz}| } |} t | | |||\}}}| || zz}t|j||zt|j||zkr|S|S)aAApproximately minimize ``1/2*|| A x + b ||^2`` inside trust-region. Approximately solve the problem of minimizing ``1/2*|| A x + b ||^2`` subject to ``||x|| < Delta`` and ``lb <= x <= ub`` using a modification of the classical dogleg approach. Parameters ---------- A : LinearOperator (or sparse matrix or ndarray), shape (m, n) Matrix ``A`` in the minimization problem. It should have dimension ``(m, n)`` such that ``m < n``. Y : LinearOperator (or sparse matrix or ndarray), shape (n, m) LinearOperator that apply the projection matrix ``Q = A.T inv(A A.T)`` to the vector. The obtained vector ``y = Q x`` being the minimum norm solution of ``A y = x``. b : array_like, shape (m,) Vector ``b``in the minimization problem. trust_radius: float Trust radius to be considered. Delimits a sphere boundary to the problem. lb : array_like, shape (n,) Lower bounds to each one of the components of ``x``. It is expected that ``lb <= 0``, otherwise the algorithm may fail. If ``lb[i] = -Inf``, the lower bound for the ith component is just ignored. ub : array_like, shape (n, ) Upper bounds to each one of the components of ``x``. It is expected that ``ub >= 0``, otherwise the algorithm may fail. If ``ub[i] = Inf``, the upper bound for the ith component is just ignored. Returns ------- x : array_like, shape (n,) Solution to the problem. Notes ----- Based on implementations described in pp. 885-886 from [1]_. References ---------- .. [1] Byrd, Richard H., Mary E. Hribar, and Jorge Nocedal. "An interior point algorithm for large-scale nonlinear programming." SIAM Journal on Optimization 9.4 (1999): 877-900. )r,r rrr zeros_liker )rYrr3rBrC newton_pointr gA_g cauchy_point origin_pointr1p_alphar7x1x2s r"r r ;si`EE!H9L\2r2   -   A %%(CFF1aL=266#s#33a7L==.L A|#A21aR3?AAui q[  .q!R/;= 5! q[ AA*1aR+79KAua U1WB AEE"IMT!%%)a-00  r#c d} tj|\} tj|\}|j| }|j|j||z}|j|}| }| r|g}|j|}t|dz}|t|z }|dkr t d|| kr"dddd}| rj |||d<||fS|/t tdtj|zd |z| }|%tj| tj }|$tj| tj}| | |z } t| | |z } | | |z } d }d }d}tj|}d}t| D]}||krd }n|d z }|j|}|dkrStj|r t d t|||||d\}} }!|!r|| |zz}t|||}d}d}n(||z } || |zz}"tj j|"|k\r6t|| |z|||\}}#}!|!r ||#| z|zz}t|||}d}d}nt#|"||rd}n|d z }|dkDr2t|| |z|||\}}#}!|!r||#| z|zz}t|||}d}|| kDrnp| rj |"|| |zz}$|j|$}%t|%dz}&|&|z }'|% |'|zz}|"}|%}|%}t|dz}|j|}t#|||s|}d}|||d}| r|d<||fS)a Solve EQP problem with projected CG method. Solve equality-constrained quadratic programming problem ``min 1/2 x.T H x + x.t c`` subject to ``A x + b = 0`` and, possibly, to trust region constraints ``||x|| < trust_radius`` and box constraints ``lb <= x <= ub``. Parameters ---------- H : LinearOperator (or sparse matrix or ndarray), shape (n, n) Operator for computing ``H v``. c : array_like, shape (n,) Gradient of the quadratic objective function. Z : LinearOperator (or sparse matrix or ndarray), shape (n, n) Operator for projecting ``x`` into the null space of A. Y : LinearOperator, sparse matrix, ndarray, shape (n, m) Operator that, for a given a vector ``b``, compute smallest norm solution of ``A x + b = 0``. b : array_like, shape (m,) Right-hand side of the constraint equation. trust_radius : float, optional Trust radius to be considered. By default, uses ``trust_radius=inf``, which means no trust radius at all. lb : array_like, shape (n,), optional Lower bounds to each one of the components of ``x``. If ``lb[i] = -Inf`` the lower bound for the i-th component is just ignored (default). ub : array_like, shape (n, ), optional Upper bounds to each one of the components of ``x``. If ``ub[i] = Inf`` the upper bound for the i-th component is just ignored (default). tol : float, optional Tolerance used to interrupt the algorithm. max_iter : int, optional Maximum algorithm iterations. Where ``max_inter <= n-m``. By default, uses ``max_iter = n-m``. max_infeasible_iter : int, optional Maximum infeasible (regarding box constraints) iterations the algorithm is allowed to take. By default, uses ``max_infeasible_iter = n-m``. return_all : bool, optional When ``true``, return the list of all vectors through the iterations. Returns ------- x : array_like, shape (n,) Solution of the EQP problem. info : Dict Dictionary containing the following: - niter : Number of iterations. - stop_cond : Reason for algorithm termination: 1. Iteration limit was reached; 2. Reached the trust-region boundary; 3. Negative curvature detected; 4. Tolerance was satisfied. - allvecs : List containing all intermediary vectors (optional). - hits_boundary : True if the proposed step is on the boundary of the trust region. Notes ----- Implementation of Algorithm 6.2 on [1]_. In the absence of spherical and box constraints, for sufficient iterations, the method returns a truly optimal result. In the presence of those constraints, the value returned is only a inexpensive approximation of the optimal value. References ---------- .. [1] Gould, Nicholas IM, Mary E. Hribar, and Jorge Nocedal. "On the solution of equality constrained quadratic programming problems arising in optimization." SIAM Journal on Scientific Computing 23.4 (2001): 1376-1395. g}:r'rz.Trust region problem does not have a solution.T)niter stop_cond hits_boundaryallvecsg{Gz?g?Fr&r(z9Negative curvature not allowed for unrestricted problems.)r4)rrr,r ValueErrorappendr/r0r-fullr+rXranger*r rVrr )(rrZrYrr3rBrCtolmax_itermax_infeasible_iter return_all CLOSE_TO_ZEROrrr rr[r_rhH_prt_g tr_distanceinforgrfcounterlast_feasible_xkipt_H_pr`rar7x_nextthetar_nextg_next rt_g_nextbetas( r"rrs`M !BA !BA qb A aeeAhlA aA A# %%(C 7A:Da(KQIJJ } $TB  NN1 %DO$w {#dRWWT]*C$J7G z WWQ  z WWQ Q38QqS!H"cMIGmmA&O A 8_T #:I  Q Q;xx % ">??'?q"b,D'B#5)E!G A-QB7 $ v U1W 99>>& !\ 1":1eAgr2;G#I Auie A %)B3AI M  !R 0G qLG Q;":1eAgr2;G#I Aui"#eEk!m"3#;?;=r#C ( (   NN6 "U3YvL!O 4HtAv    AwzeeAhiTl !B +  Y* ,D!Y d7Nr#)F)FF)__doc__ scipy.sparserrrmathrnumpyr numpy.linalgr__all__rr r r r rVr r+rr#r"rsz933 *#\&+Sn#(Rl*/(-B!J/ - ]@.0VVTtD!br#